TOMLAB Solver Reference

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This page is part of the TOMLAB Manual. See TOMLAB Manual.

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.

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

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

• clsSolve

conSolve

Solve general constrained nonlinear optimization problems.

• conSolve

cutPlane

Solve mixed integer linear programming problems (MIP).

• cutPlane

DualSolve

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

• dualSolve

expSolve

Solve exponential fitting problems for given number of terms p.

• expSolve

glbDirect

Solve box-bounded global optimization problems.

• glbDirect

glbSolve

Solve box-bounded global optimization problems.

• glbSolve

glcCluster

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

• glcCluster

glcDirect

Solve global mixed-integer nonlinear programming problems.

• glcDirect

glcSolve

Solve general constrained mixed-integer global optimization problems.

• glcSolve

infLinSolve

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

infSolve

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

• infSolve

linRatSolve

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

lpSimplex

Solve general linear programming problems.

• lpSimplex

L1Solve

Find a constrained L1 solution of a function of several variables with the use of any suitable nonlinear TOMLAB solver.

• L1Solve

MilpSolve

Solve mixed integer linear programming problems (MILP).

• MilpSolve

minlpSolve

Branch & Bound algorithm for Mixed-Integer Nonlinear Programming (MINLP) with convex or nonconvex sub problems using NLP relaxation (Formulated as minlp-IP).

• minlpSolve

mipSolve

Solve mixed integer linear programming problems (MIP).

• mipSolve

multiMin

multiMin solves general constrained mixed-integer global optimization problems. It tries to find all local minima by a multi-start method using a suitable nonlinear programming subsolver.

• multiMin

multiMINLP

multiMINLP solves general constrained mixed-integer global nonlinear optimization problems.

• multiMINLP

nlpSolve

Purpose

Solve general constrained nonlinear optimization problems.

nlpSolve 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},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 (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 (unknown function "\MATHSET"): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} .

Calling Syntax

Result = nlpSolve(Prob, varargin)

Result = tomRun('nlpSolve', 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.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	PriLevOpt	Print level: 0 Silent, 1 Final result, 2 Each iteration, short, 3 Each iteration, more info, 4 Matrix update information.
	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.
	FUNCS.d2c	Name of m-file computing the second derivatives of the constraints, weighted by an input Lagrange vector
	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.
	SolverFP	Name of the solver used for FP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: eps_g, eps_x, MaxIter, wait, size_x, method, IterPrint, xTol, bTol, cTol, and QN InitMatrix.
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.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	c_k	Value of constraints at optimum.
	H_k	Hessian value at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	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 general constraint.
	Inform	Type of convergence.
	ExitFlag	Flag giving exit status.
	ExitText	0: Convergence. Small step. Constraints fulfilled.
		1: Infeasible problem?
		2: Maximal number of iterations reached.
		3: No progress in either function value or constraint reduction.
	Inform	1: Iteration points are close.
		2: Small search direction
		3: Function value below given estimate. Restart with lower fLow if minimum not reached.
		4: Projected gradient small.
		10: Karush-Kuhn-Tucker conditions fulfilled.
	Iter	Number of iterations.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine nlpSolve implements the Filter SQP by Roger Fletcher and Sven Leyffer presented in the paper \[21\].

M-files Used

tomSolve.m, ProbCheck.m, iniSolve.m, endSolve.m

See Also

conSolve, sTrustr

pdcoTL

Purpose

pdcoTL solves linearly constrained convex nonlinear optimization problems of the kind

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

where $f(x)$ is a convex nonlinear function, Failed to parse (unknown function "\RR"): {\displaystyle x,x_L,x_U \in \RR^n} , Failed to parse (unknown function "\RR"): {\displaystyle A\in \RR^{m \times n}} and Failed to parse (unknown function "\RR"): {\displaystyle b_L, b_U \in \RR^m} .

Calling Syntax

Result=tomRun('pdco',Prob,...);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	x_0	Initial x vector, used if non-empty.
	A	The linear constraint matrix.
	b_L,b_U	Lower and upper bounds for the linear constraints.
	PriLevOpt	Print level in pdsco solver. If > 0: prints summary information.
	SOL	Structure with SOL special parameters:
	pdco	Options structure with fields as defined by pdcoSet.
	d1	Primal regularization vector. Must be a positive vector (length n) or scalar, in which case D1 = diag(d1) is used. Default: 10-4 .
	d2	Dual regularization vector. Must be a positive vector (length m) or a scalar value, in which case D2 = diag(d2) is used. Default: 10-4 .
	y0	Initial dual parameters for linear constraints (default 0)
	z0	Initial dual parameters for simple bounds (default 1/N ) xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method.
	xsize	Estimate of the biggest x at the solution. (default 1/N )
	zsize	Estimate of the biggest z at the solution. (default 1/N )
	optParam	Structure with optimization parameters. The following fields are used:
	MaxIter	Maximum number of iterations. (Prob.SOL.pdco.MaxIter).
	MinorIter	Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter).
	eps_x	Accuracy for satisfying x1 . * z1 = 0, x2 .z1 = 0, where z = z1 - z2 and z1 , z2 > 0.(Prob.SOL.pdco.OptTol)
	bTol	Accuracy for satisfying Ax + D2 r = b, AT y + z = ?f (x) and x - x1 = bL , x +x2 = bU , where x1 , x2 > 0 (Prob.SOL.pdco.FeaTol)
	wait	0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait)

Description of Outputs

Result	Structure with result from optimization. The following fields are set by pdcoTL
	x_k	Solution vector
	f_k	Function value at optimum
	g_k	Gradient of the function at the solution
	H_k	Hessian of the function at the solution, diagonal only
	x_0	Initial solution vector
	f_0	Function value at start, x = x_0
	xState	State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3;
	bState	State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
	v_k	Lagrangian multipliers (orignal bounds + constraints )
	y_k	Lagrangian multipliers (for bounds + dual solution vector) The full \[z; y\] vec- tor as returned from pdco, including slacks and extra linear constraints after rewriting constraints: -inf < b L < A * x < b U < inf ; non-inf lower AND upper bounds
	ExitFlag	Tomlab Exit status from pdco MEX
	Inform	pdcoinformation parameter: 0 = Solution found;
	0	Solution found
	1	Too many iterations
	2	Linesearch failed too often
	Iter	Number of iterations
	FuncEv	Number of function evaluations
	GradEv	Number of gradient evaluations
	HessEv	Number of Hessian evaluations
	Solver	Name of the solver ('pdco')
	SolverAlgorithm	Description of the solver

Description

pdco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).

The problem (19) is first reformulated into SOL PDCO form:

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & & & Ax & = & b \\ {} & \multicolumn{5}{l}{r \mathrm{\ unconstrained}} \\ \end{array} }

The problem actually solved by pdco is

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{lllcll} \min\limits_{x,r} & \multicolumn{5}{l}{\phi(x) + \frac{1}{2}\|D_1 x\|^2 + \frac{1}{2}\|r\|^2 } \\ \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & Ax & + & D_{2}r & = & b \\ \end{array} }

where $D_{1}$ and $D_{2}$ are positive-definite diagonal matrices defined from $d_{1}$, $d_{2}$ given in Prob.SOL.d1 and Prob.SOL.d2.

In particular, $d_{2}$ indicates the accuracy required for satisfying each row of $Ax=b$. See pdco for a detailed discussion of $D_{1}$ and $D_{2}$. Note that in pdco, the objective $f(x)$ is denoted $\phi (x)$, $bl==x\_L$ and $bu==x\_U$.

Examples

Problem 14 and 15 in tomlab/testprob/con prob.m.m are good examples of the use of pdcoTL.

M-files Used

pdcoSet.m, pdco.m, Tlsqrmat.m

See Also

pdscoTL.m

pdscoTL

Purpose

pdscoTL solves linearly constrained convex nonlinear optimization problems of the kind

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

where $f(x)$ is a convex separable nonlinear function, Failed to parse (unknown function "\RR"): {\displaystyle x,x_L,x_U \in \RR^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 \RR^{m \times n}} and Failed to parse (unknown function "\RR"): {\displaystyle b_L, b_U \in\RR^m} .

Calling Syntax

Result=tomRun('pdsco',Prob,...);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	x_0	Initial x vector, used if non-empty.
	A	The linear constraints coefficient matrix.
	b_L,b_U	Lower and upper bounds for the linear constraints.
	HessPattern	Non-zero pattern for the objective function. Only the diagonal is needed. Default if empty is the unit matrix.
	PriLevOpt	Print level in pdsco solver. If > 0: prints summary information.
	SOL	Structure with SOL special parameters:
	pdco	Options structure with fields as defined by pdscoSet.
	gamma	Primal regularization parameter.
	delta	Dual regularization parameter.
	y0	Initial dual parameters for linear constraints (default 0)
	z0	Initial dual parameters for simple bounds (default 1/N )
		xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method.
	xsize	Estimate of the biggest x at the solution. (default 1/N )
	zsize	Estimate of the biggest z at the solution. (default 1/N )
	optParam	Structure with optimization parameters. The following fields are used:
	MaxIter	Maximum number of iterations. (Prob.SOL.pdco.MaxIter).
	MinorIter	Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter).
	eps_x	Accuracy for satisfying x. * z = 0
	bTol	Accuracy for satisfying Ax + r = b, AT y + z = ?f (x) and x - x1 = bL , x + x2 =bU, where x1 , x2 > 0. (Prob.SOL.pdco.FeaTol)
	wait	0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait)

Description of Outputs

Result	Structure with result from optimization. The following fields are set by pdscoTL:
	x_k	Solution vector
	f_k	Function value at optimum
	g_k	Gradient of the function at the solution
	H_k	Hessian of the function at the solution, diagonal only
	x_0	Initial solution vector
	f_0	Function value at start, x = x_0
	xState	State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3;
	bState	State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
	v_k	Lagrangian multipliers (orignal bounds + constraints )
	y k	Lagrangian multipliers (for bounds + dual solution vector) The full \[z; y\] vec- tor as returned from pdsco, including slacks and extra linear constraints after rewriting constraints: -inf < b L < A * x < b U < inf ; non-inf lower AND upper bounds
	ExitFlag	Tomlab Exit status from pdsco MEX
	Inform	pdsco information parameter: 0 = Solution found;
	0	Solution found
	1	Too many iterations
	2	Linesearch failed too often
	Iter	Number of iterations
	FuncEv	Number of function evaluations
	GradEv	Number of gradient evaluations
	HessEv	Number of Hessian evaluations
	Solver	Name of the solver ('pdsco')
	SolverAlgorithm	Description of the solver

Description

pdsco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL). The problem (20) is first reformulated into SOL PDSCO form:

${\begin{array}{clll}\min \limits _{x}&f(x)\\\mathrm {s/t} &x&\geq &x_{U}\\{}&Ax&=&b.\\\end{array}}$

The problem actually solved by pdsco is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lllcll} \min\limits_{x,r} & \multicolumn{5}{l}{f(x) + \frac{1}{2}\|\gamma x\|^2 + \frac{1}{2}\|r / \delta \|^2 } \\ \\ \mathrm{s/t} & & & x & \geq & 0 \\ {} & Ax & + & r & = & b \\ {} & \multicolumn{5}{l}{r \mathrm{\ unconstrained}} \\ \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 \gamma} is the primal regularization parameter, typically small but 0 is allowed. Furthermore, 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 \delta} is the dual regularization parameter, typically small or 1; must be strictly greater than zero.

With positive 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 \gamma,\delta} the primal-dual solution 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,y,z)} is bounded and unique.

See pdsco for a detailed discussion of </math>\gamma</math> 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 \delta} . Note that in pdsco, the objective 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(x)} is denoted 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 \phi(x)} , 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 bl == x\_L} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bu == x\_U} .

Examples

Problem 14 and 15 in tomlab/testprob/con prob.m are good examples of the use of pdscoTL.

M-files Used

pdscoSet.m, pdsco.m, Tlsqrmat.m

See Also

pdcoTL.m

qpSolve

Purpose

Solve general quadratic programming problems.

qpSolve 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) & = & \frac{1}{2}(x)^{T}Fx + 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 F\in \MATHSET{R}^{n\times n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 = qpSolve(Prob) or

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

Description of Inputs

Prob	Problem description structure. The following fields are used:
	QP.F	Constant matrix, the Hessian.
	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.
	optParam	Structure with special fields for optimization parameters, see Table 141.Fields used are: eps f, eps Rank, MaxIter, wait, bTol and PriLev.

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.
	H_k	Hessian value at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150 .
	Iter	Number of iterations.
	ExitFlag	0: OK, see Inform for type of convergence.
		2: Can not find feasible starting point x_0.
		3: Rank problems. Can not find any solution point.
		4: Unbounded solution.
		10: Errors in input parameters.
	Inform	If ExitF lag > 0, I nf orm = ExitF lag, otherwise I nf orm show type of convergence:
		0: Unconstrained solution.
		1: ? = 0.
		2: ? = 0. No second order Lagrange mult. estimate available.
		3: ? and 2nd order Lagr. mult. positive, problem is not negative definite.
		4: Negative definite problem. 2nd order Lagr. mult. positive, but releasing variables leads to the same working set.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

Implements an active set strategy for Quadratic Programming. For negative definite problems it computes eigen- values and is using directions of negative curvature to proceed. To find an initial feasible point the Phase 1 LP problem is solved calling lpSimplex. The routine is the standard QP solver used by nlpSolve, sTrustr and conSolve.

M-files Used

ResultDef.m, lpSimplex.m, tomSolve.m, iniSolve.m, endSolve.m

See Also

qpBiggs, qpe, qplm, nlpSolve, sTrustr and conSolve

slsSolve

Purpose

Find a Sparse Least Squares (sls) solution to a constrained least squares problem with the use of any suitable TOMLAB NLP solver.

slsSolve 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 & \frac{1}{2}r(x)^T 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{m}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in\Rdim{m_1,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 c(x),c_L,c_U \in\Rdim{m_2}} .

The use of slsSolve is mainly for large, sparse problems, where the structure in the Jacobians of the residuals and the nonlinear constraints are utilized by a sparse NLP solver, e.g. SNOPT.

Calling Syntax

Result=slsSolve(Prob,PriLev)

Description of Inputs

Prob	Problem description structure. Should be created in the cls format, preferably by callingProb=clsAssign(...) if using the TQ format.
	slsSolve uses two special fields in Prob:
	SolverL2	Text string with name of the NLP solver used for solving the reformulated problem. Valid choices are conSolve, nlpSolve, sTrustr, clsSolve. Suitable SOL solvers, if available: minos, snopt, npopt.
	L2Type	Set to 1 for standard constrained formulation. Currently this is the only allowed choice.
	All other fields should be set as expected by the nonlinear solver selected. In particular:
	A	Linear constraint matrix.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	c_L	Upper bounds on the nonlinear constraints.
	c_U	Lower bounds on the nonlinear constraints.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	ConsPattern	The nonzero pattern of the constraint Jacobian.
	JacPattern	The nonzero pattern of the residual Jacobian.
		Note that Prob.LS.y must be of correct length if JacPattern is empty (but ConsPattern is not). slsSolve will create the new Prob.ConsPattern to be used by the nonlinear solver using the information in the supplied ConsPattern and JacPattern.
	PriLev	Print level in slsSolve. Default value is 2.
	0	Silent except for error messages.
	> 1	Print summary information about problem transformation. slsSolve calls Print- Result(Result,PriLev).
	2	Standard output in PrintResult.

Description of Outputs

Result	Structure with results from optimization. The contents of Result depend on which nonlinear solver was used to solved
	slsSolve transforms the following fields of Result back to the format of the original problem:
	x_k	Optimal point.
	r_k	Residual at optimum.
	J_k	Jacobian of residuals at optimum.
	c_k	Nonlinear constraint vector at optimum.
	v_k	Lagrange multipliers.
	g_k	The gradient vector is calculated as J kT · r k.
	cJac	Jacobian of nonlinear constraints at optimum.
	x_0	Starting point.
	xState	State of variables at optimum.
	cState	State of constraints at optimum.
	Result.Prob	The problem structure defining the reformulated problem.

Description

The constrained least squares problem is solved in slsSolve by rewriting the problem as a general constrained optimization problem. A set of m (the number of residuals) extra 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 z=(z_1,z_2,\ldots,z_m)} are added at the end of the vector of unknowns. The reformulated 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}{cccccc} \min \limits_x & \multicolumn{5}{l}{\frac{1}{2} z^T z} \\ \mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n) & \leq & x_U \\ {} & b_L & \leq & Ax & \leq & b_U \\ {} & c_L & \leq & c(x) & \leq & c_U \\ {} & 0 & \leq & r(x) - z & \leq & 0 \\ \end{array} }

is then solved by the solver given by Prob.SolverL2.

Examples

slsDemo.m

M-files Used

iniSolve.m, GetSolver.m

sTrustr

Purpose

Solve optimization problems constrained by a convex feasible region.

sTrustr 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} \\ \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 = sTrustr(Prob, varargin)

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 on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	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.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: eps f, eps g, eps c, eps x, eps Rank, MaxIter, wait, size x, size f, xTol, LowIts, PriLev, method and QN InitMatrix.
	PartSep	Structure with special fields for partially separable functions, see Table 142.
	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.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	c_k	Value of constraints at optimum.
	H_k	Hessian value at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	cJac	Constraint Jacobian at optimum.
	xState	State of each variable, described in Table 150.
	Iter	Number of iterations.
	ExitFlag	Flag giving exit status.
	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: Relative function value reduction low for LowIts iterations.
		5: Iteration points are close and relative function value reduction low for LowIts iterations.
		6: Projected gradient small and relative function value reduction low for LowIts iterations.
		7: Iteration points are close, projected gradient small and relative function value reduction low for LowIts iterations.
		8: Too small trust region.
		9: Trust region small. Iteration points close.
		10: Trust region and projected gradient small.
		11: Trust region and projected gradient small, iterations close.
		12: Trust region small, Relative f(x) reduction low.
		13: Trust region small, Relative f(x) reduction low. Iteration points are close.
		14: Trust region small, Relative f(x) reduction low. Projected gradient small.
		15: Trust region small, Relative f(x) reduction low. Iteration points close, Projected gradient small.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		103: Convergence to saddle point (eigenvalues computed).
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine sTrustr is a solver for general constrained optimization, which uses a structural trust region algorithm combined with an initial trust region radius algorithm (itrr). The feasible region defined by the constraints must be convex. The code is based on the algorithms in \[13\] and \[67\]. BFGS or DFP is used for the Quasi-Newton update, if the analytical Hessian is not used. sTrustr calls internal routine itrr.

M-files Used

qpSolve.m, tomSolve.m, iniSolve.m, endSolve.m

See Also

conSolve, nlpSolve, clsSolve

Tfmin

Purpose

Minimize function of one variable. Find miniumum x in [x_L, x_U] for function Func within tolerance xTol. Solves using Brents minimization algorithm. Reference: "Computer Methods for Mathematical Computations", Forsythe, Malcolm, and Moler, Prentice-Hall, 1976.

Calling Syntax

[x, nFunc] = Tfmin(Func, x_L, x_U, xTol, Prob)

Description of Inputs

Variable	Description
Func	Function of x to be minimized. Func must be defined as:
	f = Func(x) if no 5th argument Prob is given or
	f = Func(x, Prob) if 5th argument Prob is given.
x_L	Lower bound on x.
x_U	Upper bound on x.
xTol	Tolerance on accuracy of minimum.
Prob	Structure (or any Matlab variable) sent to Func. If many parameters are to be sent to Func set them in Prob as a structure. Example for parameters a and b:
	Prob.user.a = a; Prob.user.b = b;
	[x, nFunc] = Tfmin('myFunc',0,1,1E-5,Prob); In myFunc:
	function f = myFunc(x, Prob)
	a = Prob.user.a;
	b = Prob.user.b;
	f = "matlab expression dependent of x, a and b";

Description of Outputs

Variable	Description
x	Solution.
nFunc	Number of calls to Func.

Tfzero

Purpose

Tfzero, TOMLAB fzero routine.

Tfzero, TOMLAB fzero routine.\\ \\Find a zero for 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(x)} in the interval 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, x_U]} . Tfzero searches for a zero of a function 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(x)} between the given scalar values 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} 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_U} until the width of the interval (xLow, xUpp) has collapsed to within a tolerance specified by the stopping criterion, 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 abs(xLow-xUpp) <= 2.*(RelErr*abs(xLow)+AbsErr)} . The method used is an efficient combination of bisection and the secant rule and is due to T. J. Dekker.

Calling Syntax

[xLow, xUpp, ExitFlag] = Tfzero(x L, x U, Prob, x 0, RelErr, AbsErr)

Description of Inputs

Variable	Description
x_L	Lower limit on the zero x to f(x).
x_U	Upper limit on the zero x to f(x).
Prob	Structure, sent to Matlab routine ZeroFunc. The function name should be set in Prob.FUNCS.f0. Only the function will be used, not the gradient.
x_0	An initial guess on the zero to f(x). If empty, x 0 is set as the middle point in [x_L, x_U].
RelErr	Relative error tolerance, default 1E-7.
AbsErr	Absolute error tolerance, default 1E-14.

Description of Outputs

Variable	Description
xLow	Lower limit on the zero x to f(x).
xUpp	Upper limit on the zero x to f(x).
ExitFlag	Status flag 1,2,3,4,5.
	1: xLow is within the requested tolerance of a zero. The interval (xLow, xUpp) collapsed to the requested tolerance, the function changes sign in (xLow, xUpp), and f(x) decreased in magnitude as (xLow, xUpp) collapsed.
	2: f(xLow) = 0. However, the interval (xLow, xUpp) may not have collapsed to the requested tolerance.
	3: xLow may be near a singular point of f(x). The interval (xLow, xUpp) collapsed to the requested tolerance and the function changes sign in (xLow, xUpp), but f(x) increased in magnitude as (xLow, xUpp) collapsed, i.e. abs(f (xLow)) > max(abs(f (xLow - I N )), abs(f (xU pp - I N ))).
	4: No change in sign of f(x) was found although the interval (xLow, xUpp) collapsed to the requested tolerance. The user must examine this case and decide whether xLow is near a local minimum of f(x), or xLow is near a zero of even multiplicity, or neither of these.
	5: Too many (> 500) function evaluations used.

ucSolve

Purpose

Solve unconstrained nonlinear optimization problems with simple bounds on the variables.

ucSolve 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 x,x_{L},x_{U}\in \MATHSET{R} ^{n}} .

Calling Syntax

Result = ucSolve(Prob, varargin)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	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).
	f_Low	Lower bound on function value.
	Solver.Alg	Solver algorithm to be run:
		0: Gives default, either Newton or BFGS.
		1: Newton with subspace minimization, using SVD.
		2: Safeguarded BFGS with inverse Hessian updates (standard).
		3: Safeguarded BFGS with Hessian updates.
		4: Safeguarded DFP with inverse Hessian updates.
		5: Safeguarded DFP with Hessian updates.
		6: Fletcher-Reeves CG.
		7: Polak-Ribiere CG.
		8: Fletcher conjugate descent CG-method.
	Solver.Method	Method used to solve equation system:
		0: SVD (default).
		1: LU-decomposition.
		2: LU-decomposition with pivoting.
		3: Matlab built in QR.
		4: Matlab inversion.
		5: Explicit inverse.
	Solver.Method	Restart or not for C-G method:
		0: Use restart in CG-method each n:th step.
		1: Use restart in CG-method each n:th step.
	LineParam	Structure with line search parameters, see routine LineSearch and Table 140.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: eps absf, eps f, eps g, eps x, eps Rank, MaxIter, wait, size x, xTol, size f, LineSearch, LineAlg, xTol, IterPrint and QN InitMatrix.
	PriLevOpt	Print level.
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.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	H_k	Hessian value at optimum.
	B_k	Quasi-Newton approximation of the Hessian at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150.
	Iter	Number of iterations.
	ExitFlag	0 if convergence to local min. Otherwise errors.
	Inform	Binary code telling type of convergence:
		1: Iteration points are close.
		2: Projected gradient small.
		4: Relative function value reduction low for LowIts iterations.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		104: Convergence to a saddle point.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The solver ucSolve includes several of the most popular search step methods for unconstrained optimization. The search step methods included in ucSolve are: the Newton method, the quasi-Newton BFGS and DFP methods, 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 are using 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. The line search algorithm in the routine LineSearch is a modified version of an algorithm by Fletcher \[20\]. Bound constraints are treated as described in Gill, Murray and Wright \[28\]. The accuracy in the line search is critical for the performance of quasi-Newton BFGS and DFP methods and for the CG methods. If the accuracy parameter Prob.LineParam.sigma is set to the default value 0.9, ucSolve changes it automatically according to:

Prob.Solver.Alg	Prob.LineParam.sigma
4,5 (DFP)	0.2
6,7,8 (CG)	0.01

M-files Used

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

See Also

clsSolve c

Additional solvers

Documentation for the following solvers is only available at http://tomopt.com and in the m-file help.

• goalSolve - For sparse multi-objective goal attainment problems, with linear and nonlinear constraints.

• Tlsqr - Solves large, sparse linear least squares problem, as well as unsymmetric linear systems.

• lsei - For linearly constrained least squares problem with both equality and inequality constraints.

• Tnnls - Also for linearly constrained least squares problem with both equality and inequality constraints.

• qld - For convex quadratic programming problem.