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==Introduction==
[[Category:Manuals]]
 
===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  is a general purpose development, modeling and optimal control environment in Matlab for research, teaching and practical solution of optimization problems.


Line 13: Line 10:
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.
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===
==Overall Design==
 
Overall Design presents the general design of TOMLAB.
'''[[#Overall_Design|Section 2]]''' presents the general design of TOMLAB.
*[[TOMLAB  Overall Design|Overall Design]]
 
'''[[#Problem_Types_and_Solver_Routines|Section 3]]''' contains strict mathematical definitions of the optimization problem types.  All solver routines available in TOMLAB  are described.
 
'''[[#Defining_Problems_in_TOMLAB|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.
 
'''[[#Solving_Linear.2C_Quadratic_and_Integer_Programming_Problems|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.
 
===Further  Reading===
 
TOMLAB  has been discussed in several papers and at several conferences. The main paper on TOMLAB  v1.0 is \[42\].
 
The use of TOMLAB  for nonlinear programming and parameter estimation is presented in \[45\], and the use of linear and discrete optimization is discussed in \[46\]. Global optimization routines are also implemented,  one is described in \[8\].
 
In all these papers TOMLAB  was divided into two toolboxes, the NLPLIB TB and the OPERA TB. TOMLAB  v2.0 was discussed in \[43\], \[40\].  and \[41\].  TOMLAB  v4.0 and how to solve practical optimization  problems  with TOMLAB  is discussed in \[44\].
 
The use of TOMLAB  for costly global optimization with industrial applications is discussed in \[9\]; costly global optimization with financial applications in \[37, 38, 39\]. Applications of global optimization for robust control is presented in \[25, 26\]. The use of TOMLAB  for exponential fitting and nonlinear parameter estimation are discussed in e.g. \[49, 4, 22, 23, 47, 48\].
 
The manuals for the add-on solver packages are also recommended reading material.
 
==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.m''which 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'''<hr />||||Type of optimization problem<hr />
 
|-
 
|'''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'''<hr />||'''probType'''<hr />||'''Description  of test problem set'''<hr />
 
|-
 
|'''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<hr/>||Purpose '''( ''o ''is any ''probSet'', e.g. ''o''='''con''')<hr/>
 
|-
 
|''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'')''/dx''2 .
 
|-
 
|''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'')''/dx''2
 
|-
 
|}
 
 
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 ''(''x''1 '', x''2 '', . . .''), 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.
 
<pre>
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);
</pre>
 
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
 
<pre>
ucSolve <==> nlp_f <==> ls_f <==> nlp_r <==> mgh_r
</pre>
 
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''.
 
<pre>
ucSolve <==> nlp_g <==> fdng <==> nlp_r <==> uc_f
</pre>
 
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:
 
<pre>
Prob.user.a=a;
Prob.user.B=B;
Prob.user.C=C;
</pre>
 
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:
 
<pre>
a = Prob.user.a;
B = Prob.user.B;
C = Prob.user.C;
</pre>
 
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'''<hr />||'''Description'''<hr/>
 
|-
 
|''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'')''/dx''2 for '''con ''' test problems.
 
|-
 
|''con_fm''||Compute merit function ''?''(''xk '').
 
|-
 
|''con_gm''||Compute gradient of merit function ''?''(''xk '').
 
|}
 


==Problem Types and Solver Routines==
==Problem Types and Solver Routines==
Contains strict mathematical definitions of the optimization problem types.  All solver routines available in TOMLAB  are described.
*[[TOMLAB  Problem Types and Solver Routines|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.
==Defining Problems in TOMLAB==
 
*[[TOMLAB  Defining Problems in TOMLAB|Defining Problems in TOMLAB]]
===Problem Types Defined in TOMLAB===
 
The '''unconstrained optimization '''('''uc''') problem is defined as
 
 
<math>
\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}
</math>
 
 
where <math>$x, x_L, x_U \in \MATHSET{R}^n$</math> and <math>$f(x) \in \MATHSET{R}$</math> . For unbounded variables, the corresponding elements of <math>$x_L,x_U$</math> may be set to <math>$\pm \infty$</math>.
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$F \in \MATHSET{R}^{n \times n}$</math>, <math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, and <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>.
 
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
 
 
<math>
\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}
</math>
 
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$f(x) \in \MATHSET{R}$</math>, 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
 
 
<math>
\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}
</math>
 
 
where <math>$x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$f(x) \in \MATHSET{R}$</math>, <math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math> and <math>$c_L,c(x),c_U \in \MATHSET{R}^{m_2}$</math>.
The variables <math>$x \in I$</math>, the index subset of <math>$1,...,n$</math>, 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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>,
<math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, and <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>.
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$A \in \MATHSET{R}^{m_1
\times n}$</math>, and <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>. The variables <math>$x
\in I$</math>, the index subset of <math>$1,...,n$</math> are restricted to be
integers. Equality constraints are defined by setting the lower
bound equal to the upper bound, i.e. for constraint <math>$i$: $b_{L}(i)
=  b_{U}(i)$</math>.
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$d \in \MATHSET{R}^M$</math>,
<math>$C \in \MATHSET{R}^{M \times n}$</math>,
<math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math> and <math>$c_L,c(x),c_U \in \MATHSET{R}^{m_2}$</math>.
 
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$r(x) \in \MATHSET{R}^M$</math>,
<math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>
and <math>$c_L,c(x),c_U \in \MATHSET{R}^{m_2}$</math>.
 
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$A \in \MATHSET{R}^{m_l\times n}$</math>, <math>$b_L,b_U \in \MATHSET{R}^{m_l}$</math> and <math>$Q^{i}_{k}$</math> are symmetric matrices of similar dimensions in each constraint <math>$i$</math>. 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
 
 
<math>
\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 \\
</math>
 
 
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'''<hr />||'''Description'''<hr />||'''Section'''<hr />||'''Page'''<hr />
 
|- valign="top"
 
|''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
 
|- valign="top"
 
|''conSolve''||Constrained nonlinear minimization solver using two different sequential quadratic programming methods.||11.1.2||PAGE
 
|- valign="top"
 
|''cutplane''||Mixed-integer programming using a cutting plane algorithm.||11.1.3||PAGE
 
|- valign="top"
 
|''DualSolve||Solves a linear program with a dual feasible starting point.||11.1.4||PAGE
 
|- valign="top"
 
|''glbSolve||Box-bounded global optimization, using only function values.||11.1.7||PAGE
 
|- valign="top"
 
|''glcCluster||Hybrid algorithm for constrained mixed-integer global optimization. Uses a combination of ''glcFast ''(DIRECT) and a clustering algorithm.||11.1.7||PAGE
 
|- valign="top"
 
|''glcSolve||Global mixed-integer nonlinear programming, using no derivatives.||11.1.7||PAGE
 
|- valign="top"
 
|''infSolve||Constrained minimax optimization. Reformulates problem and calls any suitable nonlinear solver.||11.1.8||PAGE
 
|- valign="top"
 
|''lpSimplex||Linear programming using a simplex algorithm.||11.1.14||PAGE
 
|- valign="top"
 
|''MILPSOLVE''||Mixed-integer programming using branch and bound.||11.1.16||PAGE
|- valign="top"
 
|''L1Solve''||Constrained L1 optimization. Reformulates problem and calls any suitable nonlinear solver.||11.1.15||PAGE
 
|- valign="top"
 
|''mipSolve''||Mixed-integer programming using a branch-and-bound algorithm.||11.1.18||PAGE
 
|- valign="top"
 
|''nlpSolve''||Constrained nonlinear minimization solver using a filter SQP algorithm.||SECTION||PAGE
 
|- valign="top"
 
|''pdco''||Linearly constrained nonlinear minimization solver using a primal- dual barrier algorithm.
 
|- valign="top"
 
|''pdsco''||Linearly constrained nonlinear minimization solver using a primal- dual barrier algorithm, for separable functions.||SECTION||PAGE
 
|- valign="top"
 
|''qpSolve''||Non-convex quadratic programming.||11.1.24||PAGE
 
|- valign="top"
 
|''slsSolve''||Sparse constrained nonlinear least squares. Reformulates problem and calls any suitable sparse nonlinear solver.
 
|- valign="top"
 
|''sTrustr''||Constrained convex optimization of partially separable functions, using a structural trust region algorithm.
 
|- valign="top"
 
|''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'''<hr />||'''Description'''<hr />||'''Reference'''<hr />||'''Page'''<hr />
 
|-
 
|''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/ 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/ 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/ 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/ 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/ http://tomopt. ] [http://tomopt.com/ 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/ 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/ http:]
 
[http://tomopt.com/ //tomopt.com].
 
 
{|
 
|+Table 8: Solver routines in TOMLAB  /MINLP.
 
|-
 
|'''Function'''<hr />||'''Description'''<hr />||'''Reference'''<hr />
 
|-
 
|''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'''<hr />||'''Description'''<hr />||'''Reference'''<hr />||'''Page'''<hr />
 
|-
|''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/ 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/ http://tomopt. ] [http://tomopt.com/ 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/ http://tomopt.com].
 
{|
 
|+Table 10: Additional  solvers in TOMLAB  /PENSDP.
 
|-
 
|'''Function'''<hr />||'''Description'''<hr />||'''Reference<hr />||'''Page'''<hr />
 
|-
 
|''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/ http://tomopt.com].
 
 
{|
 
|+Table 11: Additional  solvers in TOMLAB  /PENSDP.
 
|-
 
|'''Function'''<hr />||'''Description'''<hr />||'''Reference'''<hr />||'''Page'''<hr />
 
|-
 
|''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/ 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/ http://tomopt.com].
 
{|
|+Table 12: The optimization solvers in the TOMLAB /SOL toolbox.
 
|-
 
|'''Function'''<hr />||'''Description'''<hr />||'''Reference'''<hr />||'''Page'''<hr />
 
|-
 
|''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/ 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/ 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
 
<pre>
SolverList('con')
</pre>
 
gives the output
 
<pre>
>> 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
</pre>
 
''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'').
{|
 
|-
 
| || colspan="10" align="center" border="2" | solvType
 
|-
 
|probType||'''uc'''||'''qp'''||'''con'''||'''ls'''||'''lls'''||'''cls'''||'''mip'''||'''lp'''||'''glb'''||'''glc'''
 
|-
 
| colspan="11" | <hr />
 
|-
 
|'''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
 
|-
 
| colspan="11" | <hr />
 
|-
 
|'''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:
 
<pre>
Prob = ProbCheck(Prob,  'conSolve');
Result  = conSolve(Prob);
</pre>
 
or a call to the multi-solver driver routine ''tomRun'', e.g. for constrained optimization:
 
<pre>
Result  = tomRun('conSolve', Prob);
</pre>
 
Note that ''tomRun ''handles several input formats. Step 3 could be a call to PrintResult.m:
 
<pre>
PrintResult(Result);
</pre>
 
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
 
<pre>
Result  = tomRun('conSolve',Prob, 3);
</pre>
 
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.
 
|-
 
| colspan="3" | <hr />
 
|-
 
|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'''
 
''Prob''Existing TOMLAB  problem.
 
''idx''The 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/ 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.
==Solving Linear, Quadratic and Integer Programming Problems==
Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.
*[[TOMLAB  Solving Linear Quadratic and Integer Programming Problems|Solving Linear, Quadratic and Integer Programming Problems]]


*Support for if, then, else statements.
==Solving Unconstrained and Constrained Optimization Problems==
Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.
*[[TOMLAB  Solving Unconstrained and Constrained Optimization Problems|Solving Unconstrained and Constrained Optimization Problems]]


*Automated code simplification for generated models.
==Solving Global Optimization Problems==
Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.
*[[TOMLAB Solving Global Optimization Problems|Solving Global Optimization Problems]]


*Ability to analyze most p-coded files (if code is vectorized).
==Solving Least Squares and Parameter Estimation Problems==
Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.
*[[TOMLAB Solving Least Squares and Parameter Estimation Problems|Solving Least Squares and Parameter Estimation Problems]]


====Modeling====
==Multi Layer Optimization==
Shows how to setup and define multi layer optimization problems in TOMLAB.
*[[TOMLAB Multi Layer Optimization|Multi Layer Optimization]]


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.
==Tomhelp - The Help Program==
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.  
*[[TOMLAB TomHelp|Tomhelp - The Help Program]]


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).
==TOMLAB Solver Reference==


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.
*[[TOMLAB Solver Reference]]


For vectors ''F ''and ''X '', the resulting ''J ''is the well-known Jacobian matrix.
==TOMLAB Utility Functions==
Describes the utility functions that can be used, for example tomRun and SolverList.


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).
*[[TOMLAB Utility Functions]]


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.
==Approximation of Derivatives==
Introduces the different options for derivatives, automatic differentiation.  


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.
*[[TOMLAB Approximation of Derivatives|Approximation of Derivatives]]


Example:
==Special Notes and Features==
Discusses a number of special system features such as partially separable functions and user supplied parameter information for the function computations.
*[[TOMLAB Special Notes and Features|Special Notes and Features]]


<pre>
==Appendix A: Prob - the Input Problem Structure==
>> toms y
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.


>> toms 3x1 x
*[[TOMLAB Appendix A|Appendix A: Prob - the Input Problem Structure]]


>> toms 2x3 A
==Appendix B: Result - the Output Result Structure==
Contains tables describing all elements defined in the output result structure returned from all solvers and driver routines.
*[[TOMLAB Appendix B|Appendix B: Result - the Output Result Structure]]


>> f = (A\*x).^(2\*y)
==Appendix C: TomSym - the Modeling Engine==
*[[TOMLAB Appendix C|Appendix C: TomSym - the Modeling Engine]]


==Appendix D: Global Variables and Recursive Calls==
This section is concerned with the global variables used in TOMLAB and routines for handling important global variables enabling recursive calls of any depth.
*[[TOMLAB Appendix D|Appendix D: Global Variables and Recursive Calls]]


f = tomSym(2x1):  
==Appendix E: External Interfaces==
Describes the available set of interfaces to other optimization software, such as CUTE, AMPL, and The Mathworks' Optimization Toolbox.
*[[TOMLAB Appendix E|Appendix E: External Interfaces]]


  (A\*x).^(2\*y)
==Appendix F: Motivation and Background to TOMLAB==
Gives some motivation for the development of TOMLAB.  
*[[TOMLAB Appendix F|Appendix F: Motivation and Background to TOMLAB]]


>> derivative(f,A)
==Appendix G: Performance Tests on Linear Programming Solvers==
*[[TOMLAB Appendix G|Appendix G: Performance Tests on Linear Programming Solvers]]


ans = tomSym(2x6):
==Further  Reading==


  (2\*y)\*setdiag((A\*x).^(2\*y-1))\*kron(x',eye(2))
TOMLAB  has been discussed in several papers and at several conferences. The main paper on TOMLAB  v1.0 is
</pre>
<ref>
K. Holmström.
The TOMLAB Optimization Environment in Matlab.
''Advanced Modeling and Optimization'',
1(1):47-69, 1999.
</ref>
.
The use of TOMLAB  for nonlinear programming and parameter estimation is presented in
<ref>
K. Holmström and M. Björkman.
The TOMLAB NLPLIB Toolbox for Nonlinear Programming.
''Advanced Modeling and Optimization'',
1(1):70-86, 1999.
</ref>
, and the use of linear and discrete optimization is discussed in
<ref>
K. Holmström, M. Björkman, and E. Dotzauer.
The TOMLAB OPERA Toolbox for Linear and Discrete Optimization.
''Advanced Modeling and Optimization'',
1(2):1-8, 1999.
</ref>
. Global optimization routines are also implemented,  one is described in
<ref>
M. Björkman and K. Holmström.
Global Optimization Using the DIRECT Algorithm in Matlab.
''Advanced Modeling and Optimization'',  
1(2):17-37, 1999.
</ref>.


In the above example, the 2x1 symbol ''f ''is differentiated with respect to the 2x3 symbol ''A''. The result is a 2x6 matrix, representing <math>d(vec(f))</math>.
In all these papers TOMLAB  was divided into two toolboxes, the NLPLIB TB and the OPERA TB. TOMLAB  v2.0 was discussed in
<ref>
K. Holmström.
The TOMLAB v2.0 Optimization Environment.
In E. Dotzauer, M. Björkman, and K. Holmstöm, editors, ''Sixth Meeting of the Nordic Section of the Mathematical Programming Society''.
''Proceedings'', Opuscula 49, ISSN 1400-5468, Västerås, 1999. Mälardalen University, Sweden.</ref>
<ref>K. Holmström.
New Optimization Algorithms and Software.
''Theory of Stochastic Processes'',
5(21)(1-2):55-63, 1999.
</ref>
.  and
<ref>
K. Holmström.
Solving applied optimization problems using TOMLAB.
In G. Osipenko, editor, ''Proceedings from MATHTOOLS  <nowiki>'</nowiki>99, the 2nd International Conference on Tools for Mathematical Modelling'',
pages 90-98, St.Petersburg,  Russia, 1999. St.Petersburg State Technical University.
</ref>
.  TOMLAB  v4.0 and how to solve practical optimization  problems  with TOMLAB  is discussed in
<ref>
K. Holmström. 
Practical Optimization  with the Tomlab Environment. 
In T. A. Hauge, B. Lie, R. Ergon, M. D. Diez, G.-O. Kaasa, A. Dale, B. Glemmestad, and A Mjaavatten, editors,
''Proceedings of the 42nd SIMS Conference'', 
pages 89-108, Porsgrunn, Norway, 2001. Telemark University College, Faculty of Technology.</ref>
.  


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.
The use of TOMLAB for costly global optimization with industrial applications is discussed in  
 
<ref>
<pre>
M. Björkman and K. Holmström.   
>> mcodestr(ans)
Global Optimization of Costly Nonconvex Functions Using Radial Basis Functions. ''Optimization and Engineering'',  
ans =
1(4):373-397, 2000.
setdiag((2\*y)\*(A\*x).^(2\*y-1))\*kron(x',\[1 0;0  1\])
</ref>; costly global optimization with financial applications in  
</pre>
<ref>
 
T. Hellström and K. Holmström.  
Observe that the command mcode and not mcodestr should be used when generating efficient production code.
Parameter Tuning in Trading Algorithms using ASTA.  
 
In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, ''Computational Finance (CF99) - Abstracts of the Sixth International Conference, Leonard N. Stern School of Business, January 1999'', Leonard N. Stern School of Business, New York University, 1999. Department of Statistics and Operations Research.
====Ezsolve====
</ref>
 
<ref>
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:
T. Hellström  and K. Holmström.
 
Parameter Tuning in Trading Algorithms using ASTA.
<pre>
In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, ''Computational Finance 1999'', Cambridge, MA, 1999. MIT Press.
toms integer  x
</ref>
toms       y
<ref>
 
T. Hellström and K. Holmström.  
objective = -6\*x  + 2\*x^2  + 2\*y^2  - 2\*x\*y;
Global Optimization of Costly Nonconvex Functions, with Financial Applications.  
constraints = \{x+y<=1.9,x>=0, y>=0\};
''Theory of Stochastic Processes'',  
 
7(23)(1-2):121-141, 2001.
solution = ezsolve(objective,constraints)
</ref>. Applications of global optimization for robust control is presented in
</pre>
<ref>
 
C. M. Fransson, B. Lennartson, T. Wik, and K. Holmström.
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.
Multi Criteria Controller Optimization for Uncertain MIMO Systems Using Nonconvex Global Optimization.  
 
In ''Proceedings of the 40th Conference on Decision and Control'', Orlando, FL, USA, December 2001.
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.
</ref>
 
<ref>
====Usage====
C. M. Fransson, B. Lennartson, T. Wik, K. Holmström, M. Saunders, and P.-O. Gutmann.
 
Global Controller Optimization Using Horowitz Bounds.  
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:
In ''Proceedings of the 15th IFAC Conference'', Barcelona, Spain, 21th-26th July, 2002.
 
</ref>
<pre>
. The use of TOMLAB for exponential fitting and nonlinear parameter estimation are discussed in e.g.  
toms 3x1 x
<ref>
f = sum(exp(x));
K. Holmström and J. Petersson.  
g = derivative(f,x);
A Review of the Parameter Estimation Problem of Fitting Positive Exponential Sums to Empirical Data.
</pre>
''Applied Mathematics and Computations'',
 
126(1):31-61, 2002.
results in the efficient ''g ''= ''exp''(''x'')''1''.  In contrast, the mathematically equivalent but slower code:
</ref>
 
<ref>
<pre>
Jordan M. Berg and K. Holmström.  On Parameter Estimation Using Level Sets. ''SIAM Journal on Control and Optimization'', 37(5):1372-1393, 1999.
toms 3x1 x
</ref>
f = 0;
<ref>
for i=1:length(x)
V. N. Fomin, K. Holmström, and T. Fomina.
  f = f+exp(x(i));
Least squares and Minimax methods for inorganic chemical equilibrium analysis.  
end
Research Report 2000-2, ISSN-1404-4978,   
g = derivative(f,x);
Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
</pre>
</ref>
 
<ref>
results in  
T. Fomina, K. Holmström, and V. B. Melas.  
 
Nonlinear parameter estimation for inorganic chemical equilib- rium analysis.  
<math>
Research Report 2000-3, ISSN-1404-4978,  
g=(exp(x(1))*[100]+exp(x(2))*[010])+exp(x(3))*[001]
Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
</math>
</ref>
 
<ref>
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
K. Holmström and T. Fomina.  
 
Computer Simulation for Inorganic Chemical Equilibrium Analysis.  In S.M. Ermakov, Yu. N. Kashtanov, and V.B. Melas, editors, ''Proceedings  of the 4th St.Petersburg Workshop on''  
can easily result in symbolic calculations that require more time than the actual numeric solution of the underlying optimization problem.
''Simulation'', pages 261-266, St.Petersburg, Russia, 2001.  
 
NII Chemistry St. Peterburg University Publishers.
It is thus recommended to avoid for-loops  as far as possible  when working with tomSym.
</ref>
 
<ref>
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:
K. Holmström, T. Fomina, and Michael Saunders.
 
Parameter Estimation for Inorganic Chemical Equilibria by Least Squares and Minimax Models.  
<pre>
''Optimization and Engineering'', 4, 2003. Submitted.
toms 2x1 x
</ref>
f = x(1)\*sin(x(2));
g = derivative(f,x);
</pre>
 
results in
 
<math>g=sin(x(2))*[10]+x(1)*(cos(x(2))*[01])</math>
 
Since x is never used as a 2x1 vector, it is better to use two independent 1x1 variables:
 
<pre>
toms a b
f = a\*sin(b);
g = derivative(f,\[a; b\]);
</pre>
 
which results in
 
<math>g=[sin(b) a * cos(b)]</math>.
 
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
 
<math>x=[a;b]</math>
 
====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 1''e''6, and the variable y, which is of the order of 1''e - ''6, the code:
 
<pre>
toms xScaled  yScaled
x = 1e+6\*xScaled;
y = 1e-6\*yScaled;
</pre>
 
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:
 
<pre>
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);
</pre>
 
====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.
 
<pre>
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
</pre>
 
 
 
====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:
 
<pre>
if x<2
    y = 0;
else
    y = x-2;
end
</pre>
 
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:
 
<pre>
y = ifThenElse(x<2,0,x-2)
</pre>
 
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:
 
<pre>
y = ifThenElse(x<2,0,x-2,0.1)
</pre>
 
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:
 
<pre>
toms x
 
y = x*x;
 
z = sin(y)+cos(y);
</pre>
 
In the tomSym object z, there is now a procedure, which looks something like:
 
<pre>
temp = x*x;
 
result = cos(temp)+sin(temp);
</pre>
 
Note: It is not necessary to use the intermediate symbol y. TomSym, automatically detects duplicated expressions, so the code
<math>sin(x*x)+cos(x*x)</math>
would result in the same optimized procedure for z.
 
On the other hand, the same corresponding  code using the symbolic toolbox:
 
<pre>
syms x
y = x*x;
z = sin(y)+cos(y);
</pre>
 
results in an object z that contains
<math>
cos(x^2) + sin(x^2)
</math>
, resulting in a double evaluation of
<math>
x^2
</math>
.
.


This may seem like a small difference in this simplified example, but in real-life applications, the difference can be significant.
The manuals for the add-on solver packages are also recommended reading material.
 
'''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).
 
<pre>
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.
 
<pre>
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
</pre>
 
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''.''6759''e ''+ 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:
 
<pre>
toms x
q=x;
for i=1:4
  q = x\*cos(q+2)\*cos(q);
end derivative(q,x)
</pre>
 
This yields the following tomSym procedure:
 
<pre>
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);
</pre>
Now, complete the same task using the symbolic toolbox:
 
<pre>
syms x q=x;
for i=1:4
  q = x\*cos(q+2)\*cos(q);
end
diff(q,x)
</pre>
 
This yields the following symbolic expression:
 
<pre>
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.
</pre>
 
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.
 
<pre>
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);
</pre>
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>,
<math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, and <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>.
Equality constraints are defined by setting
the lower bound equal to the upper bound, i.e.
for constraint <math>$i$: $b_{L}(i) =  b_{U}(i)$</math>.
 
To illustrate the solution of LPs consider the simple linear programming test problem
 
 
<math>
\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}
</math>
 
named ''LP Example''.
 
The following statements define this problem in Matlab
 
'''File: '''tomlab/usersguide/lpExample.m
 
<pre>
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 = [];
</pre>
 
====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
 
<pre>
lpExample;
 
Prob = lpAssign(c, A,  b_L,  b_U, x_L,  x_U, x_0,  'lpExample');
Result  = tomRun('lpSimplex', Prob,  1);
</pre>
 
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
 
<pre>
===== \* \* \* =================================================================== * * *
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.
</pre>
 
Having defined the ''Prob ''structure is it easy to call any solver that can handle linear programming problems,
 
<pre>
Result  = tomRun('qpSolve', Prob,  1);
</pre>
 
Even a nonlinear solver may be used.
 
<pre>
Result  = tomRun('nlpSolve',Prob, 3);
</pre>
 
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
 
 
<math>
\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}
</math>
 
 
where <math>$c, x, x_L, x_U \in \MATHSET{R}^n$</math>, <math>$F \in \MATHSET{R}^{n \times n}$</math>, <math>$A \in \MATHSET{R}^{m_1 \times n}$</math>, and <math>$b_L,b_U \in \MATHSET{R}^{m_1}$</math>. Equality constraints are defined by setting
the lower bound equal to the upper bound, i.e.
for constraint <math>$i$</math>:
<math>$b_{L}(i) =  b_{U}(i)$</math>.
<math>%$b_{L}(i) =  b_{U}(i), \all i \in E$, where $E$</math> 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
 
 
<math>
\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}
</math>
 
named ''QP Example''. The following statements define this problem in Matlab.


'''File: '''tomlab/usersguide/qpExample.m
==References==


<pre>
<references />
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
</pre>

Latest revision as of 08:59, 13 August 2011

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.

Overall Design

Overall Design presents the general design of TOMLAB.

Problem Types and Solver Routines

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

Defining Problems in TOMLAB

Solving Linear, Quadratic and Integer Programming Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Solving Unconstrained and Constrained Optimization Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Solving Global Optimization Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Solving Least Squares and Parameter Estimation Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Multi Layer Optimization

Shows how to setup and define multi layer optimization problems in TOMLAB.

Tomhelp - The Help Program

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.

TOMLAB Solver Reference

TOMLAB Utility Functions

Describes the utility functions that can be used, for example tomRun and SolverList.

Approximation of Derivatives

Introduces the different options for derivatives, automatic differentiation.

Special Notes and Features

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

Appendix A: Prob - the Input Problem Structure

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: Result - the Output Result Structure

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

Appendix C: TomSym - the Modeling Engine

Appendix D: Global Variables and Recursive Calls

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

Appendix E: External Interfaces

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

Appendix F: Motivation and Background to TOMLAB

Gives some motivation for the development of TOMLAB.

Appendix G: Performance Tests on Linear Programming Solvers

Further Reading

TOMLAB has been discussed in several papers and at several conferences. The main paper on TOMLAB v1.0 is [1] . The use of TOMLAB for nonlinear programming and parameter estimation is presented in [2] , and the use of linear and discrete optimization is discussed in [3] . Global optimization routines are also implemented, one is described in [4].

In all these papers TOMLAB was divided into two toolboxes, the NLPLIB TB and the OPERA TB. TOMLAB v2.0 was discussed in [5] [6] . and [7] . TOMLAB v4.0 and how to solve practical optimization problems with TOMLAB is discussed in [8] .

The use of TOMLAB for costly global optimization with industrial applications is discussed in [9]; costly global optimization with financial applications in [10] [11] [12]. Applications of global optimization for robust control is presented in [13] [14] . The use of TOMLAB for exponential fitting and nonlinear parameter estimation are discussed in e.g. [15] [16] [17] [18] [19] [20] .

The manuals for the add-on solver packages are also recommended reading material.

References

  1. ↑ K. Holmström. The TOMLAB Optimization Environment in Matlab. Advanced Modeling and Optimization, 1(1):47-69, 1999.
  2. ↑ K. Holmström and M. Björkman. The TOMLAB NLPLIB Toolbox for Nonlinear Programming. Advanced Modeling and Optimization, 1(1):70-86, 1999.
  3. ↑ K. Holmström, M. Björkman, and E. Dotzauer. The TOMLAB OPERA Toolbox for Linear and Discrete Optimization. Advanced Modeling and Optimization, 1(2):1-8, 1999.
  4. ↑ M. Björkman and K. Holmström. Global Optimization Using the DIRECT Algorithm in Matlab. Advanced Modeling and Optimization, 1(2):17-37, 1999.
  5. ↑ K. Holmström. The TOMLAB v2.0 Optimization Environment. In E. Dotzauer, M. Björkman, and K. Holmstöm, editors, Sixth Meeting of the Nordic Section of the Mathematical Programming Society. Proceedings, Opuscula 49, ISSN 1400-5468, Västerås, 1999. Mälardalen University, Sweden.
  6. ↑ K. Holmström. New Optimization Algorithms and Software. Theory of Stochastic Processes, 5(21)(1-2):55-63, 1999.
  7. ↑ K. Holmström. Solving applied optimization problems using TOMLAB. In G. Osipenko, editor, Proceedings from MATHTOOLS '99, the 2nd International Conference on Tools for Mathematical Modelling, pages 90-98, St.Petersburg, Russia, 1999. St.Petersburg State Technical University.
  8. ↑ K. Holmström. Practical Optimization with the Tomlab Environment. In T. A. Hauge, B. Lie, R. Ergon, M. D. Diez, G.-O. Kaasa, A. Dale, B. Glemmestad, and A Mjaavatten, editors, Proceedings of the 42nd SIMS Conference, pages 89-108, Porsgrunn, Norway, 2001. Telemark University College, Faculty of Technology.
  9. ↑ M. Björkman and K. Holmström. Global Optimization of Costly Nonconvex Functions Using Radial Basis Functions. Optimization and Engineering, 1(4):373-397, 2000.
  10. ↑ T. Hellström and K. Holmström. Parameter Tuning in Trading Algorithms using ASTA. In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, Computational Finance (CF99) - Abstracts of the Sixth International Conference, Leonard N. Stern School of Business, January 1999, Leonard N. Stern School of Business, New York University, 1999. Department of Statistics and Operations Research.
  11. ↑ T. Hellström and K. Holmström. Parameter Tuning in Trading Algorithms using ASTA. In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, Computational Finance 1999, Cambridge, MA, 1999. MIT Press.
  12. ↑ T. Hellström and K. Holmström. Global Optimization of Costly Nonconvex Functions, with Financial Applications. Theory of Stochastic Processes, 7(23)(1-2):121-141, 2001.
  13. ↑ C. M. Fransson, B. Lennartson, T. Wik, and K. Holmström. Multi Criteria Controller Optimization for Uncertain MIMO Systems Using Nonconvex Global Optimization. In Proceedings of the 40th Conference on Decision and Control, Orlando, FL, USA, December 2001.
  14. ↑ C. M. Fransson, B. Lennartson, T. Wik, K. Holmström, M. Saunders, and P.-O. Gutmann. Global Controller Optimization Using Horowitz Bounds. In Proceedings of the 15th IFAC Conference, Barcelona, Spain, 21th-26th July, 2002.
  15. ↑ K. Holmström and J. Petersson. A Review of the Parameter Estimation Problem of Fitting Positive Exponential Sums to Empirical Data. Applied Mathematics and Computations, 126(1):31-61, 2002.
  16. ↑ Jordan M. Berg and K. Holmström. On Parameter Estimation Using Level Sets. SIAM Journal on Control and Optimization, 37(5):1372-1393, 1999.
  17. ↑ V. N. Fomin, K. Holmström, and T. Fomina. Least squares and Minimax methods for inorganic chemical equilibrium analysis. Research Report 2000-2, ISSN-1404-4978, Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
  18. ↑ T. Fomina, K. Holmström, and V. B. Melas. Nonlinear parameter estimation for inorganic chemical equilib- rium analysis. Research Report 2000-3, ISSN-1404-4978, Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
  19. ↑ K. Holmström and T. Fomina. Computer Simulation for Inorganic Chemical Equilibrium Analysis. In S.M. Ermakov, Yu. N. Kashtanov, and V.B. Melas, editors, Proceedings of the 4th St.Petersburg Workshop on Simulation, pages 261-266, St.Petersburg, Russia, 2001. NII Chemistry St. Peterburg University Publishers.
  20. ↑ K. Holmström, T. Fomina, and Michael Saunders. Parameter Estimation for Inorganic Chemical Equilibria by Least Squares and Minimax Models. Optimization and Engineering, 4, 2003. Submitted.
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