TOMLAB: Difference between revisions

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Introduction

What is TOMLAB?

TOMLAB is a general purpose development, modeling and optimal control environment in Matlab for research, teaching and practical solution of optimization problems.

TOMLAB has grown out of the need for advanced, robust and reliable tools to be used in the development of algorithms and software for the solution of many different types of applied optimization problems.

There are many good tools available in the area of numerical analysis, operations research and optimization, but because of the different languages and systems, as well as a lack of standardization, it is a time consuming and complicated task to use these tools. Often one has to rewrite the problem formulation, rewrite the function specifications, or make some new interface routine to make everything work. Therefore the first obvious and basic design principle in TOMLAB is: Define your problem once, run all available solvers. The system takes care of all interface problems, whether between languages or due to different demands on the problem specification.

In the process of optimization one sometimes wants to graphically view the problem and the solution process, especially for ill-conditioned nonlinear problems. Sometimes it is not clear what solver is best for the particular type of problem and tests on different solvers can be of use. In teaching one wants to view the details of the algorithms and look more carefully at the different algorithmic steps. When developing new algorithms tests on thousands of problems are necessary to fully access the pros and cons of the new algorithm. One might want to solve a practical problem very many times, with slightly different conditions for the run. Or solve a control problem looping in real-time and solving the optimization problem each time slot.

All these issues and many more are addressed with the TOMLAB optimization environment. TOMLAB gives easy access to a large set of standard test problems, optimization solvers and utilities.

The Organization of This Guide

Section 2 presents the general design of TOMLAB.

Section 3 contains strict mathematical definitions of the optimization problem types. All solver routines available in TOMLAB are described.

Section 4 describes the input format and modeling environment. The functionality of the modeling engine TomSym is discussed in 4.3 and also in appendix C.

Section 5, 6, 7 and 8 contain examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Section 9 shows how to setup and define multi layer optimization problems in TOMLAB.

Section 11 contains detailed descriptions of many of the functions in TOMLAB. The TOM solvers, originally developed by the Applied Optimization and Modeling (TOM) group, are described together with TOMLAB driver routine and utility functions. Other solvers, like the Stanford Optimization Laboratory (SOL) solvers are not described, but documentation is available for each solver.

Section 12 describes the utility functions that can be used, for example tomRun and SolverList.

Section 13 introduces the different options for derivatives, automatic differentiation.

Section 14 discusses a number of special system features such as partially separable functions and user supplied parameter information for the function computations.

Appendix A contains tables describing all elements defined in the problem structure. Some subfields are either empty, or filled with information if the particular type of optimization problem is defined. To be able to set different parameter options for the optimization solution, and change problem dependent information, the user should consult the tables in this Appendix.

Appendix B contains tables describing all elements defined in the output result structure returned from all solvers and driver routines.

Appendix D is concerned with the global variables used in TOMLAB and routines for handling important global variables enabling recursive calls of any depth.

Appendix E describes the available set of interfaces to other optimization software, such as CUTE, AMPL, and The Mathworks' Optimization Toolbox.

Appendix F gives some motivation for the development of TOMLAB.

Overall Design

The scope of TOMLAB is large and broad, and therefore there is a need of a well-designed system. It is also natural to use the power of the Matlab language, to make the system flexible and easy to use and maintain. The concept of structure arrays is used and the ability in Matlab to execute Matlab code defined as string expressions and to execute functions specified by a string.

Structure Input and Output

Normally, when solving an optimization problem, a direct call to a solver is made with a long list of parameters in the call. The parameter list is solver-dependent and makes it difficult to make additions and changes to the system.

TOMLAB solves the problem in two steps. First the problem is defined and stored in a Matlab structure. Then the solver is called with a single argument, the problem structure. Solvers that were not originally developed for the TOMLAB environment needs the usual long list of parameters. This is handled by the driver routine tomRun.mwhich can call any available solver, hiding the details of the call from the user. The solver output is collected in a standardized result structure and returned to the user.

Introduction to Solver and Problem Types

TOMLAB solves a number of different types of optimization problems. The currently defined types are listed in Table 1.

The global variable probType contains the current type of optimization problem to be solved. An optimization solver is defined to be of type solvType, where solvType is any of the probType entries in Table 1. It is clear that a solver of a certain solvType is able to solve a problem defined to be of another type. For example, a constrained nonlinear programming solver should be able to solve unconstrained problems, linear and quadratic programs and constrained nonlinear least squares problems. In the graphical user interface and menu system an additional variable optType is defined to keep track of what type of problem is defined as the main subject. As an example, the user may select the type of optimization to be quadratic programming (optType == 2), then select a particular problem that is a linear programming problem (probType == 8) and then as the solver choose a constrained NLP solver like MINOS (solvType == 3).

Table 1: The different types of optimization problems defined in TOMLAB.
probType		Type of optimization problem
uc	1	Unconstrained optimization (incl. bound constraints).
qp	2	Quadratic programming.
con	3	Constrained nonlinear optimization.
ls	4	Nonlinear least squares problems (incl. bound constraints).
lls	5	Linear least squares problems.
cls	6	Constrained nonlinear least squares problems.
mip	7	Mixed-Integer programming.
lp	8	Linear programming.
glb	9	Box-bounded global optimization.
glc	10	Global mixed-integer nonlinear programming.
miqp	11	Constrained mixed-integer quadratic programming.
minlp	12	Constrained mixed-integer nonlinear optimization.
lmi	13	Semi-definite programming with Linear Matrix Inequalities.
bmi	14	Semi-definite programming with Bilinear Matrix Inequalities.
exp	15	Exponential fitting problems.
nts	16	Nonlinear Time Series.
lcp	22	Linear Mixed-Complimentary Problems.
mcp	23	Nonlinear Mixed-Complimentary Problems.

Please note that since the actual numbers used for probType may change in future releases, it is recommended to use the text abbreviations. See help for checkType for further information.

Define probSet to be a set of defined optimization problems belonging to a certain class of problems of type probType. Each probSet is physically stored in one file, an Init File. In Table 2 the currently defined problem sets are listed, and new probSet sets are easily added.

Table 2: Defined test problem sets in TOMLAB. **probSets** marked with * are not part of the standard distribution
probSet	probType	Description of test problem set
uc	1	Unconstrained test problems.
qp	2	Quadratic programming test problems.
con	3	Constrained test problems.
ls	4	Nonlinear least squares test problems.
lls	5	Linear least squares problems.
cls	6	Linear constrained nonlinear least squares problems.
mip	7	Mixed-integer programming problems.
lp	8	Linear programming problems.
glb	9	Box-bounded global optimization test problems.
glc	10	Global MINLP test problems.
miqp	11	Constrained mixed-integer quadratic problems.
minlp	12	Constrained mixed-integer nonlinear problems.
lmi	13	Semi-definite programming with Linear Matrix Inequalities.
bmi	14	Semi-definite programming with Bilinear Matrix Inequalities.
exp	15	Exponential fitting problems.
nts	16	Nonlinear time series problems.
lcp	22	Linear mixed-complimentary problems.
mcp	23	Nonlinear mixed-complimentary problems.


mgh	4	More, Garbow, Hillstrom nonlinear least squares problems.
chs	3	Hock-Schittkowski constrained test problems.
uhs	1	Hock-Schittkowski unconstrained test problems.

The names of the predefined Init Files that do the problem setup, and the corresponding low level routines are constructed as two parts. The first part being the abbreviation of the relevant probSet, see Table 2, and the second part denotes the computed task, shown in Table 3. The user normally does not have to define the more complicated functions o_d2c and o_d2r. It is recommended to supply this information when using solvers which can utilize second order information, such as TOMLAB /KNITRO and TOMLAB /CONOPT.

Table 3: Names for predefined Init Files and low level m-files in TOMLAB.
Generic name	Purpose *( o is any probSet, e.g. o=*con)
o_prob	Init File that either defines a string matrix with problem names
o_f	Compute objective function f (x).
o_g	Compute the gradient vector g(x).
o_H	Compute the Hessian matrix H (x).
o_c	Compute the vector of constraint functions c(x).
o_dc	Compute the matrix of constraint normals, ?c(x)/dx.
o_d2c	Compute the 2nd part of 2nd derivative matrix of the Lagrangian function, ?i ?i ?2 ci (x)/dx2 .
o_r	Compute the residual vector r(x).
o_J	Compute the Jacobian matrix J (x).
o_d2r	Compute the 2nd part of the Hessian matrix, ?i ri (x)?2 ri (x)/dx2

The Init File has two modes of operation; either return a string matrix with the names of the problems in the probSet or a structure with all information about the selected problem. All fields in the structure, named Prob, are presented in tables in Section A. Using a structure makes it easy to add new items without too many changes in the rest of the system. For further discussion about the definition of optimization problems in TOMLAB, see Section 4.

There are default values for everything that is possible to set defaults for, and all routines are written in a way that makes it possible for the user to just set an input argument empty and get the default.

The Process of Solving Optimization Problems

A flow-chart of the process of optimization in TOMLAB is shown in Figure 1. It is inefficient to use an interactive system. This is symbolized withthe Standard User box in the figure, which directly runs the Optimization Driver, tomRun. The direct solver call is possible for all TOMLAB solvers, if the user has executed ProbCheck prior to the call. See Section 3 for a list of the TOMLAB solvers.

Depending on the type of problem, the user needs to supply the low-level routines that calculate the objective function, constraint functions for constrained problems, and also if possible, derivatives. To simplify this coding process so that the work has to be performed only once, TOMLAB provides gateway routines that ensure that any solver can obtain the values in the correct format.

For example, when working with a least squares problem, it is natural to code the function that computes the vector of residual functions ri (x1 , x2 , . . .), since a dedicated least squares solver probably operates on the residual while a general nonlinear solver needs a scalar function, in this case f (x) = 1 rT (x)r(x). Such issues are automatically handled by the gateway functions.

Low Level Routines and Gateway Routines

Low level routines are the routines that compute:

The objective function value

The gradient vector

The Hessian matrix (second derivative matrix)

The residual vector (for nonlinear least squares problems)

The Jacobian matrix (for nonlinear least squares problems)

The vector of constraint functions

The matrix of constraint normals (the constraint Jacobian)

The second part of the second derivative of the Lagrangian function. The last three routines are only needed for constrained problems.

The names of these routines are defined in the structure fields Prob.FUNCS.f, Prob.FUNCS.g, Prob.FUNCS.H etc.

It is the task for the Assign routine to set the names of the low level m-files. This is done by a call to the routine conAssign with the names as arguments for example. There are Assign routines for all problem types handled by TOMLAB. As an example, see 'help conAssign' in MATLAB.

Prob = conAssign('f', 'g', 'H', HessPattern, x_L,  x_U, Name,x_0,... 
pSepFunc, fLowBnd, A,  b_L,  b_U, 'c', 'dc', 'd2c', ConsPattern,... 
c_L,  c_U, x_min,  x_max, f_opt, x_opt);

Only the low level routines relevant for a certain type of optimization problem need to be coded. There are dummy routines for the others. Numerical differentiation is automatically used for gradient, Jacobian and constraint gradient if the corresponding user routine is non present or left out when calling conAssign. However, the solver always needs more time to estimate the derivatives compared to if the user supplies a code for them. Also the numerical accuracy is lower for estimated derivatives, so it is recommended that the user always tries to code the derivatives, if it is possible. Another option is automatic differentiation with TOMLAB /MAD.

TOMLAB uses gateway routines (nlp f, nlp g, nlp H, nlp c, nlp dc, nlp d2c, nlp r, nlp J, nlp d2r). These routines extract the search directions and line search steps, count iterations, handle separable functions, keep track of the kind of differentiation wanted etc. They also handle the separable NLLS case and NLLS weighting. By the use of global variables, unnecessary evaluations of the user supplied routines are avoided.

To get a picture of how the low-level routines are used in the system, consider Figure 2 and 3. Figure 2 illustrates the chain of calls when computing the objective function value in ucSolve for a nonlinear least squares problem defined in mgh prob, mgh r and mgh J. Figure 3 illustrates the chain of calls when computing the numerical approximation of the gradient (by use of the routine fdng) in ucSolve for an unconstrained problem defined in uc_prob and uc_f. Information about a problem is stored in the structure variable Prob, described in detail in the tables in Appendix A. This variable is an argument to all low level routines. In the field element Prob.user, problem specific information

ucSolve <==> nlp_f <==> ls_f <==> nlp_r <==> mgh_r

Figure 2: The chain of calls when computing the objective function value in ucSolve for a nonlinear least squares problem defined in mgh prob, mgh r and mgh J.

ucSolve <==> nlp_g <==> fdng <==> nlp_r <==> uc_f

Figure 3: The chain of calls when computing the numerical approximation of the gradient in ucSolve for an unconstrained problem defined in uc prob and uc f. needed to evaluate the low level routines are stored. This field is most often used if problem related questions are asked when generating the problem. It is often the case that the user wants to supply the low-level routines with additional information besides the variables x that are optimized. Any unused fields could be defined in the structure Prob for that purpose. To avoid potential conflicts with future TOMLAB releases, it is recommended to use subfields of Prob.user. It the user wants to send some variables a, B and C, then, after creating the Prob structure, these extra variables are added to the structure:

Prob.user.a=a; 
Prob.user.B=B; 
Prob.user.C=C;

Then, because the Prob structure is sent to all low-level routines, in any of these routines the variables are picked out from the structure:

a = Prob.user.a; 
B = Prob.user.B; 
C = Prob.user.C;

A more detailed description of how to define new problems is given in sections 5, 6 and 8. The usage of Prob.user is described in Section 14.2.

Different solvers all have different demand on how information should be supplied, i.e. the function to optimize, the gradient vector, the Hessian matrix etc. To be able to code the problem only once, and then use this formulation to run all types of solvers, interface routines that returns information in the format needed for the relevant solver were developed.

Table 4 describes the low level test functions and the corresponding Init File routine needed for the predefined constrained optimization (con) problems. For the predefined unconstrained optimization (uc) problems, the global optimization (glb, glc) problems and the quadratic programming problems (qp) similar routines have been defined.

To conclude, the system design is flexible and easy to expand in many different ways.

Table 4: Generally constrained nonlinear (**con**) test problems.
Function	Description
con_prob	Init File. Does the initialization of the con test problems.
con_f	Compute the objective function f (x) for con test problems.
con_g	Compute the gradient g(x) for con test problems. x
con_H	Compute the Hessian matrix H (x) of f (x) for con test problems.
con_c	Compute the constraint residuals c(x) for con test problems.
con_dc	Compute the derivative of the constraint residuals for con test problems.
con_d2c	Compute the 2 nd part of 2 nd derivative matrix of the Lagrangian function, ?i ?i ?2 ci (x)/dx2 for con test problems.
con_fm	Compute merit function ?(xk ).
con_gm	Compute gradient of merit function ?(xk ).

Problem Types and Solver Routines

Section 3.1 defines all problem types in TOMLAB. Each problem definition is accompanied by brief suggestions on suitable solvers. This is followed in Section 3.2 by a complete list of the available solver routines in TOMLAB and the various available extensions, such as /SOL and /CGO.

Problem Types Defined in TOMLAB

The unconstrained optimization (uc) problem is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:uc} \begin{array}{ll} \min\limits_{x} & f(x) \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x, x_L, x_U \in \MATHSET{R}^n$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $f(x) \in \MATHSET{R}$} . For unbounded variables, the corresponding elements of $\$x_{L},x_{U}\$$ may be set to $\$\pm \infty \$$ .

Recommended solvers: TOMLAB /KNITRO and TOMLAB /SNOPT.

The TOMLAB Base Module routine ucSolve includes several of the most popular search step methods for unconstrained optimization. Bound constraints are treated as described in Gill et. al. \[28\]. The search step methods for unconstrained optimization included in ucSolve are: the Newton method, the quasi-Newton BFGS and DFP method, the Fletcher-Reeves and Polak-Ribiere conjugate-gradient method, and the Fletcher conjugate descent method. The quasi-Newton methods may either update the inverse Hessian (standard) or the Hessian itself. The Newton method and the quasi-Newton methods updating the Hessian use a subspace minimization technique to handle rank problems, see Lindstr¨om \[53\]. The quasi-Newton algorithms also use safe guarding techniques to avoid rank problem in the updated matrix.

Another TOMLAB Base Module solver suitable for unconstrained problems is the structural trust region algorithm sTrustr, combined with an initial trust region radius algorithm. The code is based on the algorithms in \[13\] and \[67\], and treats partially separable functions. Safeguarded BFGS or DFP are used for the quasi-Newton update, if the analytical Hessian is not used. The set of constrained nonlinear solvers could also be used for unconstrained problems.

The quadratic programming (qp) problem is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:qp} \begin{array}{ll} \min\limits_{x} & f(x) = \frac{1}{2} x^T F x + c^T x \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $F \in \MATHSET{R}^{n \times n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} .

Recommended solvers: TOMLAB /KNITRO, TOMLAB /SNOPT and TOMLAB /MINLP.

A positive semidefinite F -matrix gives a convex QP, otherwise the problem is nonconvex. Nonconvex quadratic programs are solved with a standard active-set method \[54\], implemented in the TOM routine qpSolve. qpSolve explicitly treats both inequality and equality constraints, as well as lower and upper bounds on the variables (simple bounds). It converges to a local minimum for indefinite quadratic programs.

In TOMLAB MINOS in the general or the QP-version (QP-MINOS), or the dense QP solver QPOPT may be used. In the TOMLAB /SOL extension the SQOPT solver is suitable for both dense and large, sparse convex QP and SNOPT works fine for dense or sparse nonconvex QP.

For very large-scale problems, an interior point solver is recommended, such as TOMLAB /KNITRO or TOMLAB /BARNLP.

TOMLAB /CPLEX, TOMLAB /Xpress and TOMLAB /XA should always be considered for large-scale QP problems.

The constrained nonlinear optimization problem (con) is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:con} \begin{array}{ll} \min\limits_{x} & f(x) \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ c_{L} & \leq & c(x) & \leq & c_{U} \\ \end{array} \end{array} }

Recommended solvers: TOMLAB /SNOPT, TOMLAB /NPSOL and TOMLAB /KNITRO.

For general constrained nonlinear optimization a sequential quadratic programming (SQP) method by Schittkowski \[69\] is implemented in the TOMLAB Base Module solver conSolve. conSolve also includes an implementation of the Han-Powell SQP method. There are also a TOMLAB Base Module routine nlpSolve implementing the Filter SQP by Fletcher and Leyffer presented in \[21\].

Another constrained solver in TOMLAB is the structural trust region algorithm sTrustr, described above. Currently, sTrustr only solves problems where the feasible region, defined by the constraints, is convex. TOMLAB /MINOS solves constrained nonlinear programs. The TOMLAB /SOL extension gives an additional set of general solvers for dense or sparse problems.

sTrustr, pdco and pdsco in TOMLAB Base Module handle nonlinear problems with linear constraints only.

There are many other options for large-scale nonlinear optimization to consider in TOMLAB. TOMLAB /CONOPT, TOMLAB /BARNLP, TOMLAB /MINLP, TOMLAB /NLPQL and TOMLAB /SPRNLP.

The box-bounded global optimization (glb) problem is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:glb} \begin{array}{ll} \min\limits_{x} & f(x) \\ & \\ s/t & \begin{array}{llcccll} -\infty < &x_{L} & \leq & x & \leq & x_{U}& < \infty , \\ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $f(x) \in \MATHSET{R}$} , i.e. problems of the form \ref{eq:uc} that have finite simple bounds on all variables.

Recommended solvers: TOMLAB /LGO and TOMLAB /CGO with TOMLAB /SOL.

The TOM solver glbSolve implements the DIRECT algorithm \[14\], which is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. In glbSolve no derivative information is used. For global optimization problems with expensive function evaluations the TOMLAB /CGO routine ego implements the Efficient Global Optimization (EGO) algorithm \[16\]. The idea of the EGO algorithm is to first fit a response surface to data collected by evaluating the objective function at a few points. Then, EGO balances between finding the minimum of the surface and improving the approximation by sampling where the prediction error may be high.

The global mixed-integer nonlinear programming (glc) problem is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:glc} \begin{array}{ll} \min\limits_{x} & f(x) \\ & \\ s/t & \begin{array}{llcccll} -\infty < &x_{L} & \leq & x & \leq & x_{U}& < \infty \\ &b_{L} & \leq & A x & \leq & b_{U}& \\ &c_{L} & \leq & c(x) & \leq & c_{U},& x_{j} \in \MATHSET{N}\ ~~\forall j \in $I$ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $f(x) \in \MATHSET{R}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c_L,c(x),c_U \in \MATHSET{R}^{m_2}$} . The variables $\$x\in I\$$ , the index subset of $\$1,...,n\$$ , are restricted to be integers.

Recommended solvers: TOMLAB /OQNLP.

The TOMLAB Base Module solver glcSolve implements an extended version of the DIRECT algorithm \[52\], that handles problems with both nonlinear and integer constraints.

The linear programming (lp) problem is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:LP} \begin{array}{ll} \min\limits_{x} & f(x) = c^T x \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} .

Recommended solvers: TOMLAB /CPLEX, TOMLAB /Xpress and TOMLAB /XA. The TOMLAB Base Module solver lpSimplex implements a simplex algorithm for lp problems.

When a dual feasible point is known in (6) it is efficient to use the dual simplex algorithm implemented in the TOMLAB Base Module solver DualSolve. In TOMLAB /MINOS the LP interface to MINOS, called LP-MINOS is efficient for solving large, sparse LP problems. Dense problems are solved by LPOPT. The TOMLAB /SOL extension gives the additional possibility of using SQOPT to solve large, sparse LP.

The recommended solvers normally outperforms all other solvers.

The mixed-integer programming problem (mip) is defined as

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:mip} \begin{array}{ll} \min\limits_{x} & f(x) = c^T x \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U}, ~x_{j} \in \MATHSET{N}\ ~~\forall j \in $I$ \\ \end{array} \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} . The variables $\$x\in I\$$ , the index subset of $\$1,...,n\$$ are restricted to be integers. Equality constraints are defined by setting the lower bound equal to the upper bound, i.e. for constraint Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $i$: $b_{L}(i) = b_{U}(i)$} .

Recommended solvers: TOMLAB /CPLEX, TOMLAB /Xpress and TOMLAB /XA.

Mixed-integer programs can be solved using the TOMLAB Base Module routine mipSolve that implements a standard branch-and-bound algorithm, see Nemhauser and Wolsey \[58, chap. 8\]. When dual feasible solutions are available, mipSolve is using the TOMLAB dual simplex solver DualSolve to solve subproblems, which is significantly faster than using an ordinary linear programming solver, like the TOMLAB lpSimplex. mipSolve also implements user defined priorities for variable selection, and different tree search strategies. For 0/1 - knapsack problems a round-down primal heuristic is included. Another TOMLAB Base Module solver is the cutting plane routine cutplane, using Gomory cuts. It is recommended to use mipSolve with the LP version of MINOS with warm starts for the subproblems, giving great speed improvement. The TOMLAB /Xpress extension gives access to the state-of-the-art LP, QP, MILP and MIQP solver Xpress-MP. For many MIP problems, it is necessary to use such a powerful solver, if the solution should be obtained in any reasonable time frame. TOMLAB /CPLEX is equally powerful as TOMLAB /Xpress and also includes a network solver.

The linear least squares (lls) problem is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq:lls} \begin{array}{ll} \min\limits_{x} & f(x) = \frac{1}{2} || C x - d || \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ \end{array} \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $d \in \MATHSET{R}^M$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $C \in \MATHSET{R}^{M \times n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c_L,c(x),c_U \in \MATHSET{R}^{m_2}$} .

Recommended solvers: TOMLAB /LSSOL.

Tlsqr solves unconstrained sparse lls problems. lsei solves the general dense problems. Tnnls is

a fast and robust replacement for the Matlab nnls. The general least squares solver clsSolve may also be used.

In the TOMLAB

/NPSOL or TOMLAB /SOL extension the LSSOL solver is suitable for dense linear least squares problems.

The constrained nonlinear least squares (cls) problem is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq:cls} \begin{array}{ll} \min\limits_{x} & f(x) = \frac{1}{2} r(x)^T r(x) \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ c_{L} & \leq & c(x) & \leq & c_{U} \\ \end{array} \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $r(x) \in \MATHSET{R}^M$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c_L,c(x),c_U \in \MATHSET{R}^{m_2}$} .

Recommended solvers: TOMLAB /NLSSOL.

The TOMLAB Base Module nonlinear least squares solver clsSolve treats problems with bound constraints in a similar way as the routine ucSolve. This strategy is combined with an active-set strategy to handle linear equality and inequality constraints \[7\].

clsSolve includes seven optimization methods for nonlinear least squares problems, among them: the Gauss-Newton method, the Al-Baali-Fletcher \[2\] and the Fletcher-Xu \[19\] hybrid method, and the Huschens TSSM method \[50\]. If rank problems occur, the solver uses subspace minimization. The line search algorithm used is the same as for unconstrained problems.

Another fast and robust solver is NLSSOL, available in the TOMLAB /NPSOL or the TOMLAB /SOL extension toolbox.

One important utility routine is the TOMLAB Base Module line search algorithm LineSearch, used by the solvers conSolve, clsSolve and ucSolve. It implements a modified version of an algorithm by Fletcher \[20, chap. 2\]. The line search algorithm uses quadratic and cubic interpolation, see Section 12.10. Another TOMLAB Base Module routine, preSolve, is running a presolve analysis on a system of linear qualities, linear inequalities and simple bounds. An algorithm by Gondzio \[36\], somewhat modified, is implemented in preSolve. See \[7\] for a more detailed presentation.

The linear semi-definite programming problem with linear matrix inequalities (sdp) is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq:sdp} \begin{array}{rccccl} \min\limits_{x} & \multicolumn{5}{l}{f(x) = {c^T}x} \\ & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\ & \multicolumn{5}{r}{Q^{i}_0 + \Sum{k=1}{n} Q^{i}_{k}x_{k} \preccurlyeq 0,\qquad i=1,\ldots,m.} \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_l\times n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_l}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $Q^{i}_{k}$} are symmetric matrices of similar dimensions in each constraint Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $i$} . If there are several LMI constraints, each may have it's own dimension.

Recommended solvers: TOMLAB /PENSDP and TOMLAB /PENBMI.

The linear semi-definite programming problem with bilinear matrix inequalities (bmi) is defined similarly to but with the matrix inequality

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq:bmi} Q^{i}_0 + \Sum{k=1}{n} Q^{i}_{k}x_{k} + \Sum{k=1}{n}\Sum{l=1}{n} x_{k}x_{l}K^{i}_{kl}\preccurlyeq 0 \\ }

The MEX solvers pensdp and penbmi treat sdp and bmi problems, respectively. These are available in the TOMLAB /PENSDP and TOMLAB /PENBMI toolboxes.

The MEX-file solver pensdp available in the TOMLAB /PENSDP toolbox implements a penalty method aimed at large-scale dense and sparse sdp problems. The interface to the solver allows for data input in the sparse SDPA input format as well as a TOMLAB specific format corresponding to.

The MEX-file solver penbmi available in the TOMLAB /PENBMI toolbox is similar to pensdp, with added support for the bilinear matrix inequalities.

Solver Routines in TOMLAB

TOMLAB Base Module

TOMLAB includes a large set of optimization solvers. Most of them were originally developed by the Applied Optimization and Modeling group (TOM) \[42\]. Since then they have been improved e.g. to handle Matlab sparse arrays and been further developed. Table 5 lists the main set of TOM optimization solvers in all versions of TOMLAB.

Table 5: The optimization solvers in TOMLAB Base Module.
Function	Description	Section	Page
clsSolve	Constrained nonlinear least squares. Handles simple bounds and linear equality and inequality constraints using an active-set strat- egy. Implements Gauss-Newton, and hybrid quasi-Newton and Gauss-Newton methods.	11.1.1	PAGE
conSolve	Constrained nonlinear minimization solver using two different sequential quadratic programming methods.	11.1.2	PAGE
cutplane	Mixed-integer programming using a cutting plane algorithm.	11.1.3	PAGE
DualSolve	Solves a linear program with a dual feasible starting point.	11.1.4	PAGE
glbSolve	Box-bounded global optimization, using only function values.	11.1.7	PAGE
glcCluster	Hybrid algorithm for constrained mixed-integer global optimization. Uses a combination of glcFast (DIRECT) and a clustering algorithm.	11.1.7	PAGE
glcSolve	Global mixed-integer nonlinear programming, using no derivatives.	11.1.7	PAGE
infSolve	Constrained minimax optimization. Reformulates problem and calls any suitable nonlinear solver.	11.1.8	PAGE
lpSimplex	Linear programming using a simplex algorithm.	11.1.14	PAGE
MILPSOLVE	Mixed-integer programming using branch and bound.	11.1.16	PAGE
L1Solve	Constrained L1 optimization. Reformulates problem and calls any suitable nonlinear solver.	11.1.15	PAGE
mipSolve	Mixed-integer programming using a branch-and-bound algorithm.	11.1.18	PAGE
nlpSolve	Constrained nonlinear minimization solver using a filter SQP algorithm.	SECTION	PAGE
pdco	Linearly constrained nonlinear minimization solver using a primal- dual barrier algorithm.
pdsco	Linearly constrained nonlinear minimization solver using a primal- dual barrier algorithm, for separable functions.	SECTION	PAGE
qpSolve	Non-convex quadratic programming.	11.1.24	PAGE
slsSolve	Sparse constrained nonlinear least squares. Reformulates problem and calls any suitable sparse nonlinear solver.
sTrustr	Constrained convex optimization of partially separable functions, using a structural trust region algorithm.
ucSolve	Unconstrained optimization with simple bounds on the parameters. Implements Newton, quasi-Newton and conjugate-gradient methods.	11.1.29	PAGE

Additional Fortran solvers in TOMLAB are listed in Table 6. They are called using a set of MEX-file interfaces developed in TOMLAB.

Table 6: Additional solvers in TOMLAB.
Function	Description	Reference	Page
goalSolve	Solves sparse multi-objective goal attainment problems
lsei	Dense constrained linear least squares
qld	Convex quadratic programming
Tfzero	Finding a zero to f(x) in an interval, x is one-dimensional.	\[70, 15\]
Tlsqr	Sparse linear least squares.	\[60, 59, 68\]
Tnnls	Nonnegative constrained linear least squares

TOMLAB /BARNLP

The add-on toolbox TOMLAB /BARNLP solves large-scale nonlinear programming problems using a sparse primal-dual interior point algorithm, in conjunction with a filter for globalization. The solver package was developed in co-operation with Boeing Phantom Works. The TOMLAB /BARNLP package is described in a separate manual available at http://tomopt.com.

TOMLAB /CGO

The add-on toolbox TOMLAB /CGO solves costly global optimization problems. The solvers are listed in Table 7. They are written in a combination of Matlab and Fortran code, where the Fortran code is called using a set of MEX-file interfaces developed in TOMLAB.

Table 7: Additional solvers in TOMLAB /CGO.
Function	Description	Reference
rbfSolve	Costly constrained box-bounded optimization using a RBF algorithm.	\[9\]
ego	Costly constrained box-bounded optimization using the Efficient Global Optimization (EGO) algorithm.	\[16\]

TOMLAB /CONOPT

The add-on toolbox TOMLAB /CONOPT solves large-scale nonlinear programming problems with a feasible path GRG method . The solver package was developed in co-operation with Arki Consulting and Development A/S. The TOMLAB /CONOPT is described in a separate manual available at http://tomopt.com. There is also m-file help available.

TOMLAB /CPLEX

The add-on toolbox TOMLAB /CPLEX solves large-scale mixed-integer linear and quadratic programming prob- lems. The solver package was developed in co-operation with ILOG S.A. The TOMLAB /CPLEX solver package and interface are described in a manual available at http://tomopt.com.

TOMLAB /KNITRO

The add-on toolbox TOMLAB /KNITRO solves large-scale nonlinear programming problems with interior (barrier) point trust-region methods. The solver package was developed in co-operation with Ziena Optimization Inc. The TOMLAB /KNITRO manual is available at http://tomopt.com.

TOMLAB /LGO

The add-on toolbox TOMLAB /LGO solves global nonlinear programming problems. The LGO solver package is developed by Pint´er Consulting Services, Inc. \[63\] The TOMLAB /LGO manual is available at http://tomopt. com.

TOMLAB /MINLP

The add-on toolbox TOMLAB /MINLP provides solvers for continuous and mixed-integer quadratic programming

(qp,miqp) and continuous and mixed-integer nonlinear constrained optimization (con, minlp).

All four solvers, written by Roger Fletcher and Sven Leyffer, University of Dundee, Scotland, are available in sparse and dense versions. The solvers are listed in Table 8.

The TOMLAB /MINLP manual is available at http://tomopt.com.

TOMLAB /MINOS

Another set of Fortran solvers were developed by the Stanford Optimization Laboratory (SOL). Table 9 lists the solvers included in TOMLAB /MINOS. The solvers are called using a set of MEX-file interfaces developed as part of TOMLAB. All functionality of the SOL solvers are available and changeable in the TOMLAB framework in Matlab.

TOMLAB /OQNLP

The add-on toolbox TOMLAB /OQNLP uses a multistart heuristic algorithm to find global optima of smooth constrained nonlinear programs (NLPs) and mixed-integer nonlinear programs (MINLPs). The solver package was developed in co-operation with Optimal Methods Inc. The TOMLAB /OQNLP manual is available at http:

//tomopt.com.

Table 8: Solver routines in TOMLAB /MINLP.
Function	Description	Reference
bqpd	Quadratic programming using a null-space method.	bqpdTL.m
miqpBB	Mixed-integer quadratic programming using bqpd as subsolver.miqpBBTL.m
filterSQP	Constrained nonlinear minimization using a Filtered Sequential QP method.filterSQPTL.m
minlpBB	Constrained, mixed-integer nonlinear minimization using a branch- and-bound search scheme. filterSQP is used as NLP solver.	minlpBBTL.m

Table 9: The SOL optimization solvers in TOMLAB /MINOS.
Function	Description	Reference	Page
MINOS 5.51	Sparse linear and nonlinear programming with linear and nonlin- ear constraints.	\[57\]
LP-MINOS	A special version of the MINOS 5.5 MEX-file interface for sparse linear programming.	\[57\]
QP-MINOS	A special version of the MINOS 5.5 MEX-file interface for sparse quadratic programming.	\[57\]
LPOPT 1.0-10	Dense linear programming.	\[30\]
QPOPT 1.0-10	Non-convex quadratic programming with dense constraint matrix and sparse or dense quadratic matrix.	\[30\]

TOMLAB /NLPQL

The add-on toolbox TOMLAB /NLPQL solves dense nonlinear programming problems, multi criteria optimiza- tion problems and nonlinear fitting problems. The solver package was developed in co-operation with Klaus Schittkowski. The TOMLAB /NLPQL manual is available at http://tomopt.com.

TOMLAB /NPSOL

The add-on toolbox TOMLAB /NPSOL is a sub package of TOMLAB /SOL. The package includes the MINOS solvers as well as NPSOL, LSSOL and NLSSOL. The TOMLAB /NPSOL manual is available at http://tomopt. com.

TOMLAB /PENBMI

The add-on toolbox TOMLAB /PENBMI solves linear semi-definite programming problems with linear and bilinear matrix inequalities. The solvers are listed in Table 10. They are written in a combination of Matlab and C code. The TOMLAB /PENBMI manual is available at http://tomopt.com.

Table 10: Additional solvers in TOMLAB /PENSDP.
Function	Description	Reference	Page
penbmi	Sparse and dense linear semi-definite programming using a penalty algorithm.
penfeas bmi	Feasibility check of systems of linear matrix inequalities, using penbmi.

TOMLAB /PENSDP

The add-on toolbox TOMLAB /PENSDP solves linear semi-definite programming problems with linear matrix inequalities. The solvers are listed in Table 11. They are written in a combination of Matlab and C code. The TOMLAB /PENSDP manual is available at http://tomopt.com.

Table 11: Additional solvers in TOMLAB /PENSDP.
Function	Description	Reference	Page
pensdp	Sparse and dense linear semi-definite programming using a penalty algorithm.
penfeas_sdp	Feasibility check of systems of linear matrix inequalities, using pensdp.

TOMLAB /SNOPT

The add-on toolbox TOMLAB /SNOPT is a sub package of TOMLAB /SOL. The package includes the MINOS solvers as well as SNOPT and SQOPT. The TOMLAB /SNOPT manual is available at http://tomopt.com.

TOMLAB /SOL

The extension toolbox TOMLAB /SOL gives access to the complete set of Fortran solvers developed by the Stanford Systems Optimization Laboratory (SOL). These solvers are listed in Table 9 and 12.

TOMLAB /SPRNLP

The add-on toolbox TOMLAB /SPRNLP solves large-scale nonlinear programming problems. SPRNLP is a state- of-the-art sequential quadratic programming (SQP) method, using an augmented Lagrangian merit function and safeguarded line search. The solver package was developed in co-operation with Boeing Phantom Works. The TOMLAB /SPRNLP package is described in a separate manual available at http://tomopt.com.

Table 12: The optimization solvers in the TOMLAB /SOL toolbox.
Function	Description	Reference	Page
NPSOL 5.02	Dense linear and nonlinear programming with linear and nonlinear constraints.	\[34\]
SNOPT 6.2-2	Large, sparse linear and nonlinear programming with linear and nonlinear constraints.	\[33, 31\]
SQOPT 6.2-2	Sparse convex quadratic programming.	\[32\]
NLSSOL 5.0-2	Constrained nonlinear least squares. NLSSOL is based on NPSOL. No reference except for general NPSOL reference.	\[34\]
LSSOL 1.05-4	Dense linear and quadratic programs (convex), and constrained linear least squares problems.	\[29\]

TOMLAB /XA

The add-on toolbox TOMLAB /XA solves large-scale linear, binary, integer and semi-continuous linear program- ming problems, as well as quadratic programming problems. The solver package was developed in co-operation with Sunset Software Technology. The TOMLAB /XA package is described in a separate manual available at http://tomopt.com.

3.2.19 TOMLAB /Xpress

The add-on toolbox TOMLAB /Xpress solves large-scale mixed-integer linear and quadratic programming prob- lems. The solver package was developed in co-operation with Dash Optimization Ltd. The TOMLAB /Xpress solver package and interface are described in the html manual that comes with the installation package. There is also a TOMLAB /Xpress manual available at http://tomopt.com.

3.2.20 Finding Available Solvers

To get a list of all available solvers, including Fortran, C and Matlab Optimization Toolbox solvers, for a certain solvType, call the routine SolverList with solvType as argument. solvType should either be a string ('uc', 'con' etc.) or the corresponding solvType number as given in Table 1, page 11. For example, if wanting a list of all available solvers of solvType con, then

SolverList('con')

gives the output

>> SolverList('con');

Tomlab recommended choice for Constrained Nonlinear Programming (NLP)

npsol

Other solvers for NLP 

    Licensed:

    nlpSolve
    conSolve
    sTrustr
    constr
    minos
    snopt
    fmincon
    filterSQP
    PDCO

    PDSCO

    Non-licensed: 

    NONE

Solvers also handling NLP 

    Licensed:
    
    glcSolve
    glcFast
    glcCluster
    rbfSolve
    minlpBB

    Non-licensed:

    NONE

SolverList also returns two output arguments: all available solvers as a string matrix and a vector with the corresponding solvType for each solver.

Note that solvers for a more general problem type may be used to solve the problem. In Table 13 an attempt has been made to classify these relations.

Table 13: The problem classes (probType) possible to solve with each type of solver (solvType) is marked with an x. When the solver is in theory possible to use, but from a practical point of view is probably not suitable, parenthesis are added (x).

	solvType
probType	uc	qp	con	ls	lls	cls	mip	lp	glb	glc

uc	x		x						x	(x)
qp		x	x							(x)
con			x							(x)
ls			x	x		x				(x)
lls		x	x	x	x	x				(x)
cls			x			x				(x)
mip							x			(x)
lp		x	x				x	x		(x)
glb			(x)						x	x
glc			(x)							x

exp	x		x	(x)		x				(x)

Defining Problems in TOMLAB

TOMLAB is based on the principle of creating a problem structure that defines the problem and includes all relevant information needed for the solution of the user problem. One unified format is defined, the TOMLAB format. The TOMLAB format gives the user a fast way to setup a problem structure and solve the problem from the Matlab command line using any suitable TOMLAB solver.

TOMLAB also includes a modeling engine (or advanced Matlab class), TomSym, see Section 4.3. The package uses Matlab objects and operator overloading to capture Matlab procedures, and generates source code for derivatives of any order.

In this section follows a more detailed description of the TOMLAB format.

The TOMLAB Format

The TOMLAB format is a quick way to setup a problem and easily solve it using any of the TOMLAB solvers. The principle is to put all information in a Matlab structure, which then is passed to the solver, which extracts the relevant information. The structure is passed to the user function routines for nonlinear problems, making it a convenient way to pass other types of information.

The solution process for the TOMLAB format has four steps:

Define the problem structure, often called Prob.
Call the solver or the universal driver routine tomRun.
Postprocessing, e.g. print the result of the optimization.

Step 1 could be done in several ways in TOMLAB. Recommended is to call one of the following routines dependent on the type of optimization problem, see Table 14.

Step 2, the solver call, is either a direct call, e.g. conSolve:

Prob = ProbCheck(Prob,  'conSolve'); 
Result  = conSolve(Prob);

or a call to the multi-solver driver routine tomRun, e.g. for constrained optimization:

Result  = tomRun('conSolve', Prob);

Note that tomRun handles several input formats. Step 3 could be a call to PrintResult.m:

PrintResult(Result);

The 3rd step could be included in Step 2 by increasing the print level to 1, 2 or 3 in the call to the driver routine

Result  = tomRun('conSolve',Prob, 3);

See the different demo files that gives examples of how to apply the TOMLAB format: conDemo.m, ucDemo.m, qpDemo.m, lsDemo.m, lpDemo.m, mipDemo.m, glbDemo.m and glcDemo.m.

Table 14: Routines to create a problem structure in the TOMLAB format.
Matlab call	probTypes	Type of optimization problem
Prob = bmiAssign( ... )	14	Semi-definite programming with bilinear matrix inequalities.
Prob = clsAssign( ... )	4,5,6	Unconstrained and constrained nonlinear least squares.
Prob = conAssign( ... )	1,3	Unconstrained and constrained nonlinear optimization.
Prob = expAssign( ... )	17	Exponential fitting problems.
Prob = glcAssign( ... )	9,10,15	Box-bounded or mixed-integer constrained global programming.
Prob = lcpAssign( ... )	22	Linear mixed-complimentary problems.
Prob = llsAssign( ... )	5	Linear least-square problems.
Prob = lpAssign( ... )	8	Linear programming.
Prob = lpconAssign( ... )	3	Linear programming with nonlinear constraints.
Prob = mcpAssign( ... )	23	Nonlinear mixed-complimentary problems.
Prob = minlpAssign( ... )	12	Mixed-Integer nonlinear programming.
Prob = mipAssign( ... )	7	Mixed-Integer programming.
Prob = miqpAssign( ... )	11	Mixed-Integer quadratic programming.
Prob = miqqAssign( ... )	18	Mixed-Integer quadratic programming with quadratic constraints.
Prob = qcpAssign( ... )	23	Quadratic mixed-complimentary problems.
Prob = qpblockAssign( ... )	2	Quadratic programming (factorized).
Prob = qpAssign( ... )	2	Quadratic programming.
Prob = qpconAssign( ... )	3	Quadratic programming with nonlinear constraints.
Prob = sdpAssign( ... )	13	Semi-definite programming with linear matrix inequalities.

Prob = amplAssign( ... )	1-3,7,8,11,12	For AMPL problems defined as nl -files.
Prob = simAssign( ... )	1,3-6,9-10	General routine, functions and constraints calculated at the same
		time .

Modifying existing problems

It is possible to modify an existing Prob structure by editing elements directly, however this is not recommended since there are dependencies for memory allocation and problem sizes that the user may not be aware of.

There are a set of routines developed specifically for modifying linear constraints (do not modify directly, Prob.mLin

need to be set to a proper value if so). All the static information can be set with the following routines.

add_A

Purpose

Adds linear constraints to an existing problem.

Calling Syntax

Prob = add_A(Prob, A, b L, b U)

Description of Inputs

Prob Existing TOMLAB problem. A The additional linear constraints. b_L The lower bounds for the new linear constraints. b_U The upper bounds for the new linear constraints.

Description of Outputs

Prob Modified TOMLAB problem.

keep_A

Purpose

Keeps the linear constraints specified by idx.

Calling Syntax

Prob = keep_A(Prob, idx)

Description of Inputs

Prob Existing TOMLAB problem.

idx The row indices to keep in the linear constraints.

Description of Outputs

Prob Modified TOMLAB problem.

remove A

Purpose

Removes the linear constraints specified by idx.

Calling Syntax

Prob = remove_A(Prob, idx)

Description of Inputs

ProbExisting TOMLAB problem.

idxThe row indices to remove in the linear constraints.

Description of Outputs

Prob:Modified TOMLAB problem.

replace A

Purpose

Replaces the linear constraints.

Calling Syntax

Prob = replace_A(Prob, A, b L, b U)

Description of Inputs

Prob Existing TOMLAB problem.

A New linear constraints.

b_L Lower bounds for linear constraints.

b_U Upper bounds for linear constraints.

Description of Outputs

Prob Modified TOMLAB problem.

modify b_L

Purpose

Modify lower bounds for linear constraints. If idx is not given b L will be replaced.

Calling Syntax

Prob = modify_b_L(Prob, b L, idx)

Description of Inputs

Prob Existing TOMLAB problem.

b_L New lower bounds for the linear constraints.

idx Indices for the modified constraint bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify b_U

Purpose

Modify upper bounds for linear constraints. If idx is not given b U will be replaced.

Calling Syntax

Prob = modify_b_U(Prob, b_U, idx)

Description of Inputs

Prob Existing TOMLAB problem.

b_U New upper bounds for the linear constraints.

idx Indices for the modified constraint bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify c

Purpose

Modify linear objective (LP/QP only).

Calling Syntax

Prob = modify_c(Prob, c, idx)

Description of Inputs

Prob Existing TOMLAB problem.

c New linear coefficients.

idx Indices for the modified linear coefficients (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify c_L

Purpose

Modify lower bounds for nonlinear constraints. If idx is not given c L will be replaced.

Calling Syntax

Prob = modify_c_L(Prob, c_L, idx)

Description of Inputs

Prob Existing TOMLAB problem.

c_L New lower bounds for the nonlinear constraints.

idx Indices for the modified constraint bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify c_U

Purpose

Modify upper bounds for nonlinear constraints. If idx is not given c U will be replaced.

Calling Syntax

Prob = modify_c_U(Prob, c U, idx)

Description of Inputs

Prob Existing TOMLAB problem.

c_U New upper bounds for the nonlinear constraints.

idx Indices for the modified constraint bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify x_0

Purpose

Modify starting point. If x_0 is outside the bounds an error will be returned. If idx is not given x 0 will be replaced.

Calling Syntax

Prob = modify_x_0(Prob, x 0, idx)

Description of Inputs

Prob Existing TOMLAB problem.

x_0 New starting points.

idx Indices for the modified starting points (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify x_L

Purpose

Modify lower bounds for decision variables. If idx is not given x L will be replaced. x 0 will be shifted if needed.

Calling Syntax

Prob = modify_x_L(Prob, x_L, idx)

Description of Inputs

Prob Existing TOMLAB problem.

x_L New lower bounds for the decision variables.

idx Indices for the modified lower bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

modify x_U

Purpose

Modify upper bounds for decision variables. If idx is not given x U will be replaced. x 0 will be shifted if needed.

Calling Syntax

Prob = modify_x_U(Prob, x_U, idx)

Description of Inputs

Prob Existing TOMLAB problem.

x_U New upper bounds for the decision variables.

idx Indices for the modified upper bounds (optional).

Description of Outputs

Prob Modified TOMLAB problem.

TomSym

For further information about TomSym, please visit http://tomsym.com/ - the pages contain detailed modeling examples and real life applications. All illustrated examples are available in the folder /tomlab/tomsym/examples/ in the TOMLAB installation. The modeling engine supports all problem types in TOMLAB with some minor exceptions.

A detailed function listing is available in Appendix C.

TomSym combines the best features of symbolic differentiation, i.e. produces source code with simplifications and optimizations, with the strong point of automatic differentiation where the result is a procedure, rather than an expression. Hence it does not grow exponentially in size for complex expressions.

Both forward and reverse modes are supported, however, reverse is default when computing the derivative with respect to more than one variable. The command derivative results in forward mode, and derivatives in reverse mode.

TomSym produces very efficient and fully vectorized code and is compatible with TOMLAB /MAD for situations where automatic differentiation may be the only option for parts of the model.

It should also be noted that TomSym automatically provides first and second order derivatives as well as problem sparsity patterns. With the use of TomSym the user no longer needs to code cumbersome derivative expressions and Jacobian/Hessian sparsity patterns for most optimization and optimal control problems.

The main features in TomSym can be summarized with the following list:

A full modeling environment in Matlab with support for most built-in mathematical operators.

Automatically generates derivatives as Matlab code.

A complete integration with PROPT (optimal control platform).

Interfaced and compatible with MAD, i.e. MAD can be used when symbolic modeling is not suitable.

Support for if, then, else statements.

Automated code simplification for generated models.

Ability to analyze most p-coded files (if code is vectorized).

Modeling

One of the main strength of TomSym is the ability to automatically and quickly compute symbolic derivatives of matrix expressions. The derivatives can then be converted into efficient Matlab code.

The matrix derivative of a matrix function is a fourth rank tensor - that is, a matrix each of whose entries is a matrix. Rather than using four-dimensional matrices to represent this, TomSym continues to work in two dimensions. This makes it possible to take advantage of the very efficient handling of sparse matrices in Matlab (not available for higher-dimensional matrices).

In order for the derivative to be two-dimensional, TomSym's derivative reduces its arguments to one-dimensional vectors before the derivative is computed. In the returned J , each row corresponds to an element of F , and each column corresponds to an element of X . As usual in Matlab, the elements of a matrix are taken in column-first order.

For vectors F and X , the resulting J is the well-known Jacobian matrix.

Observe that the TomSym notation is slightly different from commonly used mathematical notation. The notation used in tomSym was chosen to minimize the amount of element reordering needed to compute gradients for common expressions in optimization problems. It needs to be pointed out that this is different from the commonly used mathematical notation, where the tensor ( dF ) is flattened into a two-dimensional matrix as it is written (There are actually two variations of this in common use - the indexing of the elements of X may or may not be transposed).

For example, in common mathematical notation, the so-called self derivative matrix ( dX ) is a mn-by-mn square (or mm-by-nn rectangular in the non-transposed variation) matrix containing mn ones spread out in a random-looking manner. In tomSym notation, the self-derivative matrix is the mn-by-mn identity matrix.

The difference in notation only involves the ordering of the elements, and reordering the elements to a different notational convention should be trivial if tomSym is used to generate derivatives for applications other than for TOMLAB and PROPT.

Example:

>> toms	y

>> toms 3x1 x

>> toms 2x3 A

>> f = (A\*x).^(2\*y)


f = tomSym(2x1): 

   (A\*x).^(2\*y)

>> derivative(f,A)

ans = tomSym(2x6):

   (2\*y)\*setdiag((A\*x).^(2\*y-1))\*kron(x',eye(2))

In the above example, the 2x1 symbol f is differentiated with respect to the 2x3 symbol A. The result is a 2x6 matrix, representing Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(vec(f))} .

The displayed text is not necessarily identical to the m-code that will be generated from an expression. For example, the identity matrix is generated using speye in m-code, but displayed as eye (Derivatives tend to involve many sparse matrices, which Matlab handles efficiently). The mcodestr command converts a tomSym object to a Matlab code string.

>> mcodestr(ans)
ans =
setdiag((2\*y)\*(A\*x).^(2\*y-1))\*kron(x',\[1 0;0  1\])

Observe that the command mcode and not mcodestr should be used when generating efficient production code.

Ezsolve

TomSym provides the function ezsolve, which needs minimal input to solve an optimization problem: only the objective function and constraints. For example, the miqpQG example from the tomlab quickguide can be reduced to the following:

toms integer  x 
toms	      y

objective = -6\*x  + 2\*x^2  + 2\*y^2  - 2\*x\*y;
constraints = \{x+y<=1.9,x>=0, y>=0\};

solution = ezsolve(objective,constraints)

Ezsolve calls tomDiagnose to determine the problem type, getSolver to find an appropriate solver, then sym2prob, tomRun and getSoluton in sequence to obtain the solution.

Advanced users might not use ezsolve, and instead call sym2prob and tomRun directly. This provides for the possibility to modify the Prob struct and set flags for the solver.

Usage

TomSym, unlike most other symbolic algebra packages, focuses on vectorized notation. This should be familiar to Matlab users, but is very different from many other programming languages. When computing the derivative of a vector-valued function with respect to a vector valued variable, tomSym attempts to give a derivative as vectorized Matlab code. However, this only works if the original expressions use vectorized notation. For example:

toms 3x1 x
f = sum(exp(x));
g = derivative(f,x);

results in the efficient g = exp(x)1. In contrast, the mathematically equivalent but slower code:

toms 3x1 x 
f = 0;
for i=1:length(x)
   f = f+exp(x(i));
end
g = derivative(f,x);

results in

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=(exp(x(1))*[100]+exp(x(2))*[010])+exp(x(3))*[001] }

as each term is differentiated individually. Since tomSym has no way of knowing that all the terms have the same format, it has to compute the symbolic derivative for each one. In this example, with only three iterations, that is not really a problem, but large for-loops

can easily result in symbolic calculations that require more time than the actual numeric solution of the underlying optimization problem.

It is thus recommended to avoid for-loops as far as possible when working with tomSym.

Because tomSym computes derivatives with respect to whole symbols, and not their individual elements, it is also a good idea not to group several variables into one vector, when they are mostly used individually. For example:

toms 2x1 x
f = x(1)\*sin(x(2));
g = derivative(f,x);

results in

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=sin(x(2))*[10]+x(1)*(cos(x(2))*[01])}

Since x is never used as a 2x1 vector, it is better to use two independent 1x1 variables:

toms a b
f = a\*sin(b);
g = derivative(f,\[a; b\]);

which results in

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=[sin(b) a * cos(b)]} .

The main benefit here is the increased readability of the auto-generated code, but there is also a slight performance increase (Should the vector x later be needed, it can of course easily be created using the code

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=[a;b]}

Scaling variables

Because tomSym provides analytic derivatives (including second order derivatives) for the solvers, badly scaled problems will likely be less problematic from the start. To further improve the model, tomSym also makes it very easy to manually scale variables before they are presented to the solver. For example, assuming that an optimization problem involves the variable x which is of the order of magnitude 1e6, and the variable y, which is of the order of 1e - 6, the code:

toms xScaled  yScaled 
x = 1e+6\*xScaled;
y = 1e-6\*yScaled;

will make it possible to work with the tomSym expressions x and y when coding the optimization problem, while the solver will solve for the symbols xScaled and yScaled, which will both be in the order of 1. It is even possible to provide starting guesses on x and y (in equation form), because tomSym will solve the linear equation to obtain starting guesses for the underlying xScaled and yScaled.

The solution structure returned by ezsolve will of course only contain xScaled and yScaled, but numeric values for x and y are easily obtained via, e.g. subs(x,solution).

SDP/LMI/BMI interface

An interface for bilinear semidefinite problems is included with tomSym. It is also possible to solve nonlinear problems involving semidefinite constraints, using any nonlinear solver (The square root of the semidefinte matrix is then introduced as an extra set of unknowns).

See the examples optimalFeedbackBMI and example sdp.

Interface to MAD and finite differences

If a user function is incompatible with tomSym, it can still be used in symbolic computations, by giving it a "wrapper". For example, if the cos function was not already overloaded by tomSym, it would still be possible to do the equivalent of cos(3\*x) by writing feval('cos',3\*x).

MAD then computes the derivatives when the Jacobian matrix of a wrapped function is needed. If MAD is unavailable, or unable to do the job, numeric differentiation is used.

Second order derivatives cannot be obtained in the current implementation.

It is also possible to force the use of automatic or numerical differentiation for any function used in the code. The follow examples shows a few of the options available:

toms x1 x2 alpha  = 100;

%   1.  USE  MAD  FOR  ONE  FUNCTION.
%   Create  a wrapper  function. In  this case we  use sin, but  it could  be any
%  MAD   supported  function.
y = wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','MAD'),x1/x2);
f = alpha\*(x2-x1^2)^2 + (1-x1)^2 + y;

%   Setup and solve  the  problem 
c = -x1^2  - x2;
con = \{-1000  <= c <= 0
    -10  <= x1 <= 2
    -10  <= x2 <= 2\};

x0 = \{x1  == -1.2 
    x2 == 1\};

solution1 = ezsolve(f,con,x0);

%   2.  USE  MAD  FOR  ALL FUNCTIONS. 
options = struct; 
options.derivatives = 'automatic';
f = alpha\*(x2-x1^2)^2 + (1-x1)^2 + sin(x1/x2);
solution2 = ezsolve(f,con,x0,options);

%   3.  USE  FD  (Finite  Differences) FOR  ONE  FUNCTIONS.
%   Create  a new wrapper  function. In  this case we  use sin, but  it could  be
%   any function since  we  use numerical derivatives.
y = wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','FDJac'),x1/x2);
f = alpha\*(x2-x1^2)^2 + (1-x1)^2 + y;
solution3 = ezsolve(f,con,x0);

%   4.  USE  FD  and MAD  FOR  ONE  FUNCTION    EACH.
y1 = 0.5\*wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','MAD'),x1/x2);
y2 = 0.5\*wrap(struct('fun','sin','n',1,'sz1',1,'sz2',1,'JFuns','FDJac'),x1/x2);
f = alpha\*(x2-x1^2)^2 + (1-x1)^2 + y1 + y2;
solution4 = ezsolve(f,con,x0);

%   5.  USE  FD  FOR  ALL FUNCTIONS. 
options = struct; 
options.derivatives = 'numerical';
f = alpha\*(x2-x1^2)^2 + (1-x1)^2 + sin(x1/x2);
solution5 = ezsolve(f,con,x0,options);

%   6.  USE MAD FOR OBJECTIVE,   FD FOR CONSTRAINTS
options = struct; 
options.derivatives = 'numerical'; 
options.use_H	= 0; 
options.use_d2c = 0;
options.type	= 'con';
Prob = sym2prob(f,con,x0,options);
madinitglobals; Prob.ADObj = 1; Prob.ConsDiff = 1;
Result  = tomRun('snopt', Prob,  1);
solution6 = getSolution(Result);

Simplifications

The code generation function detects sub-expressions that occur more than once, and optimizes by creating tem- porary variables for those since it is very common for a function to share expressions with its derivative, or for the derivative to contain repeated expressions.

Note that it is not necessary to complete code generation in order to evaluate a tomSym object numerically. The subs function can be used to replace symbols by their numeric values, resulting in an evaluation.

TomSym also automatically implements algebraic simplifications of expressions. Among them are:

Multiplication by 1 is eliminated: 1 * A = A

Addition/subtraction of 0 is eliminated: 0 + A = A

All-same matrices are reduced to scalars: \[3; 3; 3\] + x = 3 + x

Scalars are moved to the left in multiplications: A * y = y * A

Scalars are moved to the left in addition/subtraction: A - y = -y + A

Expressions involving element-wise operations are moved inside setdiag: setdiag(A)+setdiag(A) = setdiag(A+ A)

Inverse operations cancel: sqrt(x)2 = x

Multiplication by inverse cancels: A * inv(A) = eye(size(A))

Subtraction of self cancels: A - A = zeros(size(A))

Among others...

Except in the case of scalar-matrix operations, tomSym does not reorder multiplications or additions, which means that some expressions, like (A+B)-A will not be simplified (This might be implemented in a later version, but must be done carefully so that truncation errors are not introduced).

Simplifications are also applied when using subs. This makes it quick and easy to handle parameterized problems. For example, if an intricate optimization problem is to be solved for several values of a parameter a, then one

might first create the symbolic functions and gradients using a symbolic a, and then substitute the different values, and generate m-code for each substituted function. If some case, like a = 0 results in entire sub-expressions being eliminated, then the m-code will be shorter in that case.

It is also possible to generate complete problems with constants as decision variables and then change the bounds for these variables to make them "real constants". The backside of this is that the problem will be slightly larger, but the problem only has to be generated once.

The following problem defines the variable alpha as a toms, then the bounds are adjusted for alpha to solve the problem for all alphas from 1 to 100.

toms x1 x2

%   Define  alpha  as a toms although it is a constant toms alpha

%   Setup and solve  the  problem
f = alpha\*(x2-x1^2)^2 + (1-x1)^2;
c = -x1^2  - x2;
con = \{-1000  <= c <= 0
    -10  <= x1 <= 2
    -10  <= x2 <= 2\};
x0 = \{x1  == -1.2; x2 == 1\}; 

Prob = sym2prob(f,con,x0);

%  Now   solve  for alpha  = 1:100, while reusing x_0 
obj  = zeros(100,1);

for  i=1:100
    Prob.x_L(Prob.tomSym.idx.alpha) = i; 
    Prob.x_U(Prob.tomSym.idx.alpha) = i; 
    Prob.x_0(Prob.tomSym.idx.alpha) = i; 
    Result  = tomRun('snopt', Prob,  1); 
    Prob.x_0  = Result.x_k;
    obj(i) = Result.f_k;
end

Special functions

TomSym adds some functions that duplicates the functionality of Matlab, but that are more suitable for symbolic treatment. For example:

setDiag and getDiag - Replaces some uses of Matlab's diag function, but clarifies whether diag(x) means "create a matrix where the diagonal elements are the elements of x" or "extract the main diagonal from the matrix x".

subsVec applies an expression to a list of values. The same effect can be achieved with a for-loop, but subsVec gives more efficient derivatives.

ifThenElse - A replacement for the if ... then ... else constructs (See below).

If ... then ... else:

A common reason that it is difficult to implement a function in tomSym is that it contains code like the following:

if x<2
    y = 0;
else
    y = x-2;
end

Because x is a symbolic object, the expression x < 2 does not evaluate to true or false, but to another symbolic object.

In tomSym, one should instead write:

y = ifThenElse(x<2,0,x-2)

This will result in a symbolic object that contains information about both the "true" and the "false" scenario. However, taking the derivative of this function will result in a warning, because the derivative is not well-defined at x = 2.

The "smoothed" form:

y = ifThenElse(x<2,0,x-2,0.1)

yields a function that is essentially the same for abs(x - 2) > 3 * 0.1, but which follows a smooth curve near x = 2, ensuring that derivatives of all orders exist. However, this introduces a local minimum which did not exist in the original function, and invalidates the convexity.

It is recommended that the smooth form ifThenElse be used for nonlinear problems whenever it replaces a dis- continuous function. However, for convex functions (like the one above) it is usually better to use code that helps

tomSym know that convexity is preserved. For example, instead of the above if ThenElse(x < 2, 0, x - 2, 0.1), the equivalent max(0, x - 2) is preferred.

Procedure vs parse-tree

TomSym works with procedures. This makes it different from many symbolic algebra packages, that mainly work with parse-trees.

In optimization, it is not uncommon for objectives and constraints to be defined using procedures that involve loops. TomSym is built to handle these efficiently. If a function is defined using many intermediate steps, then tomSym will keep track of those steps in an optimized procedure description. For example, consider the code:

toms x

y = x*x;

z = sin(y)+cos(y);

In the tomSym object z, there is now a procedure, which looks something like:

temp = x*x;

result = cos(temp)+sin(temp);

Note: It is not necessary to use the intermediate symbol y. TomSym, automatically detects duplicated expressions, so the code Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin(x*x)+cos(x*x)} would result in the same optimized procedure for z.

On the other hand, the same corresponding code using the symbolic toolbox:

syms x
y = x*x;
z = sin(y)+cos(y);

results in an object z that contains Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cos(x^2) + sin(x^2) } , resulting in a double evaluation of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 } .

This may seem like a small difference in this simplified example, but in real-life applications, the difference can be significant.

Numeric stability:

For example, consider the following code, which computes the Legendre polynomials up to the 100th order in tomSym (The calculation takes about two seconds on a modern computer).

toms x 
p\{1\}=1; 
p\{2\}=x;
for i=1:99
   p\{i+2\} = ((2\*i+1)\*x.\*p\{i+1\}-i\*p\{i\})./(i+1);
end

Replacing "toms" by "syms" on the first line should cause the same polynomials to be computed using Mathwork's Symbolic Toolbox. But after a few minutes, when only about 30 polynomials have been computed, the program crashes as it fails to allocate more memory. This is because the expression grows exponentially in size. To circumvent the problem, the expression must be algebraically simplified after each step. The following code succeeds in computing the 100 polynomials using the symbolic toolbox.

syms x 
p\{1\}=1; 
p\{2\}=x;
for i=1:99
   p\{i+2\} = simplify(((2\*i+1)\*x.\*p\{i+1\}-i\*p\{i\})./(i+1));
end

However, the simplification changes the way in which the polynomial is computed. This is clearly illustrated if we insert x = 1 into the 100th order polynomial. This is accomplished by using the command subs(p101,x,1) for both the tomSym and the Symbolic Toolbox expressions. TomSym returns the result 1.0000, which is correct. The symbolic toolbox, on the other hand, returns 2.6759e + 020, which is off by 20 orders of magnitude. The reason is that the "simplified" symbolic expressions involves subtracting very large numbers. Note: It is of course possible to get correct results from the Symbolic Toolbox using exact arithmetic instead of machine-precision floating-point, but at the cost of much slower evaluation.

In tomSym, there are also simplifications, for example identifying identical sub-trees, or multiplication by zero, but the simplifications are not as extensive, and care is taken to avoid simplifications that can lead to truncation errors. Thus, an expression computed using tomSym should be exactly as stable (or unstable) as the algorithm used to generate it.

Another example:

The following code, iteratively defines q as a function of the tomSym symbol x, and computes its derivative:

toms x 
q=x;
for i=1:4
   q = x\*cos(q+2)\*cos(q);
end derivative(q,x)

This yields the following tomSym procedure:

tempC3	= x+2;
tempC4	= cos(tempC3); 
tempC5	= x*tempC4; 
tempC10	= cos(x);
tempC12	= tempC10*(tempC4-x*sin(tempC3))-tempC5*sin(x);
tempC13	= tempC5*tempC10; 
tempC16	= tempC13+2; 
tempC17	= cos(tempC16); 
tempC18	= x*tempC17; 
tempC24	= cos(tempC13);
tempC26	= tempC24*(tempC17-x*(sin(tempC16)*tempC12))-tempC18*(sin(tempC13)*tempC12);
tempC27	= tempC18*tempC24; 
tempC30	= tempC27+2; 
tempC31	= cos(tempC30); 
tempC32	= x*tempC31; 
tempC38	= cos(tempC27);
tempC40	= tempC38*(tempC31-x*(sin(tempC30)*tempC26))-tempC32*(sin(tempC27)*tempC26);
tempC41	= tempC32*tempC38; 
tempC44	= tempC41+2; 
tempC45	= cos(tempC44);
out 	= cos(tempC41)*(tempC45-x*(sin(tempC44)*tempC40))-(x*tempC45)*(sin(tempC41)*tempC40);

Now, complete the same task using the symbolic toolbox:

syms x q=x;
for i=1:4
   q = x\*cos(q+2)\*cos(q);
end 
diff(q,x)

This yields the following symbolic expression:

cos(x*cos(x*cos(cos(x+2)*x*cos(x)+2)*cos(cos(x+2)*x*cos(x))+2)*cos(x*cos(cos(x+2)*x*cos(x)+2)*... 

cos(cos(x+2)*x*cos(x)))+2)*cos(x*cos(x*cos(cos(x+2)*x*cos(x)+2)*cos(cos(x+2)*x*cos(x))+2)*cos(x*... 

cos(cos(x+2)*x*cos(x)+2)*cos(cos(x+2)*x*cos(x))))-x*sin(x*cos(x*cos(cos(x+2)*x*cos(x)+2)*cos(...

...

and 23 more lines of  code.

And this example only had four iterations of the loop. Increasing the number of iterations, the Symbolic toolbox expressions quickly becomes unmanageable, while the tomSym procedure only grows linearly.

Problems and error messages

Warning: Directory c:\Temp\tp563415 could not be removed (or similar). When tomSym is used to automatically create m-code it places the code in a temporary directory given by Matlab's tempname function. Sometimes Matlab chooses a name that already exists, which results in this error message (The temporary directory is cleared of old files regularly by most modern operating systems. Otherwise the temporary Matlab files can easily be removed manually).

Attempting to call SCRIPT as a function (or similar). Due to a bug in the Matlab syntax, the parser cannot know if f (x) is a function call or the x:th element of the vector f. Hence, it has to guess. The Matlab parser does not understand that toms creates variables, so it will get confused if one of the names is previously used by a function or script (For example, "cs" is a script in the systems identification toolbox). Declaring toms cs and then indexing cs(1) will work at the Matlab prompt, but not in a script. The bug can be circumvented by assigning something to each variable before calling toms.

Example

A TomSym model is to a great extent independent upon the problem type, i.e. a linear, nonlinear or mixed-integer nonlinear model would be modeled with about the same commands. The following example illustrates how to construct and solve a MINLP problem using TomSym.

Name='minlp1Demo  - Kocis/Grossman.';

toms 2x1 x
toms 3x1 integer  y

objective = [2  3 1.5  2 -0.5]*[x;y];

constraints = { ... 
   x(1) >= 0,  ... 
   x(2) >= 1e-8,   ... 
   x <= 1e8,  ...
   0 <= y <=1,  ...
   [1  0 1 0 0]*[x;y] <= 1.6, ...
   1.333*x(2) + y(2) <= 3,  ... 
   [-1 -1  1]*y <= 0,  ...
   x(1)^2+y(1) == 1.25, ... 
   sqrt(x(2)^3)+1.5\*y(2) == 3,  ...
};

guess = struct('x',ones(size(x)),'y',ones(size(y)));
options = struct;
options.name  = Name;
Prob = sym2prob('minlp',objective,constraints,guess,options); 
Prob.DUNDEE.optPar(20) = 1;
Result  = tomRun('minlpBB',Prob,2);

The TomSym engine automatically completes the separation of simple bounds, linear and nonlinear constraints.

Solving Linear, Quadratic and Integer Programming Problems

This section describes how to define and solve linear and quadratic programming problems, and mixed-integer linear programs using TOMLAB. Several examples are given on how to proceed, depending on if a quick solution is wanted, or more advanced tests are needed. TOMLAB is also compatible with MathWorks Optimization TB. See Appendix E for more information and test examples.

The test examples and output files are part of the standard distribution of TOMLAB, available in directory usersguide, and all tests can be run by the user. There is a file RunAllTests that goes through and runs all tests for this section.

Also see the files lpDemo.m, qpDemo.m, and mipDemo.m, in the directory examples, where in each file a set of simple examples are defined. The examples may be ran by giving the corresponding file name, which displays a menu, or by running the general TOMLAB help routine tomHelp.m.

Linear Programming Problems

The general formulation in TOMLAB for a linear programming problem is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq2:lp} \begin{array}{ll} \min\limits_{x} & f(x) = c^T x \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ \end{array} \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} . Equality constraints are defined by setting the lower bound equal to the upper bound, i.e. for constraint Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $i$: $b_{L}(i) = b_{U}(i)$} .

To illustrate the solution of LPs consider the simple linear programming test problem

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{defLP1} \begin{array}{ccccl} \min\limits_{x_{1},x_{2}} & f(x_{1},x_{2}) & = & -7x_{1}-5x_{2} & \\s/t & x_{1} + 2x_{2} & \leq & 6 & \\ & 4x_{1} + x_{2} & \leq & 12 & \\ & x_{1},x_{2} & \geq & 0 & \end{array} }

named LP Example.

The following statements define this problem in Matlab

File: tomlab/usersguide/lpExample.m

Name	= 'lptest';
c 	= [-7 -5]'; 	%   Coefficients in linear objective function
A	= [ 1	2
            4	1 ];	%   Matrix defining linear  constraints
b_U	= [ 6 12 ]';	%   Upper bounds on the  linear inequalities 
x_L	= [ 0	0 ]';	%   Lower bounds on x

%   x_min and x_max are  only  needed if doing  plots 
x_min = [ 0  0 ]';
x_max = [10  10 ]';

%   b_L,  x_U and x_0 have default values  and need not  be defined.
%   It is possible to  call  lpAssign with  empty \[\] arguments instead b_L = [-inf -inf]';
x_U	= [];
x_0	= [];

A Quick Linear Programming Solution

The quickest way to solve this problem is to define the following Matlab statements using the TOMLAB format:

File: tomlab/usersguide/lpTest1.m

lpExample;

Prob	= lpAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  'lpExample'); 
Result  = tomRun('lpSimplex', Prob,  1);

lpAssign is used to define the standard Prob structure, which TOMLAB always uses to store all information about a problem. The three last parameters could be left out. The upper bounds will default be Inf, and the problem name is only used in the printout in PrintResult to make the output nicer to read. If x 0, the initial value, is left out, an initial point is found by lpSimplex solving a feasible point (Phase I) linear programming problem. In this test the given x 0 is empty, so a Phase I problem must be solved. The solution of this problem gives the following output to the screen

File: tomlab/usersguide/lpTest1.out

===== \* \* \* =================================================================== * * * 
TOMLAB  /SOL + /CGO  + /Xpress  MEX  + /CPLEX  Parallel 2-CPU  + 21 more - Tomlab Optimizat
===================================================================================== 
Problem:  ---	1:  lpExample	f_k 	-26.571428571428569000

Solver: lpSimplex.	EXIT=0.
Simplex  method.  Minimum  reduced  cost. 
Optimal  solution  found

FuncEv	3 Iter	3
CPU  time: 0.046800 sec.  Elapsed  time: 0.019000 sec.

Having defined the Prob structure is it easy to call any solver that can handle linear programming problems,

Result  = tomRun('qpSolve', Prob,  1);

Even a nonlinear solver may be used.

Result  = tomRun('nlpSolve',Prob, 3);

All TOMLAB solvers may either be called directly, or by using the driver routine tomRun, as in this case.

Quadratic Programming Problems

The general formulation in TOMLAB for a quadratic programming problem is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{eq2:qp} \begin{array}{ll} \min\limits_{x} & f(x) = \frac{1}{2} x^T F x + c^T x \\ & \\ s/t & \begin{array}{lcccl} x_{L} & \leq & x & \leq & x_{U}, \\ b_{L} & \leq & A x & \leq & b_{U} \\ \end{array} \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c, x, x_L, x_U \in \MATHSET{R}^n$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $F \in \MATHSET{R}^{n \times n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \MATHSET{R}^{m_1 \times n}$} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \MATHSET{R}^{m_1}$} . Equality constraints are defined by setting the lower bound equal to the upper bound, i.e. for constraint Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $i$} : Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_{L}(i) = b_{U}(i)$} . Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle %$b_{L}(i) = b_{U}(i), \all i \in E$, where $E$} is the set of equalities. Fixed variables are handled the same way.

To illustrate the solution of QPs consider the simple quadratic programming test problem

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{defQP1} \begin{array}{ll} \min\limits_{x} & f(x)=4x_{1}^2+1x_{1}x_{2}+4x_{2}^2+3x_{1}-4x_{2} \\ s/t & x_{1}+x_{2} \leq 5 \\& x_{1}-x_{2} = 0 \\& x_{1} \geq 0 \\& x_{2} \geq 0, \\ \end{array} }

named QP Example. The following statements define this problem in Matlab.

File: tomlab/usersguide/qpExample.m

Name  = 'QP Example';  % File qpExample.m
F     = [ 8   1        % Matrix F in 1/2 * x' * F * x + c' * x
          1   8 ];
c     = [ 3  -4 ]';    % Vector c in 1/2 * x' * F * x + c' * x
A     = [ 1   1        % Constraint matrix
          1  -1 ];
b_L   = [-inf  0  ]';  % Lower bounds on the linear constraints
b_U   = [ 5    0  ]';  % Upper bounds on the linear constraints
x_L   = [ 0    0  ]';  % Lower bounds on the variables
x_U   = [ inf inf ]';  % Upper bounds on the variables
x_0   = [ 0    1  ]';  % Starting point
x_min = [-1 -1 ];      % Plot region lower bound parameters
x_max = [ 6  6 ];      % Plot region upper bound parameters

A Quick Quadratic Programming solution

The quickest way to solve this problem is to define the following Matlab statements using the TOMLAB format:

File: tomlab/usersguide/qpTest1.m

qpExample;

Prob = qpAssign(F, c,  A,  b_L,  b_U, x_L,  x_U, x_0,  'qpExample');

Result  = tomRun('qpSolve', Prob,  1);

The solution of this problem gives the following output to the screen.

File: tomlab/usersguide/qpTest1.out

===== * * * ===================================================================  * * * 
TOMLAB  /SOL + /CGO  + /Xpress  MEX  + /CPLEX  Parallel 2-CPU  + 21 more - Tomlab Optimizat
===================================================================================== 
Problem:  ---	1:  qpExample	                     f_k 	-0.027777777777777790

Solver: qpSolve. 	EXIT=0.	INFORM=1. 
Active set  strategy
Optimal  point  found
First order  multipliers >= 0

Iter     4
CPU  time: 0.046800 sec.  Elapsed  time: 0.037000 sec.

qpAssign is used to define the standard Prob structure, which TOMLAB always uses to store all information about a problem. The three last parameters could be left out. The upper bounds will default be Inf, and the problem name is only used in the printout in PrintResult to make the output nicer to read. If x 0, the initial value, is left out, a initial point is found by qpSolve solving a feasible point (Phase I) linear programming problem calling the TOMLAB lpSimplex solver. In fact, the output shows that the given x0 = (0, -1)T was rejected because it was infeasible, and instead a Phase I solution lead to the initial point x0 = (0, 0)T .

Mixed-Integer Programming Problems

This section describes how to solve mixed-integer programming problems efficiently using TOMLAB. To illustrate the solution of MIPs consider the simple knapsack 0/1 test problem Weingartner 1, which has 28 binary variables and two knapsacks. The problem is defined

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{defKS1} \begin{array}{ccccl} \min\limits_{x} & c^T x \\ & \\ s/t & \begin{array}{lcccl} 0& \leq & x & \leq & 1, \\ & & A x & = & b, \\ \end{array} \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b=(600, 600)^T$} ,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rccccccccccccccl} c = -(&1898 & 440 &22507 &270 &14148 & 3100 & 4650 &30800 &615 & 4975 &1160 & 4225 & 510 & 11880 & \\ & 479 &440 & 490 & 330 & 110 & 560 &24355& 2885&11748& 4550 & 750 & 3720 &1950 &10500&)^T \end{array} }

and the A matrix is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \begin{array}{cccccccccccccccccccc} 45 & 0 & 85 &150 &65& 95& 30& 0& 170& 0 & 40 & 25 & 20 & 0 & 0& 25& 0& 0& 25& 0 \\ 165& 0 & 85 & 0 & 0& 0& 0& 100 \\ 30 & 20 & 125 & 5 & 80 & 25 & 35 & 73 & 12 &15 & 15 & 40 & 5 & 10 & 10 & 12 & 10 & 9 & 0 &20 \\ 60 & 40 & 50 & 36 & 49 & 40 & 19 & 150\\ \end{array} \right) }

The following statements define this problem in Matlab using the TOMLAB format:

File: tomlab/usersguide/mipExample.m

Name='Weingartner 1 - 2/28  0-1  knapsack';

%   Problem formulated as a minimum  problem

A = [ 45	0	85	150	65	95	30	0	170	0 ...

      40	25	20	0	0	25	0	0	25	0 ...

      165	0	85	0	0	0	0	100	; ...

      30	20	125	5	80	25	35	73	12	15 ...

      15	40	5	10	10	12	10	9	0	20 ...

      60	40	50	36	49	40	19	150];

b_U = [600;600]; % 2 knapsack capacities

c = [1898	440	22507	270	14148	3100	4650	30800	615	4975 ...
     1160	4225	510	11880	479	440	490	330	110	560 ...
     24355	2885	11748	4550	750	3720	1950	10500]'; % 28 weights

%   Make  problem  on standard  form  for  mipSolve

[m,n]	= size(A);
A	= [A  eye(m,m)];
c 	= [-c;zeros(m,1)]; %   Change  sign  to  make  a minimum  problem
[mm nn] = size(A);
x_L	= zeros(nn,1);
x_U	= [ones(n,1);b_U];
x_0	= [zeros(n,1);b_U];

fprintf('Knapsack problem.  Variables %d. Knapsacks %d\n',n,m);
fprintf('Making standard  form  with 	%d  variables\n',nn);

%   All  original variables should  be integer, but  also  slacks in this case 
IntVars = nn; 	%   Could also  be set  as:  IntVars=1:nn; or  IntVars=ones(nn,1); 
x_min	= x_L;  
x_max	= x_U; 
f_Low	= -1E7; %   f_Low <= f_optimal must hold
n	= length(c);
b_L	= b_U;
f_opt	= -141278;

The quickest way to solve this problem is to define the following Matlab statements:

'''File: '''tomlab/usersguide/mipTest1.m

<pre>
mipExample;

nProblem  = 7;  %  Use the  same  problem  number as in mip_prob.m
fIP	  = []; %  Do   not  use any prior knowledge
xIP 	  = []; %  Do   not  use any prior knowledge
setupFile = []; %  Just  define the  Prob structure, not  any permanent setup  file
x_opt 	  = []; %  The optimal integer solution is not  known
VarWeight = []; %  No variable priorities, largest fractional part will  be used
KNAPSACK  = 0; 	%  First run  without the  knapsack heuristic

Prob = mipAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  Name, setupFile, nProblem,...

IntVars, VarWeight,  KNAPSACK,  fIP, xIP, ... f_Low,  x_min,  x_max, f_opt, x_opt);

Prob.Solver.Alg	      = 2;     %   Depth First, then  Breadth  (Default Depth First)
Prob.optParam.MaxIter = 5000;  %   Must increase iterations from  default 500
Prob.optParam.IterPrint = 0; 
Prob.PriLev = 1;
Result 	= tomRun('mipSolve', Prob,  0);

%   ------------------------------------------------
%   Add priorities on the  variables
%   ------------------------------------------------ 
VarWeight = c;
%   NOTE.  Prob is  already defined,	could  skip  mipAssign,  directly set:
%   Prob.MIP.VarWeight=c;

Prob = mipAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  Name, setupFile, nProblem,...
                 IntVars, VarWeight,  KNAPSACK,  fIP, xIP, ... 
                 f_Low,  x_min,  x_max, f_opt, x_opt);

Prob.Solver.Alg	= 2;          % Depth First, then  Breadth  search
Prob.optParam.MaxIter = 5000; % Must increase number of  iterations
Prob.optParam.IterPrint = 0; 
Prob.PriLev = 1;

Result = tomRun('mipSolve', Prob,  0);

%   ----------------------------------------------
%   Use the  knapsack heuristic, but  not  priorities
%   ---------------------------------------------- 
KNAPSACK	= 1;  VarWeight = [];

%   NOTE.  Prob is  already defined,  could  skip  mipAssign,  directly set:
%   Prob.MIP.KNAPSACK=1;
%   Prob.MIP.VarWeight=[];

Prob	= mipAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  Name, setupFile, ... 
                    nProblem,  IntVars, VarWeight,  KNAPSACK,  fIP, xIP, ... 
                    f_Low,  x_min,  x_max, f_opt, x_opt);

Prob.Solver.Alg = 2;	% Depth First, then  Breadth  search
Prob.optParam.IterPrint = 0; 
Prob.PriLev = 1;

Result = tomRun('mipSolve', Prob,  0);

%   --------------------------------------------------
%  Now   use both  the  knapsack heuristic and priorities
%   -------------------------------------------------- 
VarWeight = c; KNAPSACK	= 1;
% NOTE.  Prob is  already defined,  could  skip  mipAssign,  directly set:
% Prob.MIP.KNAPSACK=1;
% Prob.MIP.VarWeight=c;

Prob = mipAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  Name, setupFile, nProblem,...
                 IntVars, VarWeight,  KNAPSACK,  fIP, xIP, ... 
                 f_Low,  x_min,  x_max, f_opt, x_opt);

Prob.Solver.Alg = 2; % Depth First, then  Breadth  search
Prob.optParam.IterPrint = 0; 
Prob.PriLev = 1;

Result = tomRun('mipSolve', Prob,  0);

To make it easier to see all variable settings, the first lines define the needed variables. Several of them are just empty arrays, and could be directly set as empty in the call to mipAssign. mipAssign is used to define the standard Prob structure, which TOMLAB always uses to store all information about a problem. After mipAssign has defined the structure Prob it is then easy to set or change fields in the structure. The solver mipSolve is using three different strategies to search the branch-and-bound tree. The default is the Depth first strategy, which is also the result if setting the field Solver.Alg as zero. Setting the field as one gives the Breadth first strategy and setting it as two gives the Depth first, then breadth search strategy. In the example our choice is the last strategy. The number of iterations might be many, thus the maximal number of iterations must be increased, the field optParam.MaxIter.

Tests show two ways to improve the convergence of MIP problems. One is to define a priority order in which the different non-integer variables are selected as variables to branch on. The field MIP.VarWeight is used to set priority numbers for each variable. Note that the lower the number, the higher the priority. In our test case the coefficients of the objective function is used as priority weights. The other way to improve convergence is to use a heuristic. For binary variables a simple knapsack heuristic is implemented in mipSolve. Setting the field MIP.KNAPSACK as true instructs mipSolve to use the heuristic.

Running the four different tests on the knapsack problem gives the following output to the screen

File: tomlab/usersguide/mipTest1.out

mipTest1
Knapsack problem.  Variables 28.  Knapsacks 2
Branch and bound.  Depth First, then  Breadth.

--- Branch &  Bound converged!  Iterations (nodes  visited) =     714 
Total LP Iterations =       713

Optimal  Objective function  =		-141278.0000000000000000 
x: 0 0 1 -0 1 1	1 1 0 1	0 1 1 1 
   0 0 0  0 1 0 1 0 1 1 0 1 0 0

B: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Branch & bound. Depth First, then Breadth. Priority weights.

--- Branch & Bound converged! 
Iterations (nodes visited) =     470 
Total LP Iterations        =     469

Optimal  Objective function  =		-141278.0000000000000000
x: 0 0 1 -0 1 1	1 1 0 1	0 1 1 1 
   0 0 0  0 1 0 1 0 1 1 0 1 0 0
B: -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
Branch and bound.  Depth First, then  Breadth. Knapsack Heuristic.
Found new BEST  Knapsack. Nodes left	0.  Nodes deleted	0. 
Best  IP  function value 	-139508.0000000000000000

Found new BEST  Knapsack. Nodes left	1.  Nodes deleted	0. 
Best  IP  function value 	-140768.0000000000000000

Found new BEST  Knapsack. Nodes left	4.  Nodes deleted	0. 
Best  IP  function value 	-141278.0000000000000000

--- Branch &  Bound converged!  Iterations (nodes  visited) =       96 
Total LP Iterations =         95

Optimal  Objective function  =		-141278.0000000000000000
x: 0 0 1 -0 1 1	1 1 0 1	0 1 1 1 
   0 0 0  0 1 0 1 0 1 1 0 1 0 0
B:  1 1 -1 1 -1 -1 -1 -1 1 -1 0 -1 -1 -1 1 1 1 1 -1 1 -1 0 -1 -1 1 -1 -1 1
Branch &  bound.  Depth First, then  Breadth. Knapsack heuristic. Priority weights.

Found new BEST  Knapsack. Nodes left	0.  Nodes deleted	0. 
Best  IP  function 
value 	-139508.0000000000000000

Found new BEST  Knapsack. Nodes left	1.  Nodes deleted	0. 
Best  IP  function value 	-140768.0000000000000000

Found new BEST  Knapsack. Nodes left	4.  Nodes deleted	0.
Best  IP  function value 	-141278.0000000000000000

--- Branch &  Bound converged!  Iterations (nodes  visited) =	94 
Total LP Iterations =	93

Optimal  Objective function  =		-141278.0000000000000000
x: 0 0 1 -0 1 1	1 1 0 1	0 1 1 1 
   0 0 0  0 1 0 1 0 1 1 0 1 0 0
B:  1 1 -1 1 -1 -1 -1 -1 1 -1 0 -1 -1 -1 1 1 1 1 -1 1 -1 0 -1 -1 1 -1 -1 1 
diary off

Note that there is a large difference in the number of iterations if the additional heuristic and priorities are used. Similar results are obtained if running the other two tree search strategies.

Solving Unconstrained and Constrained Optimization Problems

This section describes how to define and solve unconstrained and constrained optimization problems. Several examples are given on how to proceed, depending on if a quick solution is wanted, or more advanced runs are needed. See Appendix E for more information on your external solvers.

All demonstration examples that are using the TOMLAB format are collected in the directory examples. It is also possible to run each example separately. The examples relevant to this section are ucDemo and conDemo. The full path to these files are always given in the text. Throughout this section the test problem Rosenbrock's banana,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{rosen} \begin{array}{ll} \min\limits_{x} & f(x)=\alpha\left(x_{2}-x_{1}^{2}\right)^{2}+\left(1-x_{1} \right)^{2} \\ s/t & \begin{array}{lcccl} -10 & \leq & x_1 & \leq &2 \\ -10 & \leq & x_2 & \leq &2 \\ \end{array} \end{array} }

is used to illustrate the solution of unconstrained problems. The standard value is a = 100. In this formulation simple bounds are added on the variables, and also constraints in illustrative purpose. This problem is called RB BANANA in the following descriptions to avoid mixing it up with problems already defined in the problem definition files.

Defining the Problem in Matlab m-files

TOMLAB demands that the general nonlinear problem is defined in Matlab m-files, and not given as an input text string. A file defining the function to be optimized must always be supplied. For linear constraints the constraint coefficient matrix and the right hand side vector are given directly. Nonlinear constraints are defined in a separate file. First order derivatives and second order derivatives, if available, are stored in separate files, both function derivatives and constraint derivatives.

TOMLAB is compatible with MathWorks Optimization TB, which in various ways demands both functions, derivatives and constraints to be returned by the same function. TOMLAB handle all this by use of interface routines, hidden for the user.

It is generally recommended to use the TOMLAB format instead, because having defined the files in this format, all MathWorks Optimization TB solvers are accessible through the TOMLAB multi-solver driver routines.

The rest of this section shows how to make the m-files for the cases of unconstrained and constrained optimization. In Section 6.2 and onwards similar m-files are used to solve unconstrained optimization using the TOMLAB format.

The most simple way to write the m-file to compute the function value is shown for the example in (17) assuming a = 100:

File: tomlab/usersguide/rbbs f.m

%   rbbs_f   - function value  for Constrained   Rosenbrocks Banana
%
%   function f = rbbs_f(x)

function f = rbbs_f(x)

f = 100*(x(2)-x(1)^2)^2 + (1-x(1))^2;

Running TOMLAB it is recommended to use a more general format for the m-files, adding one extra parameter, the Prob problem definition structure described in detail in Appendix A. TOMLAB will handle the simpler format also, but the more advanced features of TOMLAB are not possible to use.

If using this extra parameter, then any information needed in the low-level function computation routine may be sent as fields in this structure. For single parameter values, like the above a parameter in the example, the field Prob.user is recommended.

Using the above convention, then the new m-file for the example in (17) is defined as

File: tomlab/usersguide/rbb f.m

%   rbb_f   - function value  for Rosenbrocks Banana, Problem RB  BANANA
%
%   function f = rbb_f(x,  Prob) function f = rbb_f(x,  Prob) alpha  = Prob.user.alpha;

f = alpha*(x(2)-x(1)^2)^2 + (1-x(1))^2;

The value in the field Prob.user is the a value. It is defined before calling the solver by explicit setting the Prob structure. In a similar way the gradient routine is defined as

File: tomlab/usersguide/rbb g.m

%   rbb_g  - gradient vector for Rosenbrocks Banana, Problem RB  BANANA
%
%   function g = rbb_g(x,  Prob) function g = rbb_g(x,  Prob) alpha  = Prob.user.alpha;

g = [ -4*alpha*x(1)*(x(2)-x(1)^2)-2*(1-x(1));  2*alpha*(x(2)-x(1)^2) ];

If the gradient routine is not supplied, TOMLAB will use finite differences (or automatic differentiation) if the gradient vector is needed for the particular solver. In this case it is also easy to compute the Hessian matrix, which gives the following code

File: tomlab/usersguide/rbb H.m

%   rbb_H - Hessian  matrix for Rosenbrocks Banana, Problem RB  BANANA
%
%   function H  = crbb_H(x, Prob) function H  = rbb_H(x,  Prob) alpha  = Prob.user.alpha;

H  = [ 12*alpha*x(1)^2-4*alpha*x(2)+2 , -4*alpha*x(1); -4*alpha*x(1), 2*alpha ];

If the Hessian routine is not supplied, TOMLAB will use finite differences (or automatic differentiation) if the Hessian matrix is needed for the particular solver. Often a positive-definite approximation of the Hessian matrix is estimated during the optimization , and the second derivative routine is then not used.

If using the constraints defined for the example in (17) then a constraint routine needs to defined for the single nonlinear constraint, in this case

File: tomlab/usersguide/rbb c.m

%   rbb_c  - nonlinear constraint vector for  Rosenbrocks Banana, Problem RB  BANANA
%
%   function c = rbb_c(x,  Prob) function cx  = rbb_c(x,  Prob) cx  = -x(1)^2 - x(2);

The constraint Jacobian matrix is also of interest and is defined as

File: tomlab/usersguide/rbb dc.m

%   rbb_dc  - nonlinear constraint  gradient matrix
%   for Rosenbrocks Banana, Problem RB  BANANA
%
%   function dc = crbb_dc(x,  Prob)

function dc = rbb_dc(x,  Prob)

%  One row for each constraint, one column for each variable. dc = [-2*x(1),-1];

If the constraint Jacobian routine is not supplied, TOMLAB will use finite differences (or automatic differentiation) to estimate the constraint Jacobian matrix if it is needed for the particular solver.

The solver nlpSolve is also using the second derivatives of the constraint vector. The result is returned as a weighted sum of the second derivative matrices with respect to each constraint vector element, the weights being the Lagrange multipliers supplied as input to the routine. For the example problem the routine is defined as

File: tomlab/usersguide/rbb d2c.m

%   rbb_d2c  - The second part of  the  Hessian  to  the  Lagrangian  function for the
%   nonlinear constraints for Rosenbrocks Banana, Problem RB  BANANA,   i.e.
%
%   lam' * d2c(x)
%
%   in
%
%   L(x,lam) =     f(x) - lam'  * c(x)
%   d2L(x,lam) = d2f(x) - lam'  * d2c(x)   = H(x)  - lam'  * d2c(x)
%
%   function d2c=crbb_d2c(x, lam,  Prob)

function  d2c=rbb_d2c(x, lam,  Prob)

%   The only  nonzero  element  in the  second derivative matrix for the  single
%   constraint is the  (1,1)  element, which  is a constant -2. 

d2c = lam(1)*[-2 0;  0 0];

Communication between user routines

It is often the case that mathematical expressions that occur in the function computation also is part of the gradient and Hessian computation. If these operations are costly it is natural to avoid recomputing these and reuse them when computing the gradient and Hessian.

The function routine is always called before the gradient routine in TOMLAB, and the gradient routine is always called before the Hessian routine. The constraint routine is similarly called before the computation of the constraint gradient matrix. However, the TOM solvers call the function before the constraint routine, but the SOL solvers do the reverse.

Thus it is safe to use global variables to communicate information from the function routine to the gradient and Hessian, and similarly from the constraint routine to the constraint gradient routine. Any non-conflicting name could be used as global variable, see Table 154 in Appendix D to find out which names are in use. However, the recommendation is to always use a predefined global variable named US A for this communication. TOMLAB is designed to handle recursive calls, and any use of new global variables may cause conflicts. The variable US A (and also US B) is automatically saved in a stack, and any level of recursions may safely be used. The user is free to use US A both as variable, and as a structure. If much information is to be communicated, defining US A as a structure makes it possible to send any amount of information between the user routines.

In the examples directory the constrained optimization example in conDemo is using the defined functions con1 f, con1 g and con1 H. They include an example of communicating one exponential expression between the routines.

The lsDemo example file in the examples directory communicates two exponential expressions between ls1 r and ls1 J with the use of US A and US B. In ls1 r the main part is

...
global US_A

t  = Prob.LS.t(:);

%   Exponential computations  takes  time, and may  be done once,  and
%   reused  when computing  the  Jacobian
US_A  = exp(-x(1)*t); 
US_B  = exp(-x(2)*t);

r = K*x(1)*(US_B  - US_A) / (x(3)*(x(1)-x(2))) - Prob.LS.y;

In ls1 J then US A is used

...
global US_A
%   Pick  up the  globally saved exponential computations 
e1 = US_A;
e2 = US_B;

%   Compute  the  three   columns in the  Jacobian, one for each of  variable
J = a * [ t.*e1+(e2-e1)*(1-1/b), -t.*e2+(e2-e1)/b, (e1-e2)/x(3) ];

For more discussions on global variables and the use of recursive calls in TOMLAB, see Appendix D.

In the following sections it is described how to setup problems in TOMLAB and use the defined m-files. First comes the simplest way, to use the TOMLAB format.

Unconstrained Optimization Problems

The use of the TOMLAB format is best illustrated by examples

The following is the first example in the ucDemo demonstration file. It shows an example of making a call to conAssign to create a structure in the TOMLAB format, and solve the problem with a call to ucSolve.

%   --------------------------------------------------------------------- 
function uc1Demo
%   ---------------------------------------------------------------------

format  compact 
fprintf('=====================================================\n'); 
fprintf('Rosenbrocks banana with  explicit f(x),  g(x) and H(x)\n'); 
fprintf('=====================================================\n');

Name	= 'RB Banana';
x_0	= [-1.2 1]'; 	%   Starting values  for the  optimization. x_L	= [-10;-10];	
x_L     = {-10;-10];    %   Lower bounds for x.
x_U	= [2;2];	%   Upper bounds for x. 
fLowBnd = 0; 	        %   Lower bound on function.

% Generate the problem structure using the TOMLAB format (short call) 
Prob = conAssign('uc1_f', 'uc1_g',  'uc1_H', \[\], x_L,  x_U, Name, ... 
                 x_0,  [], fLowBnd);

Result    = tomRun('ucSolve', Prob,  1);

In its more general form conAssign is used to define constrained problems. It also takes as input the nonzero pattern of the Hessian matrix, stored in the matrix HessPattern. In this case all elements of the Hessian matrix are nonzero, and either HessPattern is set as empty or as a matrix with all ones. Also the parameter pSepFunc should be set. It defines if the objective function is partially separable, see Section 14.5. Setting this parameter empty (the default), then this property is not used. In the above example the call would be

...

HessPattern  = ones(2,2);	%   The pattern of  nonzeros 
pSepFunc	= [];	        %  No   partial separability used

%   conAssign  is used to  generate  the  TOMLAB  problem  structure
Prob	= conAssign('uc1_f',  'uc1_g',  'uc1_H', HessPattern, ... 
                     x_L,  x_U, Name, x_0,  pSepFunc, fLowBnd);
...

Also see the other examples in ucDemo on how to solve the problem, when gradients routines are not present, and numerical differentiation must be used. An example on how to solve a sequence of problems is also presented.

If the gradient is not possible to define one just sets the corresponding gradient function name empty.

The example uc3Demo in file ucDemo show how to solve a sequence of problems in TOMLAB, in this case changing the steepness parameter a in (17). It is important to point out that it is only necessary to define the Prob structure once and then just change the varying parameters, in this case the a value. The a value is sent to the user routines using the field user in the Prob structure. Any field in the Prob structure could be used that is not conflicting with the predefined fields. In this example the a vector of Result structures are saved for later preprocessing.

%   --------------------------------------------------------------------- 
function uc3Demo  - Sequence  of  Rosenbrocks banana functions
%   ---------------------------------------------------------------------

%   conAssign  is used to  generate  the  TQ  problem  structure
%   Prob = conAssign(f,g,H,  HessPattern, x_L,  x_U, Name, x_0,  pSepFunc, fLowBnd); 
Prob = conAssign('uc3_f',\[\],\[\],\[\],\[-10;-10\], \[2;2\], \[-1.2;1\], 'RB Banana',\[\],0)

%   The different steepness  parameters  to  be tried
Steep = [100  500 1000 10000];

for i = 1:4
    Prob.user.alpha	= Steep(i);
    Result(i)	= tomRun('ucSolve', Prob,  1);
end

Direct Call to an Optimization Routine

When wanting to solve a problem by a direct call to an optimization routine there are two possible ways of doing it. The difference is in the way the problem dependent parameters are defined. The most natural way is to use a Init File, like the predefined TOMLAB Init Files o prob (e.g. uc prob if the problem is of the type unconstrained) to define those parameters. The other way is to call the routine conAssign. In this subsection, examples of two different approaches are given.

First, solve the problem RB BANANA in (17) as an unconstrained problem. In this case, define the problem in the files ucnew prob, ucnew f, ucnew g and ucnew H. Using the problem definition files in the working directory solve the problem and print the result by the following calls.

File: tomlab/usersguide/ucnewSolve1.m

probFile = 'ucnew_prob';           %   Problem definition file.
P  = 18;                           %   Problem number for the  added RB  Banana. 
Prob = probInit(probFile, P); 	   %   Setup Prob structure.

Result  = tomRun('ucSolve', Prob,  1);

Now, solve the same problem as in the example above but define the parameters x 0, x L and x L by calling the routine conAssign. Note that in this case the file ucnew prob is not used, only the files ucnew f and ucnew g. The file ucnew H is not needed because a quasi-Newton BFGS algorithm is used.

File: tomlab/usersguide/ucnewSolve2.m

x_0	= [-1.2;1];	           %   Starting values for the optimization. 
x_L	= [-10;-10];	           %   Lower bounds for x.
x_U     = [2;2];                   %   Upper bounds for x.

Prob = conAssign('ucnew_f','ucnew_g', \[\], \[\], x_L,  x_U,...
                 'ucNew', x_0);

Prob.P 	= 18; 	                   %   Problem number. 
Prob.Solver.Alg=1;	           %   Use quasi-Newton  BFGS 
Prob.user.uP = 100; 	           %   Set  alpha  parameter 
Result = tomRun('ucSolve',Prob,1);

Constrained Optimization Problems

Study the following constrained exponential problem, Exponential problem III,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \label{ExpIII} \begin{array}{ll} \min\limits_{x} & f(x)= exp(x_1) (4 x_1^2+ 2 x_2^2+ 4 x_1 x_2+2 x_2+1) \\ s/t & \begin{array}{lcccl} -10 & \leq & x_1 & \leq &10 \\-10 & \leq & x_2 & \leq &10 \\ 0 & \leq & x_1 + x_2 & \leq &0 \\ 1.5 & \leq & -x_1 x_2+x_1+x_2 & & \\ -10 & \leq & x_1 x_2 & & \\ \end{array} \end{array}. }

The first two constraints are simple bounds, the third is a linear equality constraint, because lower and upper bounds are the same. The last two constraints are nonlinear inequality constraints. To solve the problem, define the following statements, available as con1Demo in file conDemo.

Name    = 'Exponential problem III';
A       = [1  1];	% One    linear constraint
b_L	= 0;		% Lower bound on linear constraint 
b_U	= 0;		% b_L == b_U implies equality
c_L     = [1.5;-10]     % Two nonlinear inequality constraints
c_U	= [];		% Empty means  Inf (default) for the  two constraints 
x_0	= [-5;5];	% Initial value  for  x
x_L	= [-10;-10];	% Lower bounds on x 
x_U	= [10;10];	% Upper bounds on x
fLowBnd = 0; 		% A lower bound on the optimal function value 
x_min	= [-2;-2];	% Used for plotting, lower  bounds
x_max    = [4;4];       % Used for plotting, upper  bounds

x_opt = [-3.162278, 3.162278; -1.224745, 1.224745]; % Two stationary points 
f_opt = [1.156627; 1.8951];


HessPattern  = [];      % All elements  in Hessian  are  nonzero.
ConsPattern  = [];      % All elements  in the  constraint Jacobian  are  nonzero. 
pSepFunc     = [];      % The function f is not  defined as separable

%   Generate the  problem  structure using  the  TOMLAB  format
Prob = conAssign('con1_f',  'con1_g',  'con1_H', HessPattern, x_L,  x_U, ...
       Name, x_0,  pSepFunc, fLowBnd, A,  b_L,  b_U,	'con1_c', 'con1_dc',... \[\],  
       ConsPattern, c_L,  c_U, x_min,  x_max, f_opt, x_opt);

Result 	= tomRun('conSolve',Prob); PrintResult(Result);

The following example, con2Demo in file conDemo, illustrates numerical estimates of the gradient and constrained Jacobian matrix. Only the statements different from the previous example is given. Note that the gradient routine is not given at all, but the constraint Jacobian routine is given. Setting Prob.ConsDiff greater than zero will overrule the use of the constraint Jacobian routine. The solver conSolve is in this case called directly.

%   Generate the  problem  structure using  the  TOMLAB  format
Prob = conAssign('con1_f', \[\], \[\], HessPattern,	x_L,  x_U, Name, x_0, ...
                 pSepFunc, fLowBnd, A,  b_L,  b_U, 'con1_c', 'con1_dc', \[\], ...
                 ConsPattern, c_L,  c_U, x_min,  x_max, f_opt, x_opt);

Prob.NumDiff  = 1;      %   Use standard  numerical differences
Prob.ConsDiff = 5;      %   Use the  complex variable method to estimate derivatives

Prob.Solver.Alg = 0;    %   Use default algorithm in conSolve

Result    = tomRun('conSolve', Prob,  1);

The third example, con3Demo in file conDemo, shows how to solve the same problem for a number of different initial values on x. The initial values are stored in the matrix X 0, and in each loop step Prob.x 0 is set to one of the columns in X 0. In a similar way any of the values in the Prob structure may be changed in a loop step, if e.g. the loop is part of a control loop. The Prob structure only needs to be defined once. The different initial points reveal that this problem is nasty, and that several points fulfill the convergence criteria. Only the statements different from the previous example is given. A different solver is called dependent on which TOMLAB version is used.

X0              = [ -1  -5    1   0  -5 ;
                     1	 7   -1   0   5];

%   Generate the  problem  structure using  the  TOMLAB  format
Prob = conAssign('con1_f',  'con1_g',  'con1_H', HessPattern, x_L,  x_U, Name, ...
                 X0(:,1), pSepFunc, fLowBnd, A,  b_L,  b_U, 'con1_c', 'con1_dc',...
                 [], ConsPattern, c_L,  c_U, x_min,  x_max, f_opt, x_opt);

Prob.Solver.Alg = 0;
TomV            = tomlabVersion;

for i = 1:size(X0,2)
   Prob.x_0  = X0(:,i);

   if TomV(1:1) ~= 'M'
      % Users of v3.0 may instead call MINOS (or SNOPT, NPSOL in v3.0 /SOL)    
      Result 	= tomRun('minos',Prob, 2);
   else
      Result 	= tomRun('conSolve',Prob, 2);
   end
end

The constrained optimization solvers all have proven global convergence to a local minimum. If the problem is not convex, then it is always difficult to assure that a global minimum has been reached. One way to make it more likely that the global minimum is found is to optimize very many times with different initial values. The fifth example, con5Demo in file conDemo illustrates this approach by solving the exponential problem 50 times with randomly generated initial points.

If the number of variables are not that many, say fifteen, another approach is to use a global optimization solver like glcSolve to crunch the problem and search for the global minimum. If letting it run long enough, it is very likely to find the global minimum, but maybe not with high precision. To run glcSolve the problem must be box-bounded, and the advise is to try to squeeze the size of the box down as much as possible. The sixth example, con6Demo in file conDemo, illustrates a call to glcSolve. It is very simple to do this call if the problem has been defined in the TOMLAB format. The statements needed are the following

Prob.optParam.MaxFunc = 5000; % Define  maximal number of  function evaluations
Result                = tomRun('glcSolve',Prob,2);

A more clever approach, using warm starts and successive checks on the best function value obtained, is discussed in Section 7. It is also better to use glcAssign and not conAssign if the intension is to use global optimization.

Efficient use of the TOMLAB solvers

To follow the iterations in the TOMLAB Base Module solvers, it is useful to set the IterPrint parameter as true. This gives one line of information for each iteration. This parameter is part of the optParam subfield:

Prob.optParam.IterPrint = 1;

Note that ucSolve implements a whole set of methods for unconstrained optimization. If the user explicitly wants Newtons method to be used, utilizing second order information, then set

Prob.Solver.Alg=1;                    %   Use Newtons Method

But ucSolve will switch to the default BFGS method if no routine has been given to return the Hessian matrix. If the user still wants to run Newtons method, then the Hessian routine must be defined and return an empty Hessian. That triggers a numerical estimation of the Hessian. Do help ucSolve to see the different algorithmic options and other comments on how to run the solver.

Both ucSolve and conSolve use line search based methods. The parameter s influences the accuracy of the line search each step. The default value is

Prob.LineParam.sigma  = 0.9;    %   Inaccurate line  search

However, using the conjugate gradient methods in ucSolve, they benefit from a more accurate line search

Prob.LineParam.sigma  = 0.01;    %   Default accurate line  search  for C-G methods

as do quasi-Newton DFP methods (default s = 0.2). The test for the last two cases are made for s = 0.9. If the user really wishes these methods to use s = 0.9, the value must be set slightly different to fool the test:

Prob.LineParam.sigma  = 0.9001;  %   Avoid  the  default value  for C-G methods

The choice of line search interpolation method is also important, a cubic or quadratic interpolation. The default is to use cubic interpolation.

Prob.LineParam.LineAlg = 1;  %   0 = quadratic, 1 = cubic

Solving Global Optimization Problems

Global Optimization deals with optimization problems that might have more than one local minimum. To find the global minimum out of a set of local minimum demands other types of methods than for the problem of finding local minimum. The TOMLAB routines for global optimization are based on using only function or constraint values, and no derivative information. Two different types are defined, Box-bounded global optimization glb and global mixed-integer nonlinear programming glc. For the second case, still the problem should be box-bounded.

All demonstration examples that are using the TOMLAB format are collected in the directory examples. Running the menu program tomMenu, it is possible to run all demonstration examples. It is also possible to run each example separately. The examples relevant to this section are glbDemo and glcDemo.

Box-Bounded Global Optimization Problems

Box-bounded global optimization problems are simple to define, only one function routine is needed, because the global optimization routines in TOMLAB does not utilize information about derivatives. To define the Shekel 5 test problem in a routine glb1_f, the following statements are needed

function f = glb1_f(x,  Prob)

A  = [4 4 4 4;  1 1 1 1;  8 8 8 8;  6 6 6 6;  3 7 3 7]';
f=0;    c = [.1 .2  .2  .4  .4]';

for i = 1:5
   z = x-A(:,i);
   f = f - 1/(z'\*z + c(i) ); % Shekel  5
end

To solve the Shekel 5 test problem define the following statements, available as glb1Demo in glbDemo.

function glb1Demo

Name       = 'Shekel 5';
x_L	= [ 0  0  0  0]';     % Lower bounds in the  box 
x_U	= [10 10 10 10]'; % Upper bounds in the  box

% Generate the  problem  structure using  the  TOMLAB  format  (short call) 
Prob	= glcAssign('glb1_f', x_L,  x_U, Name);
Result  = tomRun('glbSolve', Prob,  1); % Solve  using  the  default of  200 iterations

If the user knows the optimal function value or some good approximation, it could be set as a target for the optimization, and the solver will stop if the target value is achieved within a relative tolerance. For the Shekel 5 problem, the optimal function value is known and could be set as target value with the following statements.

Prob.optParam.fGoal = -10.1532; %   The optimal value  set  as target
Prob.optParam.eps_f =  0.01;    %   Convergence tolerance one percent

Convergence will occur if the function value sampled is within one percent of the optimal function value.

Without additional knowledge about the problem, like the function value at the optimum, there is no convergence criteria to be used. The global optimization routines continues to sample points until the maximal number of function evaluations or the maximum number of iteration cycles are reached. In practice, it is therefore important to be able to do warm starts, starting once again without having to recompute the past history of iterations and function evaluations. Before doing a new warm start, the user can evaluate the results and determine if to continue or not. If the best function value has not changed for long it is a good chance that there are no better function value to be found.

In TOMLAB warm starts are automatically handled, the only thing the user needs to do is to set one flag, Prob.WarmStart, as true. The solver glbSolve defines a binary mat-file called glbSave.mat to store the information needed for a warm start. It is important to avoid running other problems with this solver when doing warm starts. The warm start information would then be overwritten. The example glb3Demo in glbDemo shows how to do warm starts. The number of iterations per call is set very low to be able to follow the process.

Name      = 'Shekel 5';
x_L       = [ 0   0   0   0]';
x_U       = [10  10  10  10]';

%   Generate the  problem  structure using  the  TOMLAB  format  (short call) 
Prob	= glcAssign('glb1_f', x_L,  x_U, Name); 
Prob.optParam.MaxIter = 5;  % Do only  five iterations per  call

Result 	= tomRun('glbSolve',Prob,2);  
pause(1) 
Prob.WarmStart 	= 1;  %   Set  the  flag for warm  start

for i = 1:6   %  Do   6 warm  starts
   Result  = tomRun('glbSolve',Prob,2);  pause(1)
end

The example glb4Demo in glbDemo illustrates how to send parameter values down to the function routine from the calling routine. Change the Shekel 5 test problem definition so that A and c are given as input to the function routine

function f = glb4_f(x,  Prob)

%  A   and c info are  sent  using  Prob structure 
f = 0;  A  = Prob.user.A; c = Prob.user.c;
for i = 1:5
   z = x-A(:,i);
   f = f - 1/(z'\*z + c(i) ); 	%   Shekel  5
end

Then the following statements solve the ''Shekel 5 ''test problem.

Name	= 'Shekel 5';
x_L	= [0  0  0  0]';
x_U	= [10 10 10 10]';

%   Generate the  problem  structure using  the  TOMLAB  format  (short call) 
Prob	= glcAssign('glb4_f', x_L,  x_U, Name);

%   Add information to  be sent  to  glb4_f. Used in f(x) computation 
Prob.user.A = [4  4 4 4;1  1 1 1;8  8 8 8;6  6 6 6;3  7 3 7]'; 
Prob.user.c = [.1 .2  .2  .4  .4]';

Result  = tomRun('glbSolve',Prob,2);

Global Mixed-Integer Nonlinear Problems

To solve global mixed-integer nonlinear programming problems with the TOMLAB format, only two routines need to be defined, one routine that defines the function and one that defines the constraint vector. No derivative information is utilized by the TOMLAB solvers. To define the Floudas-Pardalos 3.3 test problem, one routine glc1 f

function f = fp3_3f(x,  Prob)
f = -25*(x(1)-2)^2-(x(2)-2)^2-(x(3)-1)^2-(x(4)-4)^2-(x(5)-1)^2-(x(6)-4)^2;

and one routine glc1 c

function c = fp3_3c(x,  Prob)
c = [(x(3)-3)^2+x(4); (x(5)-3)^2+x(6)]; % Two nonlinear constraints (QP)

is needed. Below is the example glc1Demo in glcDemo that shows how to solve this problem doing ten warm starts. The warm starts are automatically handled, the only thing the user needs to do is to set one flag as true, Prob.WarmStart. The solver glcSolve defines a binary mat-file called glcSave.mat to store the information needed for the warm start. It is important to avoid running other problems with glcSolve when doing warm starts. Otherwise the warm start information will be overwritten with the new problem. The original Floudas-Pardalos 3.3 test problem, has no upper bounds on x1 and x2 , but such bounds are implicit from the third linear constraint, x1 + x2 = 6. This constraint, together with the simple bounds x1 = 0 and x2 = 0 immediately leads to x1 = 6 and x2 = 6. Using these inequalities a finite box-bounded problem can be defined.

Name	= 'Floudas-Pardalos 3.3'; % This  example is number 16 in  glc_prob.m 

x_L	= [ 0  0 1 0 1 0]';   % Lower bounds on x
A       = [ 1 -3 0 0 0 0
           -1  1 0 0 0 0
            1  1 0 0 0 0];    % Linear equations
b_L	= [-inf -inf	2 ]'; % Upper bounds for linear equations
b_U	= [  2	  2	6 ]'; % Lower bounds for linear equations
x_U	= [6  6 5 6 5 10]';   % Upper bounds after x(1),x(2) values  inserted 
c_L	= [4  4]';	      % Lower bounds on two nonlinear constraints
c_U	= [];	              % Upper bounds are  infinity for nonlinear constraints 
x_opt   = [5  1 5 0 5 10]';   % Optimal x value
f_opt   = -310; 	      % Optimal f(x) value 
x_min = x_L;  x_max = x_U;    % Plotting bounds

% Set the rest of  the  arguments as empty
IntVars = []; VarWeight = [];

fIP = []; xIP = []; fLowBnd = []; x_0 = [];

%IntVars  = [1:5]; % Indices of  the  variables that should  be integer  valued

Prob = glcAssign('glc1_f', x_L,  x_U, Name, A,  b_L,  b_U, 'glc1_c', ... 
          c_L,  c_U, x_0,  IntVars, VarWeight,  ...
          fIP, xIP, fLowBnd, x_min,  x_max, f_opt, x_opt);

% Increase the default max number of function evaluations in glcSolve
Prob.optParam.MaxFunc = 500;

Result  = tomRun('glcSolve', Prob,  3); 

Prob.WarmStart  = 1;
% Do 10 restarts, call driver tomRun, PriLev  = 2 gives call to PrintResult 
for i=1:10
   Result  = tomRun('glcSolve',Prob,2);
end

Solving Least Squares and Parameter Estimation Problems

This section describes how to define and solve different types of linear and nonlinear least squares and parameter estimation problems. Several examples are given on how to proceed, depending on if a quick solution is wanted, or more advanced tests are needed. TOMLAB is also compatible with MathWorks Optimization TB. See Appendix E for more information and test examples.

All demonstration examples that are using the TOMLAB format are collected in the directory examples. The examples relevant to this section are lsDemo and llsDemo. The full path to these files are always given in the text.

Section 8.5 (page 81) contains information on solving extreme large-scale ls problems with Tlsqr.

Linear Least Squares Problems

This section shows examples how to define and solve linear least squares problems using the TOMLAB format. As a first illustration, the example lls1Demo in file llsDemo shows how to fit a linear least squares model with linear constraints to given data. This test problem is taken from the Users Guide of LSSOL \[29\].

Name='LSSOL  test  example';

%  In TOMLAB it is best to use Inf and -Inf, not  big  numbers. 
n = 9; 	% Number  of  unknown  parameters
x_L = [-2 -2  -Inf, -2*ones(1,6)]';
x_U = 2*ones(n,1);

A       = [ ones(1,8) 4;  1:4,-2,1 1 1 1;  1 -1  1 -1, ones(1,5)];
b_L = [2     -Inf  -4]';
b_U = [Inf     -2  -2]';

y = ones(10,1);
C  = \[ ones(1,n); 1 2 1 1 1 1 2 0 0; 1 1 3 1 1 1 -1  -1  -3; ...
1 1 1 4 1 1 1 1 1;1 1 1 3 1 1 1 1 1;1 1 2 1 1 0 0 0 -1; ...
1 1 1 1 0 1 1 1 1;1 1 1 0 1 1 1 1 1;1 1 0 1 1 1 2 2 3;  ...
1 0 1 1 1 1 0 2 2\];

x_0 = 1./[1:n]';

t           = [];   %  No time set for y(t) (used  for plotting)
weightY     = [];   %  No weighting 
weightType  = [];   %  No weighting type  set

x_min	    = [];   %  No lower bound for plotting 
x_max	    = [];   %  No upper bound for plotting

Prob = llsAssign(C, y, x_L,  x_U, Name, x_0,  t, weightType,   weightY, ...
                 A,  b_L,  b_U,	x_min,  x_max); Result 	= tomRun('lsei',Prob,2);

It is trivial to change the solver in the call to tomRun to a nonlinear least squares solver, e.g. clsSolve, or a general nonlinear programming solver.

Linear Least Squares Problems using the SOL Solver LSSOL

The example lls2Demo in file llsDemo shows how to fit a linear least squares model with linear constraints to given data using a direct call to the SOL solver LSSOL. The test problem is taken from the Users Guide of LSSOL \[29\].

%   Note that when calling the  LSSOL   MEX  interface directly,  avoid  using
%   Inf and -Inf. Instead use big  numbers that indicate Inf.
%   The standard  for the  MEX  interfaces is 1E20 and -1E20,  respectively.

n = 9; % There are  nine  unknown  parameters, and 10 equations 
x_L = [-2 -2  -1E20,  -2*ones(1,6)]';
x_U = 2*ones(n,1);

A   = [ ones(1,8) 4;  1:4,-2,1 1 1 1;  1 -1  1 -1, ones(1,5)];
b_L = [2 	-1E20 -4]';
b_U = [1E20	   -2 -2]';
% Must put lower and upper bounds on variables and constraints together 
bl = \[x_L;b_L];
bu = \[x_U;b_U\];

H   = [ ones(1,n); 1 2 1 1 1 1 2 0 0;  1 1 3 1 1 1 -1  -1  -3; ...
        1 1 1 4 1 1 1 1 1;1  1 1 3 1 1 1 1 1;1  1 2 1 1 0 0 0 -1; ...
        1 1 1 1 0 1 1 1 1;1  1 1 0 1 1 1 1 1;1  1 0 1 1 1 2 2 3;  ...
        1 0 1 1 1 1 0 2 2];
y   = ones(10,1);

x_0 = 1./[1:n]';

% Set  empty indicating default values  for most variables
c         = [];	          % No linear coefficients, they are for LP/QP 
Warm	  = [];	          % No  warm  start
iState	  = [];	          % No  warm  start
Upper	  = [];	          % C   is not  factorized 
kx 	  = [];	          % No  warm  start
SpecsFile = [];	          % No   parameter  settings in a SPECS  file 
PriLev 	  = [];	          % PriLev  is not  really used in LSSOL 
ProbName  = [];           % ProbName  is not  really used in LSSOL 
optPar(1) = 50;           % Set  print level at  maximum
PrintFile  = 'lssol.txt'; % Print result on the  file with  name  lssol.txt

z0 = (y-H*x_0);
f0  = 0.5*z0'*z0;
fprintf('Initial function value  %f\n',f0);

[x,  Inform, iState, cLamda, Iter, fObj, r, kx]  = ... 
     lssol( A,  bl, bu,  c, x_0,  optPar, H,  y, Warm, ...
     iState, Upper,  kx,  SpecsFile, PrintFile,  PriLev, ProbName );

%  We could equally well call with the following shorter call:
%  [x,  Inform, iState, cLamda, Iter, fObj, r, kx]  = ...
%       lssol( A,  bl, bu,  c, x, optPar, H,  y);

z = (y-H*x);
f = 0.5*z'*z;
fprintf('Optimal  function value  %f\n',f);

Nonlinear Least Squares Problems

This section shows examples how to define and solve nonlinear least squares problems using the TOMLAB format. As a first illustration, the example ls1Demo in file lsDemo shows how to fit a nonlinear model of exponential type with three unknown parameters to experimental data. This problem, Gisela, is also defined as problem three in ls prob. A weighting parameter K is sent to the residual and Jacobian routine using the Prob structure. The solver clsSolve is called directly. Note that the user only defines the routine to compute the residual vector and the Jacobian matrix of derivatives. TOMLAB has special routines ls f, ls g and ls H that computes the nonlinear least squares objective function value, given the residuals, as well as the gradient and the approximative Hessian, see Table 39. The residual routine for this problem is defined in file ls1 r in the directory example with the statements

function r = ls_r(x, Prob)

% Compute  residuals to  nonlinear least squares  problem  Gisela

% US_A is the  standard  TOMLAB  global parameter  to  be used in the
% communication  between the  residual and the  Jacobian  routine 

global US_A

% The extra weight parameter  K  is sent as part of the structure
K = Prob.user.K;
t = Prob.LS.t(:); % Pick  up the  time  points

% Exponential  expressions to be later used when computing the Jacobian
US_A.e1  = exp(-x(1)\*t); US_A.e2  = exp(-x(2)\*t);

r = K\*x(1)\*(US_A.e2  - US_A.e1) / (x(3)\*(x(1)-x(2))) - Prob.LS.y;

Note that this example also shows how to communicate information between the residual and the Jacobian routine. It is best to use any of the predefined global variables US A and US B, because then there will be no conflicts with respect to global variables if recursive calls are used. In this example the global variable US A is used as structure array storing two vectors with exponential expressions. The Jacobian routine for this problem is defined in the file ls1 J in the directory example. The global variable US A is accessed to obtain the exponential expressions, see the statements below.

function J = ls1_J(x,  Prob)

% Computes  the  Jacobian  to  least squares  problem  Gisela. J(i,j) is dr_i/d_x_j
% Parameter K  is input in the  structure 
Prob a = Prob.user.K * x(1)/(x(3)*(x(1)-x(2)));
b      = x(1)-x(2);
t      = Prob.LS.t;

% Pick up the globally saved exponential computations 
global US_A
e1 = US_A.e1; e2 = US_A.e2;

% Compute  the  three columns in the Jacobian, one for each of variable
J = a * [ t.*e1+(e2-e1)*(1-1/b), -t.*e2+(e2-e1)/b, (e1-e2)/x(3) ];

The following statements solve the ''Gisela ''problem.
%   --------------------------------------------------------------------- 
function ls1Demo - Nonlinear parameter  estimation with  3 unknowns
%   --------------------------------------------------------------------- 

Name='Gisela';

% Time values
t = [0.25; 0.5; 0.75; 1;  1.5; 2;  3;  4;  6;  8;  12;  24;  32;  48;  54;  72;  80;...
     96;  121;  144;  168;  192;  216;  246;  276;  324;  348;  386];

%   Observations
y = [30.5; 44;  43;  41.5; 38.6; 38.6; 39;  41;  37;  37;  24;  32;  29;  23;  21;...
      19;  17;  14;  9.5; 8.5; 7;  6;  6;  4.5; 3.6; 3;  2.2; 1.6];

x_0 = \[6.8729,0.0108,0.1248\]'; %   Initial values  for unknown  x

%   Generate the  problem  structure using  the  TOMLAB  format  (short call)
%   Prob = clsAssign(r, J, JacPattern, x_L,  x_U, Name, x_0,  ...
%	             y, t, weightType,   weightY, SepAlg,  fLowBnd, ...
%	             A,  b_L,  b_U, c, dc,  ConsPattern, c_L,  c_U, ...
%	             x_min,  x_max, f_opt, x_opt);

Prob = clsAssign('ls1_r', 'ls1_J', \[\], \[\], \[\], Name, x_0,  y, t);

% Weighting  parameter  K  in model is sent  to  r and J computation  using  Prob
Prob.user.K = 5;

Result 	= tomRun('clsSolve', Prob,  2);

The second example ls2Demo in file lsDemo solves the same problem as ls1Demo, but using numerical differences to compute the Jacobian matrix in each iteration. To make TOMLAB avoid using the Jacobian routine, the variable Prob.NumDiff has to be set nonzero. Also in this example the flag Prob.optParam.IterPrint is set to enable one line of printing for each iteration. The changed statements are

...
Prob.NumDiff            = 1; % Use standard  numerical differences
Prob.optParam.IterPrint = 1; % Print one line each iteration

Result    = tomRun('clsSolve',Prob,2);

The third example ls3Demo in file lsDemo solves the same problem as ls1Demo, but six times for different values of the parameter K in the range \[3.8, 5.0\]. It illustrates that it is not necessary to remake the problem structure Prob for each optimization, but instead just change the parameters needed. The Result structure is saved as an vector of structure arrays, to enable post analysis of the results. The changed statements are

for i=1:6
   Prob.user.K = 3.8  + 0.2\*i;
   Result(i)	= tomRun('clsSolve',Prob,2);
end

fprintf('\nWEIGHT PARAMETER  K  is %9.3f\n\n\n',Prob.user.K);

Table 39 describes the low level routines and the initialization routines needed for the predefined constrained nonlinear least squares (cls) test problems. Similar routines are needed for the nonlinear least squares (ls) test problems (here no constraint routines are needed).

Table 39: Constrained nonlinear least squares (**cls**) test problems.
Function	Description
cls prob	Initialization of cls test problems.
cls r	Compute the residual vector ri (x), i = 1, ..., m. x ? Rn for cls test problems.
cls J	Compute the Jacobian matrix Jij (x) = ?ri /dxj , i = 1, ..., m, j = 1, ..., n for cls test problems.
cls c	Compute the vector of constraint functions c(x) for cls test problems.
cls dc	Compute the matrix of constraint normals ?c(x)/dx for cls test problems.
cls d2c	Compute the second part of the second derivative of the Lagrangian function for cls test problems.
ls_f	General routine to compute the objective function value f (x) = 1 r(x)T r(x) for nonlinear least squares type of problems.
ls_g	General routine to compute the gradient g(x) = J (x)T r(x) for nonlinear least squares type of problems.
ls_H	General routine to compute the Hessian approximation H (x) = J (x)T J* (x) for nonlinear least squares type of problems.

Fitting Sums of Exponentials to Empirical Data

In TOMLAB the problem of fitting sums of positively weighted exponential functions to empirical data may be formulated either as a nonlinear least squares problem or a separable nonlinear least squares problem \[66\]. Several empirical data series are predefined and artificial data series may also be generated. There are five different types of exponential models with special treatment in TOMLAB, shown in Table 40. In research in cooperation with Todd Walton, Vicksburg, USA, TOMLAB has been used to estimate parameters using maximum likelihood in simulated Weibull distributions, and Gumbel and Gamma distributions with real data. TOMLAB has also been useful for parameter estimation in stochastic hydrology using real-life data.

Table 40: Exponential models treated in TOMLAB.
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p \alpha_i e^{-\beta_i t}$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... <\beta_p$} .
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p \alpha_i e^{-\beta_i t}$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... <\beta_p$} .
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p \alpha_i(1-e^{-\beta_i t})$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... <\beta_p$} .
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p t \alpha_i e^{-\beta_i t}$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... <\beta_p$} .
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p (t \alpha_i-\gamma_i) e^{-\beta_i t}$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i,\gamma_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... >\beta_p$} .
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f(t) = \sum\limits_i^p t \alpha_i e^{-\beta_i (t - \gamma_i)}$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\alpha_i \geq 0$} ,	Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $0\leq\beta_1<\beta_2< ... <\beta_p$} .

Algorithms to find starting values for different number of exponential terms are implemented. Test results show that these initial value algorithms are very close to the true solution for equidistant problems and fairly good for non-equidistant problems, see the thesis by Petersson \[61\]. Good initial values are extremely important when solving real life exponential fitting problems, because they are so ill-conditioned. Table 41 shows the relevant routines.

TOMLAB: Difference between revisions

Revision as of 17:47, 16 June 2011

Introduction

What is TOMLAB?

The Organization of This Guide

Further Reading

Overall Design

Structure Input and Output

Introduction to Solver and Problem Types

The Process of Solving Optimization Problems

Low Level Routines and Gateway Routines

Problem Types and Solver Routines

Problem Types Defined in TOMLAB

Solver Routines in TOMLAB

TOMLAB Base Module

TOMLAB /BARNLP

TOMLAB /CGO

TOMLAB /CONOPT

TOMLAB /CPLEX

TOMLAB /KNITRO

TOMLAB /LGO

TOMLAB /MINLP

TOMLAB /MINOS

TOMLAB /OQNLP

TOMLAB /NLPQL

TOMLAB /NPSOL

TOMLAB /PENBMI

TOMLAB /PENSDP

TOMLAB /SNOPT

TOMLAB /SOL

TOMLAB /SPRNLP

TOMLAB /XA

3.2.19 TOMLAB /Xpress

3.2.20 Finding Available Solvers

Defining Problems in TOMLAB

The TOMLAB Format

Modifying existing problems

add_A

keep_A

remove A

replace A

modify b_L

modify b_U

modify c

modify c_L

modify c_U

modify x_0

modify x_L

modify x_U

TomSym

Modeling

Ezsolve

Usage

Scaling variables

SDP/LMI/BMI interface

Interface to MAD and finite differences

Simplifications

Special functions

Procedure vs parse-tree

Problems and error messages

Example

Solving Linear, Quadratic and Integer Programming Problems

Linear Programming Problems

A Quick Linear Programming Solution

Quadratic Programming Problems

A Quick Quadratic Programming solution

Mixed-Integer Programming Problems

Solving Unconstrained and Constrained Optimization Problems

Defining the Problem in Matlab m-files

Communication between user routines

Unconstrained Optimization Problems

Direct Call to an Optimization Routine

Constrained Optimization Problems

Efficient use of the TOMLAB solvers

Solving Global Optimization Problems

Box-Bounded Global Optimization Problems

Global Mixed-Integer Nonlinear Problems

Solving Least Squares and Parameter Estimation Problems

Linear Least Squares Problems

Linear Least Squares Problems using the SOL Solver LSSOL