TOMLAB Solver Reference

TOMLAB Solver Reference

Detailed descriptions of the TOMLAB solvers, driver routines and some utilities are given in the following sections. Also see the M-file help for each solver. All solvers except for the TOMLAB Base Module are described in separate manuals.

TOMLAB Base Module

For a description of solvers called using the MEX-file interface, see the M-file help, e.g. for the MINOS solver minosTL.m. For more details, see the User's Guide for the particular solver.

clsSolve

Purpose

Solves dense and sparse nonlinear least squares optimization problems with linear inequality and equality con- straints and simple bounds on the variables.

clsSolve solves problems of the form

${\begin{array}{cccccc}\min \limits _{x}&f(x)&=&{\frac {1}{2}}r(x)^{T}r(x)&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $r(x)\in \MATHSET{R} ^{N}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{m_{1}\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_{L},b_{U}\in \MATHSET{R}^{m_{1}}$} .

Calling Syntax

Result = clsSolve(Prob, varargin)

Result = tomRun('clsSolve', Prob);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	Solver.Alg	Solver algorithm to be run:
		0: Gives default, the Fletcher - Xu hybrid method;
		1: Fletcher - Xu hybrid method; Gauss-Newton/BFGS.
		2: Al-Baali - Fletcher hybrid method; Gauss-Newton/BFGS.
		3: Huschens method. SIAM J. Optimization. Vol 4, No 1, pp 108-129 jan 1994.
		4: The Gauss-Newton method.
		5: Wang, Li, Qi Structured MBFGS method.
		6: Li-Fukushima MBFGS method.
		7: Broydens method.
		Recommendations: Alg=5 is theoretically best, and seems best in practice as well. Alg=1 and Alg=2 behave very similar, and are robust methods. Alg=4 may be good for ill-conditioned problems. Alg=3 and Alg=6 may sometimes fail. Alg=7 tries to minimize Jacobian evaluations, but might need more resid- ual evaluations. Also fails more often that other algorithms. Suitable when analytic Jacobian is missing and evaluations of the Jacobian is costly. The problem should not be too ill-conditioned.
	Solver.Method	Method to solve linear system:
		0: QR with pivoting (both sparse and dense).
		1: SVD (dense).
		2: The inversion routine (inv) in Matlab (Uses QR).
		3: Explicit computation of pseudoinverse, using pinv(Jk ).
		Search method technique (if Prob.LargeScale = 1, then Method = 0 always): Prob.Solver.Method = 0 Sparse iterative QR using Tlsqr.
	LargeScale	If = 1, then sparse iterative QR using Tlsqr is used to find search directions
	x_0	Starting point.
	x_L	Lower bounds on the variables.
	x U	Upper bounds on the variables.
	b_L	Lower bounds on the linear constraints. b
	b_U	Upper bounds on the linear constraints. A Constraint matrix for linear constraints.
	c_L	Lower bounds on the nonlinear constraints.
	c_U	Upper bounds on the nonlinear constraints.
	f_Low	A lower bound on the optimal function value, see LineParam.fLowBnd below.
	SolverQP	Name of the solver used for QP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m.
	PriLevOpt	Print Level.
	optParam	Structure with special fields for optimization parameters, see Table 141. Fields used are: bTol, eps absf, eps g, eps Rank, eps x, IterPrint, MaxIter, PreSolve, size f, size x, xTol, wait, and QN InitMatrix (Initial Quasi-Newton matrix, if not empty, otherwise use identity matrix).
	LineParam	Structure with line search parameters. Special fields used:
	LineAlg	If Alg = 7
		0 = Fletcher quadratic interpolation line search
		3 = Fletcher cubic interpolation line search
		otherwise Armijo-Goldstein line search (LineAlg == 2)
		If Alg! = 7
		0 = Fletcher quadratic interpolation line search
		1 = Fletcher cubic interpolation line search
		2 = Armijo-Goldstein line search
		otherwise Fletcher quadratic interpolation line search (LineAlg == 0)
		If Fletcher, see help LineSearch for the LineParam parameters used. Most important is the accuracy in the line search: sigma - Line search accuracy tolerance, default 0.9.
	If LineAlg == 2, then the following parameters are used
	agFac	Armijo Goldsten reduction factor, default 0.1
	sigma	Line search accuracy tolerance, default 0.9
	fLowBnd	A lower bound on the global optimum of f(x). NLLS problems always have f(x) values >= 0 The user might also give lower bound estimate in Prob.f Low clsSolve computes LineParam.fLowBnd as: LineParam.fLowBnd = max(0,Prob.f Low,Prob.LineParam.fLowBnd) fLow = LineParam.fLowBnd is used in convergence tests.
varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	v_k	Lagrange multipliers (not used).
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	x_0	Starting point.
	f_0	Function value at start.
	r_k	Residual at optimum.
	J_k	Jacobian matrix at optimum.
	xState	State of each variable, described in Table 150.
	bState	State of each linear constraint, described in Table 151.
	Iter	Number of iterations.
	ExitFlag	Flag giving exit status. 0 if convergence, otherwise error. See Inform.
	Inform	Binary code telling type of convergence:
		1: Iteration points are close.
		2: Projected gradient small.
		3: Iteration points are close and projected gradient small.
		4: Function value close to 0.
		5: Iteration points are close and function value close to 0.
		6: Projected gradient small and function value close to 0.
		7: Iteration points are close, projected gradient small and function value close to 0.
		8: Relative function value reduction low for LowI ts = 10 iterations.
		11: Relative f(x) reduction low for LowIts iter. Close Iters.
		16: Small Relative f(x) reduction.
		17: Close iteration points, Small relative f(x) reduction.
		18: Small gradient, Small relative f(x) reduction.
		32: Local minimum with all variables on bounds.
		99: The residual is independent of x. The Jacobian is 0.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		104: x_k not feasible, constraint violated.
		105: The residual is empty, no NLLS problem.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The solver clsSolve includes seven optimization methods for nonlinear least squares problems: the Gauss-Newton method, the Al-Baali-Fletcher \[3\] and the Fletcher-Xu \[19\] hybrid method, the Huschens TSSM method \[50\] and three more. If rank problem occur, the solver is using subspace minimization. The line search is performed using the routine LineSearch which is a modified version of an algorithm by Fletcher \[20\]. Bound constraints are partly treated as described in Gill, Murray and Wright \[28\]. clsSolve treats linear equality and inequality constraints using an active set strategy and a null space method.

M-files Used

ResultDef.m, preSolve.m, qpSolve.m, tomSolve.m, LineSearch.m, ProbCheck.m, secUpdat.m, iniSolve.m, endSolve.m

See Also

conSolve, nlpSolve, sTrustr

Limitations

When using the LargeScale option, the number of residuals may not be less than 10 since the sqr2 algorithm may run into problems if used on problems that are not really large-scale.

Warnings

Since no second order derivative information is used, clsSolve may not be able to determine the type of stationary point converged to.

conSolve

Purpose

Solve general constrained nonlinear optimization problems.

conSolve solves problems of the form

${\begin{array}{cccccc}\min \limits _{x}&f(x)&&&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&c_{L}&\leq &c(x)&\leq &c_{U}\\\end{array}}$

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x,x_{L}, <math>x_{U}\in \MATHSET{R}^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{m_{2}\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_{L},b_{U}\in \MATHSET{R}^{m_{2}}$} .

Calling Syntax

Result = conSolve(Prob, varargin)

Result = tomRun('conSolve', Prob);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	Solver.Alg	Choice of algorithm. Also affects how derivatives are obtained.
		See following fields and the table on page 92.
		0,1,2: Schittkowski SQP.
		3,4: Han-Powell SQP.
	x_0	Starting point.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	A	Constraint matrix for linear constraints.
	c_L	Lower bounds on the general constraints.
	c_U	Upper bounds on the general constraints.
	NumDiff	How to obtain derivatives (gradient, Hessian).
	ConsDiff	How to obtain the constraint derivative matrix.
	SolverQP	Name of the solver used for QP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m.
	f_Low	A lower bound on the optimal function value, see LineParam.fLowBnd below.
		Used in convergence tests, f k(x k) <= f Low. Only a feasible point x k is accepted.
	FUNCS.f	Name of m-file computing the objective function f (x).
	FUNCS.g	Name of m-file computing the gradient vector g(x).
	FUNCS.H	Name of m-file computing the Hessian matrix H (x).
	FUNCS.c	Name of m-file computing the vector of constraint functions c(x).
	FUNCS.dc	Name of m-file computing the matrix of constraint normals ?c(x)/dx.
	PriLevOpt	Print level.
	optParam	Structure with optimization parameters, see Table 141. Fields that are used: bTol, cTol, eps absf, eps g, eps x, eps Rank, IterPrint, MaxIter, QN InitMatrix, size f, size x, xTol and wait.
	LineParam	Structure with line search parameters. See Table 140.
	fLowBnd	A lower bound on the global optimum of f(x). The user might also give lower bound estimate in Prob.f Low conSolve computes LineParam.fLowBnd as: LineParam.fLowBnd = max(Prob.f Low,Prob.LineParam.fLowBnd).
varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	v_k	Lagrange multipliers.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	H_k	Hessian value at optimum.
	x_0	Starting point.
	f_0	Function value at start.
	c_k	Value of constraints at optimum.
	cJac	Constraint Jacobian at optimum.
	xState	State of each variable, described in Table 150 .
	bState	State of each linear constraint, described in Table 151.
	cState	State of each nonlinear constraint.
	Iter	Number of iterations.
	ExitFlag	Flag giving exit status.
	ExitText'	'Text string giving ExitFlag and Inform information.
	Inform	Code telling type of convergence:
		1: Iteration points are close. Result Structure with result from optimization. The following fields are changed:, continued
		2: Small search direction.
		3: Iteration points are close and Small search direction.
		4: Gradient of merit function small.
		5: Iteration points are close and gradient of merit function small.
		6: Small search direction and gradient of merit function small.
		7: Iteration points are close, small search direction and gradient of merit func-tion small.
		8: Small search direction p and constraints satisfied.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		103: Iteration points are close, but constraints not fulfilled. Too large penalty weights to be able to continue. Problem is maybe infeasible.
		104: Search direction is zero and infeasible constraints. The problem is very likely infeasible.
		105: Merit function is infinity.
		106: Penalty weights too high.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine conSolve implements two SQP algorithms for general constrained minimization problems. One imple- mentation, Solver.Alg = 0, 1, 2, is based on the SQP algorithm by Schittkowski with Augmented Lagrangian merit function described in \[69\]. The other, Solver.Alg = 3, 4, is an implementation of the Han-Powell SQP method.

The Hessian in the QP subproblems are determined in one of several ways, dependent on the input parameters. The following table shows how the algorithm and Hessian method is selected.

Solver.Alg	NumDiff	AutoDiff	isempty(FUNCS.H)	Hessian computation	Algorithm
0	0	0	0	Analytic Hessian	Schittkowski SQP
0	any	any	any	BFGS	Schittkowski SQP
1	0	0	0	Analytic Hessian	Schittkowski SQP
1	0	0	1	Numerical differences H	Schittkowski SQP
1	> 0	0	any	Numerical differences g,H	Schittkowski SQP
1	< 0	0	any	Numerical differences H	Schittkowski SQP
1	any	1	any	Automatic differentiation	Schittkowski SQP
2	0	0	any	BFGS	Schittkowski SQP
2	= 0	0	any	BFGS, numerical gradient g	Schittkowski SQP
2	any	1	any	BFGS, automatic diff gradient	Schittkowski SQP
3	0	0	0	Analytic Hessian	Han-Powell SQP
3	0	0	1	Numerical differences H	Han-Powell SQP
3	> 0	0	any	Numerical differences g,H	Han-Powell SQP
3	< 0	0	any	Numerical differences H	Han-Powell SQP
3	any	1	any	Automatic differentiation	Han-Powell SQP
4	0	0	any	BFGS	Han-Powell SQP
4	= 0	0	any	BFGS, numerical gradient g	Han-Powell SQP
4	any	1	any	BFGS, automatic diff gradient	Han-Powell SQP

M-files Used

ResultDef.m, tomSolve.m, LineSearch.m, iniSolve.m, endSolve.m, ProbCheck.m.

See Also

nlpSolve, sTrustr

cutPlane

Purpose

Solve mixed integer linear programming problems (MIP).

cutplane solves problems of the form

Failed to parse (unknown function "\MATHSET"): {\displaystyle \begin{array}{ccccccl} \min\limits_{x} & f(x) & = & c^{T}x & & \\\mbox{subject to} & 0 & \leq & x & \leq & x_{U} & \\& & & Ax & = & b, & ~x_{j} \in \MATHSET{N}\, \forall j \in $I$ \\ \end{array} }

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $c, x, x_U \in \MATHSET{R}^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{m\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b \in \MATHSET{R}^{m}$} . The variables $\$x\in I\$$, the index subset of $\$1,...,n\$$ are restricted to be integers.

Calling Syntax

Result = cutplane(Prob); or

Result = tomRun('cutplane', Prob);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	c	Constant vector.
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	x_L	Lower bounds on the variables (assumed to be 0).
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	QP.B	Active set B_0 at start:
		B(i) = 1: Include variable x(i) in basic set.
		B(i) = 0: Variable x(i) is set on it's lower bound.
		B(i) = -1: Variable x(i) is set on it's upper bound.
		B empty: lpSimplex solves Phase I LP to find a feasible point.
	Solver.Method	Variable selection rule to be used:
		0: Minimum reduced cost. (default)
		1: Bland's anti-cycling rule.
		2: Minimum reduced cost, Dantzig's rule.
	MIP.IntVars	Which of the n variables are integers.
	SolverLP	Name of the solver used for initial LP subproblem.
	SolverDLP	Name of the solver used for dual LP subproblems.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: MaxIter, PriLev, wait, eps f, eps Rank, xTol and bTol.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum, c.
	v_k	Lagrange multipliers.
	QP.B	Optimal active set. See input variable QP.B.
	xState	State of each variable, described in Table 150 .
	x_0	Starting point.
	f_0	Function value at start.
	Iter	Number of iterations.
	FuncEv	Number of function evaluations. Equal to Iter.
	ConstrEv	Number of constraint evaluations. Equal to Iter.
	ExitFlag	0: OK.
		1: Maximal number of iterations reached.
		4: No feasible point x 0 found.
	Inform	If ExitFlag > 0, Inform = ExitFlag.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine cutplane is an implementation of a cutting plane algorithm with Gomorov cuts. cutplane normally uses the linear programming routines lpSimplex and DualSolve to solve relaxed subproblems. By changing the setting of the structure fields Prob.Solver.SolverLP and Prob.Solver.SolverDLP, different sub-solvers are possible to use.

cutplane can interpret Prob.MIP.IntVars in two different ways:

• Vector of length less than dimension of problem: the elements designate indices of integer variables, e.g. $\$IntVars=[1\;3\;5]\$$ restricts $\$x_{1},x_{3}\$$ and $\$x_{5}\$$ to take integer values only.

• Vector of same length as $\$c\$$: non-zero values indicate integer variables, e.g. with five variables (Failed to parse (unknown function "\MATHSET"): {\displaystyle $x\in \MATHSET{R}^5$} ), $\$IntVars=[1\;1\;0\;1\;1]\$$ demands all but $\$x_{3}\$$ to take integer values.

M-files Used

lpSimplex.m, DualSolve.m

See Also

mipSolve, balas, lpsimpl, lpsimp2, lpdual, tomSolve.

DualSolve

Purpose

Solve linear programming problems when a dual feasible solution is available.

DualSolve solves problems of the form

${\begin{array}{cccccl}\min \limits _{x}&f(x)&=&c^{T}x&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&&&Ax&=&b_{U}\\\end{array}}$

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $c \in \MATHSET{R}^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{m\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_{U} \in \MATHSET{R}^{m}$} .

Finite upper bounds on x are added as extra inequality constraints. Finite nonzero lower bounds on x are added as extra inequality constraints. Fixed variables are treated explicitly. Adding slack variables and making necessary sign changes gives the problem in the standard form

${\begin{array}{ccccl}\min \limits _{x}&f_{P}(x)&=&c^{T}x&\\s/t&{\hat {A}}x&=&b&\\&x&\geq &0&\end{array}}$

and the following dual problem is solved,

${\begin{array}{ccccl}\max \limits _{y}&f_{D}(y)&=&b^{T}y\\s/t&{\hat {A}}^{T}y&\leq &c\\&y&urs&\end{array}}$

with Failed to parse (unknown function "\MATHSET"): {\displaystyle $x,c \in \MATHSET{R}^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{\hat{m}\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b,y \in \MATHSET{R}^{m}$} .

Calling Syntax

[Result] = DualSolve(Prob)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	QP.c	Constant vector.
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point, must be dual feasible.
	y_0	Dual parameters (Lagrangian multipliers) at x _0.
	QP.B	Active set B_0 at start:
		B(i) = 1: Include variable x(i) is in basic set.
		B(i) = 0: Variable x(i) is set on its lower bound.
		B(i) = -1: Variable x(i) is set on its upper bound.
	Solver.Alg	Variable selection rule to be used:
		0: Minimum reduced cost (default).
		1: Bland's anti-cycling rule.
		2: Minimum reduced cost. Dantzig's rule.
	PriLevOpt	Print Level.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: MaxIter, wait, eps f, eps Rank and xTol.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal primal solution x.
	f_k	Function value at optimum.
	v_k	Optimal dual parameters. Lagrange multipliers for linear constraints.
	x_0	Starting point.
	Iter	Number of iterations.
	QP.B	Optimal active set.
	ExitFlag	Exit flag:
		0: Optimal solution found.
		1: Maximal number of iterations reached. No primal feasible solution found.
		2: Infeasible Dual problem.
		4: Illegal step length due to numerical difficulties. Should not occur.
		6: No dual feasible starting point found.
		7: Illegal step length due to numerical difficulties.
		8: Convergence because fk >= QP.DualLimit.
		9: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x_L(i) > x_U(i) + xTol$} for some i. No solution exists.
Solver	Solver used.
SolverAlgorithm	Solver algorithm used.
	c	Constant vector in standard form formulation.
	A	Constraint matrix for linear constraints in standard form.
	b	Right hand side in standard form.

Description

When a dual feasible solution is available, the dual simplex method is possible to use. DualSolve implements this method using the algorithm in \[35, pages 105-106\]. There are three rules available for variable selection. Bland's cycling prevention rule is the choice if fear of cycling exist. The other two are variants of minimum reduced cost variable selection, the original Dantzig's rule and one which sorts the variables in increasing order in each step (the default choice).

M-files Used

cpTransf.m

See Also

lpSimplex

expSolve

Purpose

Solve exponential fitting problems for given number of terms p.

Calling Syntax

Prob = expAssign( ... );

Result = expSolve(Prob, PriLev); or

Result = tomRun('expSolve', PriLev);

Description of Inputs

Prob.SolverL2||Name of solver to use. If empty, TOMLAB selects dependent on license.

Prob	Problem created with expAssign.
PriLev	Print level in tomRun call.

Description of Outputs

Result TOMLAB	Result structure as returned by solver selected by input argument Solver.
LS	Statistical information about the solution. See Table 153, page 239.

Global Parameters Used

Description

expSolve solves a cls (constrained least squares) problem for exponential fitting formulates by expAssign. The problem is solved with a suitable or given cls solver.

The aim is to provide a quicker interface to exponential fitting, automating the process of setting up the problem structure and getting statistical data.

M-files Used

GetSolver, expInit, StatLS and expAssign

Examples

Assume that the Matlab vectors t, y contain the following data:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{|r|rrrrrrrrr|} t_i & 0 & 1.00 & 2.00 & 4.00 & 6.00 & 8.00 & 10.00 & 15.00 & 20.00\\y_i & 905.10 & 620.36 & 270.17 & 154.68 & 106.74 & 80.92 & 69.98 & 62.50 & 56.29\\ \end{array} }

To set up and solve the problem of fitting the data to a two-term exponential model

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \alpha_1 e^{-\beta_1 t} + \alpha_2 e^{-\beta_2 t}} ,

give the following commands:

>> p      = 2;                         % Two terms
>> Name   = 'Simple two-term exp fit'; % Problem name, can be anything
>> wType  = 0;                         % No weighting
>> SepAlg = 0;                         % Separable problem
>> Prob = expAssign(p,Name,t,y,wType,[],SepAlg);

>> Result = tomRun('expSolve',Prob,1);
>> x = Result.x_k'

x =
          0.01          0.58         72.38        851.68

The Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x$} vector contains the parameters as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x=[\beta_1,\beta_2,\alpha_1,\alpha_2]$} so the solution may be visualized with

>> plot(t,y,'-*', t,x(3)*exp(-t*x(1)) + x(4)*exp(-t*x(2)) );

Figure 6: Results of fitting experimental data to two-term exponential model. Solid line: final model, dash-dot: data.

glbDirect

Purpose

Solve box-bounded global optimization problems.

glbDirect solves problems of the form

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

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f \in \MATHSET{R}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} .

glbDirect is a Fortran MEX implementation of glbSolve.

Calling Syntax

Result = glbDirectTL(Prob,varargin)

Result = tomRun('glbDirect', Prob);

Description of Inputs

options||Structure with options. These options have precedence over all other options in the Prob struct. Available options are: MAXITER: Eq. to Prob.optParam.MaxIter. Default: 200

Prob	Problem description structure. The following fields are used:
	x_L	Lower bounds for x, must be given to restrict the search space.
	x_U	Upper bounds for x, must be given to restrict the search space.
	Name	Name of the problem. Used for security if doing warm start.
	FUNCS.f	Name of m-file computing the objective function f (x).
	PriLevOpt	Print Level. 0 = Silent. 1 = Errors. 2 = Termination message and warm start info. 3 = Option summary.
	WarmStart	If true, > 0, glbDirect reads the output from the last run from Prob.glbDirect.WarmStartInto if it exists. If it doesn't exist, glbDirect attempts to open and read warm start data from mat-file glbDirectSave.mat. glbDirect uses this warm start information to continue from the last run.
	optParam	Structure in Prob, Prob.optParam. Defines optimization parameters. Fields used:
	IterPrint	Print iteration log every IterPrint iteration. Set to 0 for no iteration log. PriLev must be set to at least 1 to have iteration log to be printed.
	MaxIter	Maximal number of iterations, default 200.
	MaxFunc	Maximal number of function evaluations, default 10000 (roughly).
	EpsGlob	Global/local weight parameter, default 1E-4.
	fGoal	Goal for function value, if empty not used.
	eps f	Relative accuracy for function value, f T ol == epsf . Stop if abs(f - f Goal)<= abs(f Goal) * f T ol , if f Goal = 0. Stop if abs(f - f Goal)<= f T ol , iff Goal == 0.
	eps x	Convergence tolerance in x. All possible rectangles are less than this tolerance(scaled to (0,1) ). See the output field maxTri.
	glbDirect	Structure in Prob, Prob.glbDirect. Solver specific.
		PRILEV: Equivalent to Prob.PrilevOpt. Default: 0
		MAXFUNC: Eq. to Prob.optParam.MaxFunc. Default: 10000
		PARALLEL: Set to 1 in order to have glbDirect to call Prob.FUNCS.f with a matrix x of points to let the user function compute function values in parallel. Default: 0
		WARMSTART: Eq. to Prob.WarmStart. Default: 0
		ITERPRINT: Eq. to Prob.optParam.IterPrint. Default: 0
		FUNTOL: Eq. to Prob.optParam.eps f. Default: 1e-2
		VARTOL: Eq. to Prob.optParam.eps x. Default: 1e-13
		GLWEIGHT: Eq. to Prob.optParam.EpsGlob. Default: 1e-4
		Structure with WarmStartInfo. Use WarmDefDIRECT.m to define it.
	WarmStartInfo

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Matrix with optimal points as columns.
	f_k	Function value at optimum.
	Iter	Number of iterations.
	FuncEv	Number function evaluations.
	ExitText	Text string giving ExitFlag and Inform information.
	ExitFlag	Exit code.
		0 = Normal termination, max number of iterations /func.evals reached.
		1 = Some bound, lower or upper is missing.
		2 = Some bound is inf, must be finite.
		4 = Numerical trouble determining optimal rectangle, empty set and cannot continue.
	Inform	Inform code.
		1 = Function value f is less than fGoal.
		2 = Absolute function value f is less than fTol, only if fGoal = 0 or Relative error in function value f is less than fTol, i.e. abs(f - f Goal)/abs(f Goal) <= f T ol.
		3 = Maximum number of iterations done.
		4 = Maximum number of function evaluations done.
		91= Numerical trouble, did not find element in list.
		92= Numerical trouble, No rectangle to work on.
		99= Other error, see ExitFlag.
	glbDirect	Substructure for glbDirect specific result data.
	nextIterFunc	If optimization algorithm was stopped because of maximum number of function evaluations reached, this is the number of function evaluations required to complete the next iteration.
	maxTri	Maximum size of any triangles. Structure containing warm start data. Could be used to continue optimization
	WarmStartInfo	where glbDirect stopped.
	To make a warm start possible, glbDirect saves the following information in the structure Result.glbDirect.WarmStartInfo and file glbDirectSave.mat (for internal solver use only):
	points	Matrix with all rectangle centerpoints, in [0,1]-space.
	dRect	Vector with distances from centerpoint to the vertices.
	fPoints	Vector with function values.
	nIter	Number of iterations.
	lRect	Matrix with all rectangle side lengths in each dimension.
	Name	Name of the problem. Used for security if doing warm start.
	dMin	Row vector of minimum function value for each distance.
	ds	Row vector of all different distances, sorted.
	glbfMin	Best function value found at a feasible point.
	iMin	The index in D which has lowest function value, i.e. the rectangle which mini- mizes (F -glbf M in+E)./D where E = max(EpsGlob*abs(glbf M in), 1E -8).
	ign	Rectangles to be ignored in the rect. selection procedure.

Description

The global optimization routine glbDirect is an implementation of the DIRECT algorithm presented in \[14\]. The algorithm in glbDirect is a Fortran MEX implementation of the algorithm in glbSolve. DIRECT is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. Since no such constant is used, there is no natural way of defining convergence (except when the optimal function value is known). Therefore glbDirect runs a predefined number of iterations and considers the best function value found as the optimal one. It is possible for the user to restart glbDirect with the final status of all parameters from the previous run, a so called warm start. Assume that a run has been made with glbDirect on a certain problem for 50 iterations. Then a run of e.g. 40 iterations more should give the same result as if the run had been using 90 iterations in the first place. To do a warm start of glbDirect a flag Prob.WarmStart should be set to one and WarmDefDIRECT run. Then glbDirect is using output previously obtained to make the restart. The m-file glbSolve also includes the subfunction conhull (in MEX) which is an implementation of the algorithm GRAHAMHULL in \[65, page 108\] with the modifications proposed on page 109. conhull is used to identify all points lying on the convex hull defined by a set of points in the plane.

M-files Used

iniSolve.m, endSolve.m glbSolve.m.

glbSolve

Purpose

Solve box-bounded global optimization problems. glbSolve solves problems of the form

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

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $f \in \MATHSET{R}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} .

Calling Syntax

Result = glbSolve(Prob,varargin)

Result = tomRun('glbSolve', Prob);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	x_L	Lower bounds for x, must be given to restrict the search space.
	x_U	Upper bounds for x, must be given to restrict the search space.
	Name	Name of the problem. Used for security if doing warm start.
	FUNCS.f	Name of m-file computing the objective function f (x).
	PriLevOpt	Print Level. 0 = silent. 1 = some printing. 2 = print each iteration.
	WarmStart	If true, > 0, glbSolve reads the output from the last run from the mat-file glbSave.mat, and continues from the last run.
	MaxCPU	Maximal CPU Time (in seconds) to be used.
	optParam	Structure in Prob, Prob.optParam. Defines optimization parameters. Fields used:
	IterPrint	Print iteration \#, \# of evaluated points and best f(x) each iteration.
	MaxIter	Maximal number of iterations, default max(5000, n * 1000).
	MaxFunc	Maximal number of function evaluations, default max(10000, n * 2000).
	EpsGlob	Global/local weight parameter, default 1E-4.
	fGoal	Goal for function value, if empty not used.
	eps_f	Relative accuracy for function value, f T ol == epsf . Stop if abs(f - f Goal)<= abs(f Goal) * f T ol , if f Goal = 0. Stop if abs(f - f Goal) <= f T ol , if f Goal == 0.
	If warm start is chosen, the following fields saved to glbSave.mat are also used and contains information from the previous run:
	C	Matrix with all rectangle centerpoints, in [0,1]-space.
	D	Vector with distances from centerpoint to the vertices. DMin Row vector of minimum function value for each distance. DSort Row vector of all different distances, sorted.
	E	Computed tolerance in rectangle selection.
	F	Vector with function values.
	L	Matrix with all rectangle side lengths in each dimension. Name Name of the problem. Used for security if doing warm start. glbfMin Best function value found at a feasible point.
	iMin	The index in D which has lowest function value, i.e. the rectangle which mini- mizes (F -glbf M in+E)./D where E = max(EpsGlob*abs(glbf M in), 1E -8).
	varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Matrix with all points giving the function value f k.
	f_k	Function value at optimum.
	Iter	Number of iterations.
	FuncEv	Number function evaluations.
	maxTri	Maximum size of any triangle.
	ExitText	Text string giving ExitFlag and Inform information.
	ExitFlag	Exit code.
		0 = Normal termination, max number of iterations /func.evals reached.
		1 = Some bound, lower or upper is missing.
		2 = Some bound is inf, must be finite.
		4 = Numerical trouble determining optimal rectangle, empty set and cannot continue.
	Inform	Inform code.
		0 = Normal Exit.
		1 = Function value f is less than fGoal.
		2 = Absolute function value f is less than fTol, only if fGoal = 0 or Relative error in function value f is less than fTol, i.e. abs(f - f Goal)/abs(f Goal) <= f T ol.
		9 = Max CPU Time reached.
	Solver	Solver used, 'glbSolve'.

Description

The global optimization routine glbSolve is an implementation of the DIRECT algorithm presented in \[14\]. DIRECT is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. Since no such constant is used, there is no natural way of defining convergence (except when the optimal function value is known). Therefore glbSolve runs a predefined number of iterations and considers the best function value found as the optimal one. It is possible for the user to restart glbSolve with the final status of all parameters from the previous run, a so called warm start Assume that a run has been made with glbSolve on a certain problem for 50 iterations. Then a run of e.g. 40 iterations more should give the same result as if the run had been using 90 iterations in the first place. To do a warm start of glbSolve a flag Prob.WarmStart should be set to one. Then glbSolve is using output previously written to the file glbSave.mat to make the restart. The m-file glbSolve also includes the subfunction conhull (in MEX) which is an implementation of the algorithm GRAHAMHULL in \[65, page 108\] with the modifications proposed on page 109. conhull is used to identify all points lying on the convex hull defined by a set of points in the plane.

M-files Used

iniSolve.m, endSolve.m

glcCluster

Purpose

Solve general constrained mixed-integer global optimization problems using a hybrid algorithm.

glcCluster solves problems of the form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\ & c_{L} & \leq & c(x) & \leq & c_{U} \\& & & \multicolumn{3}{c}{x_i \in \MATHSET{N} \; \forall i \in I} \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R}^{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A\in \MATHSET{R}^{m_{2}\times n}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_{L},b_{U}\in \MATHSET{R}^{m_{2}}$} .

Calling Syntax

Result = glcCluster(Prob, maxFunc1, maxFunc2, maxFunc3, ProbL)

Result = tomRun('glcCluster', Prob, PriLev) (driver call)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	c_L	Lower bounds on the general constraints.
	c_U	Upper bounds on the general constraints.
	x_L	Lower bounds for x, must be given to restrict the search space.
	x_U	Upper bounds for x, must be given to restrict the search space.
	FUNCS.f	Name of m-file computing the objective function f (x).
	FUNCS.c	Name of m-file computing the vector of constraint functions c(x).
	PriLevOpt	Print level. 0=Silent. 1=Some output from each glcCluster phase. 2=More output from each phase. 3=Further minor output from each phase. 6=Use PrintResult( ,1) to print summary from each global and local run. 7 = Use PrintResult( ,2) to print summary from each global and local run. 8 = Use PrintResult( ,3) to print summary from each global and local run.
	WarmStart	If true, > 0, glcCluster warm starts the DIRECT solver. The DIRECT solver will utilize all points sampled in last run, from one or two calls, dependent on the success in last run. Note: The DIRECT solver may not be changed if doing WarmStart mat-file glcFastSave.mat, and continues from the last run.
	Name	Name of the problem. glcCluster uses the warmstart capability in glcFast and needs the name for security reasons.
	GO	Structure in Prob, Prob.GO. Fields used:
	maxFunc1	Number of function evaluations in 1st call to glcFast. Should be odd number (automatically corrected). Default 100 * dim(x) + 1.
	maxFunc2	Number of function evaluations in 2nd call to glcFast.
	maxFunc3	If glcFast is not feasible after maxFunc1 function evaluations, it will be repeat- edly called (warm start) doing maxFunc1 function evaluations until maxFunc3 function evaluations reached.
	ProbL	Structure to be used in the local search. By default the same problem structure as in the global search is used, Prob (see below). Using a second structure is important if optimal continuous variables may take values on bounds. glcFast used for the global search only converges to the simple bounds in the limit, and therefore the simple bounds may be relaxed a bit in the global search. Also, if the global search has difficulty fulfilling equality constraints exactly, the lower and upper bounds may be slightly relaxed. But being exact in the local search. Note that the local search is using derivatives, and can utilize given analytic derivatives. Otherwise the local solver is using numerical derivatives or automatic differentiation. If routines to provide derivatives are given in ProbL, they are used. If only one structure Prob is given, glcCluster uses the derivative routines given in the this structure.
	localSolver	Optionally change local solver used ('snopt' or 'npsol' etc.).
	DIRECT	DIRECT subsolver, either glcSolve or glcFast (default).
	localTry	Maximal number of local searches from cluster points. If <= 0, glcCluster stops after clustering. Default 100.
	maxDistmin	The minimal number used for clustering, default 0.05.
	optParam	Structure with special fields for optimization parameters, see Table 141. Fields used are: PriLev, cTol, IterPrint, MaxIter, MaxFunc, EpsGlob, fGoal, eps f, eps x.
	MIP.IntVars	Structure in Prob, Prob.MIP. If empty, all variables are assumed non-integer (LP problem). If length(I ntV ars) > 1 ==> length(I ntV ars) == length(c) should hold. Then I ntV ars(i) == 1 ==> x(i) integer. I ntV ars(i) == 0 ==> x(i) real. If length(I ntV ars) < n, IntVars is assumed to be a set of indices. It is advised to number the integer values as the first variables, before the continuous. The tree search will then be done more efficiently.
varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Matrix with all points giving the function value f_k.
	f_k	Function value at optimum.
	c_k	Nonlinear constraints values at x_k.
	Iter	Number of iterations.
	FuncEv	Number function evaluations.
	maxTri	Maximum size of any triangle.
	ExitText	Text string giving ExitFlag and Inform information.
	Cluster	Subfield with clustering information
	x_k	Matrix with best cluster points.
	f_k	Row vector with f(x) values for each column in Cluster.x_k.
	maxDist	maxDist used for clustering.
	minDist	vector of all minimal distances between points.

Description

The routine glcCluster implements an extended version of DIRECT, see \[52\], that handles problems with both nonlinear and integer constraints.

DIRECT is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. Since no such constant is used, there is no natural way of defining convergence (except when the optimal function value is known). Therefore glcCluster is run for a predefined number of function evaluations and considers the best function value found as the optimal one. It is possible for the user to restart glcCluster with the final status of all parameters from the previous run, a so called warm start Assume that a run has been made with glcCluster on a certain problem for 500 function evaluations. Then a run of e.g. 200 function evaluations more should give the same result as if the run had been using 700 function evaluations in the first place. To do a warm start of glcCluster a flag Prob.WarmStart should be set to one. Then glcCluster is using output previously written to the file glcSave.mat to make the restart.

DIRECT does not explicitly handle equality constraints. It works best when the integer variables describe an ordered quantity and is less effective when they are categorical.

M-files Used

iniSolve.m, endSolve.m, glcFast.m

glcDirect

Purpose

Solve global mixed-integer nonlinear programming problems.

glcDirect solves problems of the form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\& c_{L} & \leq & c(x) & \leq & c_{U} \\& & & x_i \mathrm{\ \ integer} & & i \in I \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R}^{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A\in \MATHSET{R}^{m_{2}\times n}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_{L},b_{U}\in \MATHSET{R}^{m_{2}}$} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x \in I$, the index subset of $1,...,n$} are restricted to be integers. Recommendation: Put the integers as the first variables. Put low range integers before large range integers. Linear constraints are specially treated. Equality constraints are added as penalties to the objective. Weights are computed automatically, assuming f(x) scaled to be roughly 1 at optimum. Otherwise scale f(x).

glcDirect is a Fortran MEX implementation of glcSolve.

Calling Syntax

Result = glcDirectTL(Prob,varargin)

Result = tomRun('glcDirect', Prob);

Description of Inputs

1 demands||f cALL == 0. This option could save some time if f(x) is a bit costly, however overall performance could on some problems be dramatically worse.

Prob	Problem description structure. The following fields are used:
	Name	Problem name. Used for safety when doing warm starts.
	FUNCS.f	Name of m-file computing the objective function f (x).
	FUNCS.c	Name of m-file computing the vector of constraint functions c(x).
	A	Linear constraints matrix.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	c_L	Lower bounds on the general constraints.
	c U	Upper bounds on the general constraints.
	x_L	Lower bounds for x, must be finite to restrict the search space.
	x_U	Upper bounds for x, must be finite to restrict the search space.
	PriLevOpt	Print Level. This controls both regular printing from glcDirect and the amount of iteration log information to print.
		0 = Silent.
		1 = Warnings and errors printed. Iteration log on iterations im- proving function value.
		2 = Iteration log on all iterations.
		3 = Log for each function evaluation.
		4 = Print list of parameter settings.
		See optParam.IterPrint for more information on iteration log printing.
	WarmStart	If true, > 0, glcDirect reads the output from the last run from Prob.glcDirect.WarmStartInfo if it exists. If it doesn't exist, glcDirect attempts to open and read warm start data from mat-file glcDirectSave.mat. glcDirect uses this warm start information to continue from the last run.
	MaxCPU	Maximum CPU Time (in seconds) to be used.
	MIP	Structure in Prob, Prob.MIP.
	Intvars	If empty, all variables are assumed non-integer (LP problem) If length(IntVars) > 1 ==> length(IntVars) == length(c) should hold Then IntVars(i) == 1 ==> x(i) integer. IntVars(i) == 0 ==> x(i) real If length(IntVars) < n, IntVars is assumed to be a set of indices. It is advised to number the integer values as the first variables, before the continuous. The tree search will then be done more efficiently.
	fIP	An upper bound on the optimal f(x) value. If empty, set as Inf.
	xIP	The x-values giving the fIP value. If fIP empty and xIP given, fIP will be computed if xIP nonempty, its feasibility is checked
	glcDirect	Structure with DIRECT algorithm specific parameters. Fields used:
	fcALL	=0 (Default). If linear constraints cannot be feasible anywhere inside rectangle, skip f(x) and c(x) computation for middle point.
		=1 Always compute f(x) and c(x), even if linear constraints are not feasible anywhere in rectangle. Do not update rates of change for the constraints.
		=2 Always compute f(x) and c(x), even if linear constraints are not feasible anywhere in rectangle. Update rates of change constraints.
	useRoC	=1 (Default). Use original Rate of Change (RoC) for constraints to weight the constraint violations in selecting which rectangle divide.
		=0 Avoid RoC, giving equal weights to all constraint violations. Suggested if difficulty to find feasible points. For problems where linear constraints have been added among the nonlinear (NOT RECOMMENDED; AVOID!!!), then option useRoc=0 has been successful, whereas useRoC completely fails.
		=2 Avoid RoC for linear constraints, giving weight one to these constraint violations, whereas the nonlinear constraints use RoC.
		=3 Use RoC for nonlinear constraints, but linear constraints are not used to determine which rectangle to use.
	BRANCH	=0 Divide rectangle by selecting the longest side, if ties use the lowest index. This is the Jones DIRECT paper strategy.
		=1 First branch the integer variables, selecting the variable with the least splits. If all integer variables are split, split on the continuous variables as in BRANCH=0. DEFAULT! Normally much more efficient than =0 for mixed- integer problems.
		=2 First branch the integer variables with 1,2 or 3 possible values, e.g \[0,1\],\[0,2\] variables, selecting the variable with least splits. Then branch the other integer variables, selecting the variable with the least splits. If all integer variables are split, split on the continuous variables as in BRANCH=0.
		=3 Like =2, but use priorities on the variables, similar to mipSolve, see Prob.MIP.VarWeight.
	RECTIE	When minimizing the measure to find which new rectangle to try to get feasible, there are often ties, several rectangles have the same minimum. RECTIE = 0 or 1 seems reasonable choices. Rectangles with low index are often larger then the rectangles with higher index. Selecting one of each type could help, but often =0 is fastest.
		=0 Use the rectangle with value a, with lowest index (original).
		=1 (Default): Use 1 of the smallest and 1 of largest rectangles.
		=2 Use the last rectangle with the same value a, not the 1st.
		=3 Use one of the smallest rectangles with same value a.
		=4 Use all rectangles with the same value a, not just the 1st.
	EqConFac	Weight factor for equality constraints when adding to objective function f(x) (Default value 10). The weight is computed as EqConFac/"right or left hand side constant value", e.g. if the constraint is Ax <= b, the weight is EqCon- Fac/b If DIRECT just is pushing down the f(x) value instead of fulfilling the equality constraints, increase EqConFac.
	AxFeas	Set nonzero to make glcDirect skip f(x) evaluations, when the linear constraints are infeasible, and still no feasible point has been found. The default is 0. Value
	fEqual	All points with function values within tolerance fEqual are considered to be global minima and returned. Default 1E-10.
	LinWeight	\|a(i, :)\|\| for linear constraints. Balance be- tween linear and nonlinear constraints. Default 0.1. The higher value, the less influence from linear constraints.
	alpha	Exponential forgetting factor in RoC computation, default 0.9.
	AvIter	How many values to use in startup of RoC computation before switching to exponential smoothing with forgetting factor alpha. Default 50.
	optParam	Structure with special fields for optimization parameters, see Table 141 on page 229.
		Fields used by glcDirect are: IterPrint, bTol, cTol, MaxIter, MaxFunc, EpsGlob, fGoal, eps f, eps x.
	varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Matrix with all points giving the function value f k.
	f_k	Function value at optimum.
	c_k	Nonlinear constraints values at x k.
	Iter	Number of iterations.
	FuncEv	Number function evaluations.
	maxTri	Maximum size of any triangle.
	ExitText	Text string giving ExitFlag and Inform information.
	ExitFlag	0 = Normal termination, max number of iterations func.evals reached.
		2 = Some upper bounds below lower bounds.
		4 = Numerical trouble, and cannot continue.
		7 = Reached maxFunc or maxIter, NOT feasible.
		8 = Empty domain for integer variables.
		10= Input errors.
	Inform	1 = Function value f is less than fGoal.
		2 = Absolute function value f is less than fTol, only if fGoal = 0 or Relative error in function value f is less than fTol, i.e. abs(f-fGoal)/abs(fGoal) <= fTol.
		3 = Maximum number of iterations done.
		4 = Maximum number of function evaluations done.
		5 = Maximum number of function evaluations would most likely be too many in the next iteration, save warm start info, stop.
		6 = Maximum number of function evaluations would most likely be too many in the next iteration, because 2 * sLen >= maxFDim - nFunc, save warm start info, stop.
		7 = Space is dense.
		8 = Either space is dense, or MIP is dense.
		10= No progress in this run, return solution from previous one.
		91= Infeasible.
		92= No rectangles to work on.
		93= sLen = 0, no feasible integer solution exists.
		94= All variables are fixed.
		95= There exist free constraints.
	glcDirect	Substructure for glcDirect specific result data.
	convFlag	Converge status flag from solver.
		Structure with warm start information. Use WarmDefDIRECT to reuse this
	WarmStartInfo	information in another run.
	glcDirectSave.mat	To make a warm start possible, glcDirect saves the following information in the structure Result.glcDirect.WarmStartInfo and file glcDirectSave.mat (for internal solver use only):
	C	Matrix with all rectangle centerpoints, in \[0,1\]-space.
	D	Vector with distances from centerpoint to the vertices.
	F	Vector with function values.
	G	Matrix with constraint values for each point.
	Iter	Number of iterations.
	Name	Name of the problem. Used for security if doing warm start.
	Split	Split(i, j) is the number of splits along dimension i of rectangle j.
	Tr	T r(i) is the number of times rectangle i has been trisected.
	fMinIdx	Indices of the currently best points.
	fMinEQ	sum(abs(infeasibilities)) for minimum points, 0 if no equalities.
	glcfMin	Best function value found at a feasible point.
	feasible	Flag indicating if a feasible point has been found.
	ignoreidx	Rectangles to be ignored in the rectangle selection procedure.
	roc	Rate of change s, for each constraint.
	s0	Sum of observed rate of change s0 in the objective.
	t	t(i) is the total number of splits along dimension i.

Description

The routine glcDirect implements an extended version of DIRECT, see \[52\], that handles problems with both nonlinear and integer constraints. The algorithm in glcDirect is a Fortran MEX implementation of the algorithm in glcSolve.

DIRECT is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. Since no such constant is used, there is no natural way of defining convergence (except when the optimal function value is known). Therefore glcDirect is run for a predefined number of function evaluations and considers the best function value found as the optimal one. It is possible for the user to restart glcDirect with the final status of all parameters from the previous run, a so called warm start. Assume that a run has been made with glcDirect on a certain problem for 500 function evaluations. Then a run of e.g. 200 function evaluations more should give the same result as if the run had been using 700 function evaluations in the first place. To do a warm start of glcDirect a flag Prob.WarmStart should be set to one. Then glcDirect will use output previously written to the file glcDirectSave.mat (or the warm start structure) to make the restart.

DIRECT does not explicitly handle equality constraints. It works best when the integer variables describe an ordered quantity and is less effective when they are categorical.

M-files Used

iniSolve.m, endSolve.m and glcSolve.m.

Warnings

A significant portion of glcDirect is coded in Fortran MEX format. If the solver is aborted, it may have allocated memory for the computations which is not returned. This may lead to unpredictable behavior if glcDirect is started again. To reduce the risk of trouble, do "clear mex" if a run has been aborted.

glcSolve

Purpose

Solve general constrained mixed-integer global optimization problems.

glcSolve solves problems of the form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\& c_{L} & \leq & c(x) & \leq & c_{U} \\& & & x_i \mathrm{\ \ integer} & & i \in I \\\end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R}^{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c(x),c_{L},c_{U}\in\MATHSET{R}^{m_{1}}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A\in \MATHSET{R}^{m_{2}\times n}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_{L},b_{U}\in \MATHSET{R}^{m_{2}}$} .

The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x \in I$} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $1,...,n$} are restricted to be integers. Recommendation: Put the integers as the first variables. Put low range integers before large range integers. Linear constraints are specially treated. Equality constraints are added as penalties to the objective. Weights are computed automatically, assuming f(x) scaled to be roughly 1 at optimum. Otherwise scale f(x).

Calling Syntax

Result = glcSolve(Prob,varargin)

Result = tomRun('glcSolve', Prob);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	c_L	Lower bounds on the general constraints.
	c_U	Upper bounds on the general constraints.
	MIP	Structure in Prob, Prob.MIP.
	Intvars	If empty, all variables are assumed non-integer (LP problem) If length(IntVars)> 1 ==> length(IntVars) == length(c) should hold Then IntVars(i) == 1 ==> x(i) integer. IntVars(i) == 0 ==> x(i) real If length(IntVars) < n, IntVars is assumed to be a set of indices. It is advised to number the integer values as the first variables, before the continuous. The tree search will then be done more efficiently.
	fIP	An upper bound on the optimal f(x) value. If empty, set as Inf.
	xIP	The x-values giving the fIP value. If fIP empty and xIP given, fIP will be computed if xIP nonempty, its feasibility is checked
	x_L	Lower bounds for x, must be given to restrict the search space. Any lower bounds that are inf are changed to -10000.
	x_U	Upper bounds for x, must be given to restrict the search space. Any upper bounds that are inf are changed to 10000.
	FUNCS.f	Name of m-file computing the objective function f (x).
	FUNCS.c	Name of m-file computing the vector of constraint functions c(x).
	Name	Name of the problem. Used for security if doing warm start.
	PriLevOpt	Print level. 0 = silent. 1 = some printing. 2 = print each iteration.
	WarmStart	If true (> 0), glcSolve reads the output from the last run from the mat-file glcSave.mat, and continues from the last run. NOTE: All rectangles that are fathomed in the previous run are deleted. This saves space and computational time and enables solving larger problems and more function evaluations to be done.
	MaxCPU	Maximal CPU Time (in seconds) to be used.
	glcDirect	Structure with DIRECT algorithm specific parameters. Fields used:
	fcALL	=0 (Default). If linear constraints cannot be feasible anywhere inside rectangle, skip f(x) and c(x) computation for middle point.
		=1 Always compute f(x) and c(x), even if linear constraints are not feasible anywhere in rectangle. Do not update rates of change for the constraints.
		=2 Always compute f(x) and c(x), even if linear constraints are not feasible anywhere in rectangle. Update rates of change constraints.
	useRoC	=1 (Default). Use original Rate of Change (RoC) for constraints to weight the constraint violations in selecting which rectangle divide.
		=0 Avoid RoC, giving equal weights to all constraint violations. Suggested if difficulty to find feasible points. For problems where linear constraints have been added among the nonlinear (NOT RECOMMENDED; AVOID!!!), then option useRoc=0 has been successful, whereas useRoC completely fails.
		=2 Avoid RoC for linear constraints, giving weight one to these constraint violations, whereas the nonlinear constraints use RoC.
		=3 Use RoC for nonlinear constraints, but linear constraints are not used to determine which rectangle to use.
	BRANCH	=0 Divide rectangle by selecting the longest side, if ties use the lowest index. This is the Jones DIRECT paper strategy.
		=1 First branch the integer variables, selecting the variable with the least splits. If all integer variables are split, split on the continuous variables as in BRANCH=0. DEFAULT! Normally much more efficient than =0 for mixed- integer problems.
		=2 First branch the integer variables with 1,2 or 3 possible values, e.g \[0,1\],\[0,2\] variables, selecting the variable with least splits. Then branch the other integer variables, selecting the variable with the least splits. If all integer variables are split, split on the continuous variables as in BRANCH=0.
		=3 Like =2, but use priorities on the variables, similar to mipSolve, see Prob.MIP.VarWeight.
	RECTIE	When minimizing the measure to find which new rectangle to try to get feasible, there are often ties, several rectangles have the same minimum. RECTIE = 0 or 1 seems reasonable choices. Rectangles with low index are often larger then the rectangles with higher index. Selecting one of each type could help, but often =0 is fastest.
		=0 Use the rectangle with value a, with lowest index (original).
		=1 (Default): Use 1 of the smallest and 1 of largest rectangles.
		=2 Use the last rectangle with the same value a, not the 1st.
		=3 Use one of the smallest rectangles with same value a.
		=4 Use all rectangles with the same value a, not just the 1st.
	EqConFac	Weight factor for equality constraints when adding to objective function f(x) (Default value 10). The weight is computed as EqConFac/"right or left hand side constant value", e.g. if the constraint is Ax <= b, the weight is EqCon- Fac/b If DIRECT just is pushing down the f(x) value instead of fulfilling the equality constraints, increase EqConFac.
	AxFeas	Set nonzero to make glcSolve skip f(x) evaluations, when the linear constraints are infeasible, and still no feasible point has been found. The default is 0. Value 1 demands f cALL == 0. This option could save some time if f(x) is a bit costly, however overall performance could on some problems be dramatically worse.
	fEqual	All points with function values within tolerance fEqual are considered to be global minima and returned. Default 1E-10.
	LinWeight	RateOfChange = LinWeight * ||a(i, :)|| for linear constraints. Balance be- tween linear and nonlinear constraints. Default 0.1. The higher value, the less influence from linear constraints.
	alpha	Exponential forgetting factor in RoC computation, default 0.9.
	AvIter	How many values to use in startup of RoC computation before switching to exponential smoothing with forgetting factor alpha. Default 50.
	If WarmStart is chosen, the following fields in glcSave.mat are also used and contains information from the previous run:
	C	Matrix with all rectangle centerpoints.
	D	Vector with distances from centerpoint to the vertices.
	F	Vector with function values.
	G	Matrix with constraint values for each point.
	Name	Name of the problem. Used for security if doing warm start.
	Split	Split(i, j) is the number of splits along dimension i of rectangle j.
	T	T (i) is the number of times rectangle i has been trisected.
	fMinEQ	sum(abs(infeasibilities)) for minimum points, 0 if no equalities.
	fMinIdx	Indices of the currently best points.
	feasible	Flag indicating if a feasible point has been found.
	glcfmin	Best function value found at a feasible point.
	iL	iL(i, j) is the lower bound for rectangle j in integer dimension I (i).
	iU	iU (i, j) is the upper bound for rectangle j in integer dimension I (i).
	ignoreidx	Rectangles to be ignored in the rectangle selection procedure.
	s	s(j) is the sum of observed rates of change for constraint j.
	s_0	s_0 is used as s(0).
	t	t(i) is the total number of splits along dimension i.
	SubRes	Additional output from nlp f, if suboptimization done.
	optParam	Structure with special fields for optimization parameters, see Table 141 on page 229.
		Fields used by glcSolve are: IterPrint, bTol, cTol, MaxIter (defaultmax(5000, n * 1000)), MaxFunc (default max(10000, n * 2000)), EpsGlob, fGoal, eps_f, eps_x.
varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Matrix with all points giving the function value f_k.
	f_k	Function value at optimum.
	c_k	Nonlinear constraints values at x_k.
	glcSave.mat	Special file containing:
	C	Matrix with all rectangle centerpoints.
	D	Vector with distances from centerpoint to the vertices.
	F	Vector with function values.
	G	Matrix with constraint values for each point.
	Name	Name of the problem. Used for security if doing warm start.
	Split	Split(i, j) is the number of splits along dimension i of rectangle j.
	T	T (i) is the number of times rectangle i has been trisected.
	fMinEQ	sum(abs(infeasibilities)) for minimum points, 0 if no equalities.
	fMinIdx	Indices of the currently best points.
	feasible	Flag indicating if a feasible point has been found.
	glcf min	Best function value found at a feasible point.
	iL	iL(i, j) is the lower bound for rectangle j in integer dimension I (i).
	iU	iU (i, j) is the upper bound for rectangle j in integer dimension I (i).
	ignoreidx	Rectangles to be ignored in the rectangle selection procedure.
	s	s(j) is the sum of observed rates of change for constraint j.
	s_0	s_0 is used as s(0).
	t	t(i) is the total number of splits along dimension i.
	Iter	Number of iterations.
	FuncEv	Number function evaluations.
	maxTri	Maximum size of any triangle.
	ExitText	Text string giving ExitFlag and Inform information.
	ExitFlag	0 - Reached maxFunc or maxIter.
		2 - Some upper bounds below lower bounds.
		7 - Reached maxFunc or maxIter, NOT feasible.
		8 - Empty domain for integer variables.
	Inform	1 = Function value f is less than fGoal. 2 = Absolute function value f is less than fTol, only if fGoal = 0 or Relative error in function value f is less than fTol, i.e. abs(f-fGoal)/abs(fGoal) <= fTol. 3 = Maximum number of iterations done. 4 = Maximum number of function evaluations done. 9 = Max CPU Time reached. 91= Infeasible. 99= Input error, see ExitFlag.

Description

The routine glcSolve implements an extended version of DIRECT, see \[52\], that handles problems with both nonlinear and integer constraints.

DIRECT is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. Since no such constant is used, there is no natural way of defining convergence (except when the optimal function value is known). Therefore glcSolve is run for a predefined number of function evaluations and considers the best function value found as the optimal one. It is possible for the user to restart glcSolve with the final status of all parameters from the previous run, a so called warm start Assume that a run has been made with glcSolve on a certain problem for 500 function evaluations. Then a run of e.g. 200 function evaluations more should give the same result as if the run had been using 700 function evaluations in the first place. To do a warm start of glcSolve a flag Prob.WarmStart should be set to one. Then glcSolve is using output previously written to the file glcSave.mat to make the restart.

DIRECT does not explicitly handle equality constraints. It works best when the integer variables describe an ordered quantity and is less effective when they are categorical.

M-files Used

iniSolve.m, endSolve.m

infLinSolve

Purpose

Finds a linearly constrained minimax solution of a function of several variables with the use of any suitable TOMLAB solver. The decision variables may be binary or integer.

infLinSolve solves problems of the type:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{\max Dx} \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\\end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_L,x_U \in \Rdim{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \Rdim{m_1}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in\Rdim{m_1 \times n}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $D \in \Rdim{m_2 \times n}$} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x \in I$} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $1,...,n$} are restricted to be integers. The different objectives are stored in D row-wise.

Calling Syntax

Result=infLinSolve(Prob,PriLev)

Description of Inputs

Prob	Structure Prob. Prob must be defined. Best is to use Prob = lp/mipAssign(.....), if using the TQ format. Prob.QP.D matrix should then be set to the rows (Prob.QP.c ignored).
	PriLev	Print level in infLinSolve.
		= 0 Silent except for error messages.
		> 0 Print summary information about problem transformation.
		Calls PrintResult with specified PriLev.
		= 2 Standard output from PrintResult (default).
	Extra fields used in Prob:
	SolverInf	Name of the TOMLAB solver. Valid names are: cplex, minos, snopt, xa and more. SeeSolverList('lp'); or SolverList('mip');
	QP.D	The rows with the different objectives.
	f_Low	Lower bound on the objective (optional).
	f_Upp	Upper bound on the objective (optional).

Description of Outputs

Result	Structure with results from optimization. Output depends on the solver used.
		The fields x_k, f_k, x_0, xState, bState, v_k are transformed back to match the original problem.
		The output in Result.Prob is the result after infLinSolve transformed the problem, i.e. the altered Prob structure

Description

The linear minimax problem is solved in infLinSolve by rewriting the problem as a linear optimization problem. One additional variable Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $z\in \MATHSET{R}$} , stored as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x_{n+1}$} is added and the problem is rewritten as:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x z}\\ \\\mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n)^T & \leq & x_U \\& -\infty & \leq & z & \leq & \infty \\& b_L & \leq & A x &\leq & b_U \\& -\infty & \leq & D x - z e & \leq & 0 \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $e \in \Rdim{N},\; e(i)=1 \ \forall i$} .

To handle cases where a row in D\*x is taken the absolute value of: minmax\|D * x\|, expand the problem with extra residuals with the opposite sign: \[D * x; -D * x\].

See Also

lpAssign.

infSolve

Purpose

Find a constrained minimax solution with the use of any suitable TOMLAB solver.

infSolve solves problems of the type:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{\max r(x)} \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\& c_L & \leq & c(x) & \leq & c_U \\ \end{array} }

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

Calling Syntax

Result=infSolve(Prob,PriLev)

Description of Inputs

Prob	Problem description structure. Should be created in the cls format. infSolve uses two special fields in Prob:
	SolverInf	Name of solver used to solve the transformed problem.
		Valid choices are conSolve, nlpSolve, sTrustr and clsSolve.
		If TOMLAB /SOL is installed: minos, snopt, npopt.
	InfType	1 - constrained formulation (default).
		2 - LS penalty approach (experimental).
	The remaining fields of Prob should be defined as required by the selected subsolver.
	PriLev	Print level in infSolve.
		= 0 Silent except for error messages.
		> 0 Print summary information about problem transformation.
		Calls PrintResult with specified PriLev.
		= 2Standard output from PrintResult (default).

Description of Outputs

Result	Structure with results from optimization. Output depends on the solver used.
	The fields x_k, r_k, J_k, c_k, cJac, x_0, xState, cState, v_k are transformed back to match the original problem.
	g_k is calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \VAR{J\_k} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $^T$} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle }} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\cdot$} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \VAR{r\_k}} .
	The output in Result.Prob is the result after infSolve transformed the problem, i.e. the altered Prob structure

Description

The minimax problem is solved in infSolve by rewriting the problem as a general constrained optimization problem. One additional variable Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $z\in \MATHSET{R}$} , stored as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x_{n+1}$} is added and the problem is rewritten as:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x z}\\\\\mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n)^T & \leq & x_U \\& -\infty & \leq & z & \leq & \infty \\& b_L & \leq & A x & \leq & b_U \\& c_L & \leq & c(x) & \leq & c_U \\& -\infty & \leq & r(x) - z e & \leq & 0 \\\end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $e \in \Rdim{N},\; e(i)=1 \ \forall i$} .

To handle cases where an element Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $r_i(x)$} in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $r(x)$} appears in absolute value: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $\min \max |r_i(x)|$} , expand the problem with extra residuals with the opposite sign: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $ [ r_i(x); -r_i(x) ] $}

Examples

minimaxDemo.m.

See Also

clsAssign.

linRatSolve

Purpose

Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables are not supported.

linRatSolve solves problems of the type:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c1,c2,x,x_L,x_U \in \Rdim{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \Rdim{m_1}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A \in \Rdim{m_1 \times n}$} .

Calling Syntax

Result=linRatSolve(Prob,PriLev)

Description of Inputs

Prob	Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored).
PriLev	Print level in linRatSolve.
	= 0Silent except for error messages.
	> 0 Print summary information about problem transformation.
	Calls PrintResult with specified PriLev.
	= 2 Standard output from PrintResult (default).
Extra fields used in Prob:
SolverRat	Name of the TOMLAB solver. Valid names are: cplex, minos, snopt, xa and more. See SolverList('lp');
QP.c1	The numerator in the objective.
QP.c2	The denominator in the objective.
z1_L	Lower bound for z1 (default 1e-5). See description below

Description of Outputs

Result	Structure with results from optimization. Output depends on the solver used.
	The fields x_k, f_k, x_0, xState, bState, v_k are transformed back to match the original problem.
	The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure

Description

The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{ccc} \\ z1 & = & 1 / (c2 x) \\ z2 & = & x z1 \\ z1 (c1 x) & = & (c1 z1 x) = c1 z2\\ \end{array} }

The problem then becomes:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x c1 z2}\\\\\mbox{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\ \end{array} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A1 \in \Rdim{N},\; A1=speye(N)$} .

OBSERVE the denominator c2x must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).

See Also

lpAssign.

lpSimplex

Purpose

Solve general linear programming problems.

lpSimplex solves problems of the form

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

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $c \in \MATHSET{R}^{n}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A\in \MATHSET{R}^{m\times n}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_{L},b_{U} \in \MATHSET{R}^{m}$} .

Calling Syntax

Result = lpSimplex(Prob) or

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

Description of Inputs

Prob	Problem description structure. The following fields are used:
	QP.c	Constant vector.
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	Solver.Alg	Variable selection rule to be used:
		0: Minimum reduced cost.
		1: Bland's rule (default).
		2: Minimum reduced cost. Dantzig's rule.
	QP.B	Active set B 0 at start:
		B(i) = 1: Include variable x(i) is in basic set.
		B(i) = 0: Variable x(i) is set on its lower bound.
		B(i) = -1: Variable x(i) is set on its upper bound.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: MaxIter, PriLev, wait, eps_f, eps_Rank, xTol and bTol.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum, c.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150.
	ExitFlag	0: Optimal solution found.
		1: Maximal number of iterations reached.
		2: Unbounded feasible region.
		5: Too many active variables in given initial point.
		6: No feasible point found with Phase 1.
		10: Errors in input parameters.
		11: Illegal initial x as input.
	Inform	If ExitF lag > 0, I nf orm = ExitF lag.
	QP.B	Optimal active set. See input variable QP.B.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Iter	Number of iterations.
	FuncEv	Number of function evaluations. Equal to Iter. ConstrEv Number of constraint evaluations. Equal to Iter.
	Prob	Problem structure used.

Description

The routine lpSimplex implements an active set strategy (Simplex method) for Linear Programming using an additional set of slack variables for the linear constraints. If the given starting point is not feasible then a Phase I objective is used until a feasible point is found.

M-files Used

ResultDef.m

See Also

qpSolve