TOMLAB Solver Reference: Difference between revisions

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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

Purpose

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

The parameter Convex (see below) determines if to assume the NLP subproblems are convex or not.

minlpSolve depends on a suitable NLP solver.

minlpSolve solves problems of the form

Failed to parse (unknown function "\multicolumn"): {\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}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \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 (unknown function "\MATHSET"): {\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 $x\in I$ , the index subset of $1,...,n$ are restricted to be integers.

Calling Syntax

Result = tomRun('minlpSolve',Prob,...)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	x_L	Lower bounds on x.
	x_U	Upper bounds on x.
	A	The linear constraint matrix.
	b_L	Lower bounds on linear constraints.
	b_U	Upper bounds on linear constraints.
	c_L	Lower bounds on nonlinear constraints.
	c_U	Upper bounds on nonlinear constraints.
	x_0	Starting point.
	Convex	If Convex==1, assume NLP problems are convex, and only one local NLP solver call is used at each node. If Convex==0 (Default), multiMin is used to do many calls to a local solver to determine the global minima at each node. The global minimum with most components integer valued is chosen.
	MaxCPU	Maximal CPU Time (in seconds) to be used by minlpSolve, stops with best point found
	PriLev	Print level in minlpSolve (default 1). Also see optParam.IterPrint
	PriLevOpt	Print level in sub solvers (SNOPT and other NLP solvers):
		=0 No output; >0 Convergence results
		>1 Output every iteration, >2 Output each step in the NLP alg
		For other NLP solvers, see the documentation for the solver
	WarmStart	If true, >0, minlpSolve reads the output from the last run from Prob.minlpSolve, if it exists. If it doesn't exist, minlpSolve attempts to open and read warm start data from mat-file minlpSolveSave.mat. minlpSolve uses the warm start information to continue from the last run. The mat-file minlp- SolveSave.mat is saved every Prob.MIP.SaveFreq iteration.
	SolverNLP	Name of the solver used for NLP subproblems. If empty, the default solver is found calling GetSolver('con',1); If TOMLAB /SOL installed, SNOPT is the default solver. If SolverNLP is a SOL solver (SNOPT, MINOS or NPSOL), the SOL.optPar and SOL.PrintFile is used: See help minosTL.m, npsolTL.m or snoptTL.m for how to set these parameters
	RandState	If Convex == 0, RandState is sent to multiMin to initialize the random gen- erator. RandState is used as follows:
		If > 0, rand('state',RandState) is set to initialize the pseudo-random generator
		if < 0, rand('state',sum(100\*clock)) is set to give a new set of random values each run
		if RandState == 0, rand('state',) is not called. Default RandState = -1
	MIP	Structure in Prob, Prob.MIP. Defines integer optimization parameters. Fields used:
	IntVars	If empty, all variables are assumed non-integer. If islogical(IntVars) (=all el- ements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element >1, IntVars is the indices for integer variables.
	VarWeight	Weight for each variable in the variable selection phase. A lower value gives higher priority. Setting Prob.MIP.VarWeight might improve convergence.
	DualGap	minlpSolve stops if the duality gap is less than DualGap. DualGap = 1, stop at first integer solution e.g. DualGap = 0.01, stop if solution < 1% from optimal solution.
	fIP	An upper bound on the IP value wanted. Makes it possible to cut branches and avoid node computations. Used even if xIP not given.
	xIP	The x-values giving the fIP value, if a solution (xIP,fIP) is known.
	NodeSel	Node selection method in branch and bound
		= 0 Depth First. Priority on nodes with more integer components
		= 1 Breadth First. Priority on nodes with more integer components
		= 2 Depth First. When integer solution found, use NodeSel = 1 (default)
		= 3 Pure LIFO (Last in, first out) Depth First
		= 4 Pure FIFO (First in, first out) Breadth First
		= 5 Pure LIFO Depth First. When integer solution found, use NodeSel 4
	VarSel	Variable selection method in branch and bound:
		= 1 Use variable with most fractional value
		= 2 Use gradient and distance to nearest integer value
	KNAPSACK	If = 1, use a knapsack heuristic. Default 0.
	ROUNDH	If = 1, use a rounding heuristic. Default 0.
	SaveFreq	Warm start info saved on minlpSolveSave.mat every SaveFreq iteration (default -1, i.e. no warm start info is saved)
	optParam	Structure in Prob. Fields used in Prob.optParam, also in sub solvers:
	MaxIter	Maximal number of iterations, default 10000
	IterPrint	Print short information each iteration (PriLev > 0 ==> IterPrint = 1). Iter- ation number: Depth in tree (symbol L\[\] - empty list, symbol Gap - Dual Gap convergence), fNLP (Optimal f(x) current node), fIPMin (Best integer feasi- ble f(x) found), LowBnd (Lower bound on optimal integer feasible f(x)), Dual Gap in absolut value and percent, The length of the node list L, -L-, The Inform and ExitFlag the solver returned at the current node, FuEv (Number of function evaluations used by solver at current node), date/time stamp.
	bTol	Linear constraint violation convergence tolerance.
	cTol	Constraint violation convergence tolerance.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	Iter	Number of iterations.
	ExitFlag	0: Global optimal solution found, or integer solution with duality gap less than user tolerance.
		1: Maximal number of iterations reached.
		2: Empty feasible set, no integer solution found.
		4: No feasible point found running NLP relaxation.
		5: Illegal x 0 found in NLP relaxation.
		99: Maximal CPU Time used (cputime > Prob.MaxCPU).
	Inform	Code telling type of convergence, returned from subsolver.
	ExitText	Text string giving ExitFlag and Inform information.
	DualGap	Relative duality gap, max(0,fIPMin-fLB)/-fIPMin-, if fIPMin =0; max(0,fIPMin-fLB) if fIPMin == 0. If fIPMin =0: Scale with 100, 100\Dual- Gap, to get the percentage duality gap. For absolute value duality gap: scale with fIPMin, fIPMin \ DualGap
	x_k	Solution.
	v_k	Lagrange multipliers. Bounds, Linear and Nonlinear Constraints, n + mLin + mNonLin.
	f_k	Function value at optimum.
	g_k	Gradient vector at optimum.
	x_0	Starting point x 0.
	f_0	Function value at start.
	c_k	Constraint values at optimum.
	cJac	Constraint derivative values at optimum.
	xState	State of each variable, described in Table 150.
	bState	State of each constraint, described in Table 151.
	cState	State of each general constraint, described in Table 152.
	Solver	Solver used ('mipSolve').
	SolverAlgorithm	Text description of solver algorithm used.
	Prob	Problem structure used.
	minlpSolve	A structure with warm start information. Use with WarmDefGLOBAL, see example below.

Description

To make a restart (warm start), just set the warm start flag, and call minlpSolve once again:

Prob.WarmStart = 1;

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

minlpSolve will read warm start information from the minlpSolveSave.mat file. Another warm start (with same MaxFunc) is made by just calling tomRun again:

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

To make a restart from the warm start information in the Result structure, make a call to WarmDefGLOBAL before calling minlpSolve. WarmDefGLOBAL moves information from the Result structure to the Prob structure and sets the warm start flag, Prob.WarmStart = 1;

Prob = WarmDefGLOBAL('minlpSolve', Prob, Result);

where Result is the result structure returned by the previous run. A warm start (with same MaxIter) is done by just calling tomRun again:

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

To make another warm start with new MaxIter 100, say, redefine MaxIter as:

Prob.optParam.MaxIter = 100;

Then repeat the two lines:

Prob = WarmDefGLOBAL('minlpSolve', Prob, Result);

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

mipSolve

Purpose

Solve mixed integer linear programming problems (MIP).

mipSolve solves problems of the form

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & = & c^{T}x & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\ & & & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \end{array} }

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

Starting with TOMLAB version 4.0, mipSolve accepts upper and lower bounds on the linear constraints like most other TOMLAB solvers. Thus it is no longer necessary to use slack variables to handle inequality constraints.

Calling Syntax

Result = tomRun('mipSolve',Prob,...)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	c	The vector c in $c^{T}x$ .
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints. If empty, Prob.b U is used.
	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.
	MaxCPU	Maximal CPU Time (in seconds) to be used by mipSolve, stops with best point found.
	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.
	SolverLP	Name of solver used for initial LP subproblem. Default solver is used if empty, see GetSolver.m and tomSolve.m.
	SolverDLP	Name of solver used for the dual LP subproblems. Default solver is used if empty, see GetSolver.m and tomSolve.m.
	PriLevOpt	Print level in lpSimplex and DualSolve:
		0: No output; > 0: Convergence result;
		> 1: Output every iteration; > 2: Output each step in simplex algorithm.
	PriLev	Print level in mipSolve.
	SOL.optPar	Parameters for the SOL solvers, if they are used as subsolvers.
	SOL.PrintFile	Name of print file for SOL solvers, if they are used as subsolvers.
	MIP	Structure with fields for integer optimization The following fields are used:
	IntVars	The set of integer variables.
		If empty, all variables are assumed non-integer (LP problem)
	VarWeight	Weight for each variable in the variable selection phase.
		A lower value gives higher priority. Setting Prob.MIP.VarWeight = Prob.c improves convergence for knapsack problems.
	DualGap	mipSolve stops if the duality gap is less than DualGap. To stop at the first found integer solution, set DualGap =1. For example, DualGap = 0.01 makes the algorithm stop if the solution is < 1% from the optimal solution.
	fIP	An upper bound on the IP value wanted. Makes it possible to cut branches and avoid node computations.
	xIP	The x-value giving the fIP value.
	KNAPSACK	If solving a knapsack problem, set to true (1) to use a knapsack heuristic.
	optParam	Structure with special fields for optimization parameters, see Table 141.Fields used are: IterPrint, MaxIter, PriLev, wait, eps f and eps Rank.
	Solver	Structure with fields for algorithm choices:
	Alg	Node selection method:
		0: Depth first
		1: Breadth first
		2: Depth first. When integer solution found, switch to Breadth.
	method	Rule to select new variables in DualSolve/lpSimplex:
		0: Minimum reduced cost, sort variables increasing. (Default)
		1: Bland's rule (default).
		2: Minimum reduced cost. Dantzig's rule.

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, \[Constraints + lower + upper bounds\].
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150, page 239.
	Inform	If ExitF lag > 0, I nf orm = ExitF lag.
	QP.B	Optimal active set. See input variable QP.B.
	QP.y	Dual parameters y (also part of Result.v_k.
	p_dx	Search steps in x.
	alphaV	Step lengths for each search step.
	ExitFlag	0: OK.
		1: Maximal number of iterations reached.
		2: Empty feasible set, no integer solution found.
		3: Rank problems. Can not find any solution point.
		4: No feasible point found running LP relaxation.
		5: Illegalx_0 found in LP relaxation.
		99: Maximal CPU Time used (cputime > Prob.MaxCPU).
	Iter	Number of iterations.
	Solver	Solver used ('mipSolve').
	SolverAlgorithm	Text description of solver algorithm used.
	Prob	Problem structure used.

Description

The routine mipSolve is an implementation of a branch and bound algorithm from Nemhauser and Wolsey \[58, chap. 8.2\]. mipSolve normally uses the linear programming routines lpSimplex and DualSolve to solve relaxed subproblems. mipSolve calls the general interface routines SolveLP and SolveDLP. By changing the setting of the structure fields Prob.Solver.SolverLP and Prob.Solver.SolverDLP, different sub-solvers are possible to use, see the help for the interface routines.

Algorithm

See \[58, chap. 8.2\] and the code in mipSolve.m.

Examples

See exip39, exknap, expkorv.

M-files Used

lpSimplex.m, DualSolve.m, GetSolver.m, tomSolve.m

See Also

cutplane, balas, SolveLP, SolveDLP

multiMin

Purpose

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 solves problems of the form

Failed to parse (unknown function "\multicolumn"): {\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 (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 (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}}} .

The variables $x\in I$ , the index subset of $1,...,n$ are restricted to be integers.

The integer components of every x_0 is rounded to the nearest integer value inside simple bounds, and these components are fixed during the nonlinear local search.

If generating random points and there are linear constraints, multiMin checks feasibility with respect to the linear constraints, and for each initial point tries 100 times to get linear feasibility before accepting the initial point.

Calling Syntax

Result = multiMin(Prob, xInit)

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

Description of Inputs

3 = tomRun (PrintResult) output from every optimization, print level 1.

Prob	Problem description structure. The following fields are used:
xInit	Either, 1x1 number - The number of random initial points, default 10\*Prob.N dxm matrix - Every column is an initial point (of length d=Prob.N). May also be set as Prob.xInit.
	fCut	If initial f(x_0) > fCut, no local optimization is done.
	WarmStart	If true, > 0, multiMin assumes the field multiMin defined with the output from a previous run on the same problem. See the Output fields of Result.multiMin. Use WarmDefGLOBAL to set the correct fields in the Prob structure. Nec- essary fields are fOpt and xOpt. If any of the other fields are missing, the corresponding variables are initialized to 0. These other fields are: localTry, Iter, FuncEv, GradEv, HessEv, ConstrEv Inform (is set to zeros(length(fOpt,1) if not defined).
		In WarmDefGLOBAL the Result structure for the optimal run will be fed back to multiMin as Prob.multiMin.ResOpt In case this field is not defined, and no better point is found during the runs, a special call to the localSolver is used to generate a proper Result structure.
	RandState	If WarmStart and isscalar(xInit), RandState is used as follows: If > 0, rand('state',RandState) is set to initialize the pseudo-random generator if < 0, rand('state',sum(100\*clock)) is set to give a new set of random values each run if RandState == 0, rand('state',) is not called Default RandState = -1
	xEqTol	Tolerance to test if new point x_k already defined as optimum: norm(xk - xOpt(:, i)) <= xEqTol max(1, norm(xk))* If test fulfilled x_k is assumed to be too close to xOpt(:,i) Default xEqTol = 1E-5
	x_L	Lower bounds for each element in x. If generating random points, -inf elements of x L are set to -10000.
	x_U	Upper bounds for each element in x. If generating random points, inf elements of x U are set to 10000.
	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.
	PriLevOpt	0 = silent.
		1 = Display one row for each unique local minima found. The minima are sorted, lowest value first (possibly the global minima)
		The following 4 information values are displayed:
		1. Order \#
		2. Function value f(x) at local minima
		3. Point x at local minima. Only up to 10 values are displayed
		4. Inform value returned from local Solver (normally 0)
		2 = One row of output from each multiMin local optimization trial
		The following 6 information values are displayed:
		1. Step \#
		2. Text Old (previously found local minima), FAIL (solver failed to verify local minima) or blank (solver success, new local minima found)
		3. Inform value from local solver
		4. f(x_0) - function value f(x_0) for initial x_0
		5. f(x) - best f(x) value found in this local search
		6. x - point for which best f(x) value was found in this local search. Only up to 10 values are displayed.
		4 = tomRun (PrintResult) output from every optimization, print level 2. For constrained problems output of sum(-constr-) and information if optimal point was accepted w.r.t. feasibility.
		5 = tomRun (PrintResult) output from every optimization, print level 3.
	GO	Structure in Prob, Prob.GO. Fields used:
	localSolver	The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0).
	optParam	Defines optimization parameters. Fields used:
	fGoal	Goal for function value f(x), if empty not used. If fGoal is reached, no further local optimizations are done.
	eps_f	Relative accuracy for function value, fTol == eps_f. Stop if abs(f-fGoal) <= abs(fGoal) \* fTol , if fGoal = 0. Stop if abs(f-fGoal) <= fTol , if fGoal ==0.
	bTol	The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0).
	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:
	The following fields in Result are changed by multiMin before return:
	ExitFlag	= 0 normal output, of if fGoal set and found.
		= 1 A solution reaching the user defined fGoal was not found
	The Solver, SolverAlgorithm and ExitText fields are also reset.
	A special field in Result is also returned, Result.multiMin:
	xOpt	Prob.N x k matrix with k distinct local optima, the test being norm(xk -xOpt(:, i)) <= xEqT ol max*(1, norm(xk )) that if fulfilled assumes x k to be to closeto xOpt(:,i)
	fOpt	The k function values in the local optima xOpt(:,i),i=1,...,k.
	Inform	The Inform value returned by the local solver when finding each of the local optima xOpt(:,i); i=1,...,k. The Inform value can be used to judge the validity of the local minimum reported.
	localTry	Total number of local searches.
	Iter	Total number of iterations.
	FuncEv	Total number of function evaluations.
	GradEv	Total number of gradient evaluations.
	HessEv	Total number of Hessian evaluations.
	ConstrEv	Total number of constraint function evaluations.
	ExitText	Text string giving ExitFlag and Inform information.

multiMINLP

Purpose

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

It is aimed for problems where the number of integer combinations nMax is huge and relaxation of the integer constraint is possible.

If no integer variables, multiMINLP calls multiMin. If nMax <= min(Prob.optParam.MaxFunc,5000), glcDirect is used. Otherwise, multiMINLP first finds a set M of local minima calling multiMin with no integer restriction on any variable. The best local minimum is selected. glcDirect is called to find the best integer feasible solution fIP in a small area (< +- 2 units) around the best local minimum found.

The other local minima are pruned, if fOpt(i) > fIP, no integer feasible solution could be found close to this local minimum i.

The area close to the remaining candidate local minima are searched one by one by calling glcDirect to find any fIPi < fIP.

multiMINLP solves problems of the form

Failed to parse (unknown function "\multicolumn"): {\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}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \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 (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}}} . The variables $x\in I$ , the index subset of $1,...,n$ are restricted to be integers.

Calling Syntax

Result = tomRun('multiMINLP',Prob,...)

Description of Inputs

Prob	Problem description structure. The following fields are used:
		Prob is a structure, defined as to solve a standard MINLP problem. The Prob structure is fed to the localSolver. See e.g. minlpAssign.
		See multiMin and glcDirect for input to the subsolvers e.g. Prob.xInit is used in multiMin (and fCut, RandState, xEQTol).
	x_L	Lower bounds for each element in x. If generating random points, -inf elements of x L are set to min(-L,xMin,x U-L) xMin is the minimum of the finite x L values.
	x_U	Upper bounds for each element in x. If generating random points, inf elements of x U are set to max(L,xMax,x L+L) xMax is the maximum of the finite x U values.
		L is 100 for nonlinear least squares, otherwise 1000.
	b_L	Lower bounds on linear constraints.
	b_U	Upper bounds on linear constraints.
	A	The linear constraint matrix.
	c_L	Lower bounds on nonlinear constraints.
	c_U	Upper bounds on nonlinear constraints.
	PriLev	Print Level:
		0 = Silent
		1 = Display 2 summary rows
		2 = Display some extra summary rows
		5 = Print level 1 in tomRun call
		6 = Print level 2 in tomRun call
		7 = Print level 3 in tomRun call
	xInit	Used in multiMin. See help for multiMin.
	GO.localSolver	The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0).
	optParam	Structure in Prob, Prob.optParam. Defines optimization parameters. Fields used:
	MaxFunc	Max number of function evaluations in each subproblem
	fGoal	Goal for function value f(x), if empty not used. If fGoal is reached, no further local optimizations are done
	eps_f	Relative accuracy for function value, fTol == eps_f. Stop if abs(f-fGoal) <= abs(fGoal) \* fTol , if fGoal = 0. Stop if abs(f-fGoal) <= fTol , if fGoal ==0. Default 1e-8.
	bTol	Linear constraint feasibility tolerance. Default 1e-6
	cTol	Nonlinear constraint feasibility tolerance. Default 1e-6
	MIP	Structure in Prob, Prob.MIP. Defines integer optimization parameters. Fields used:
	IntVars	If empty, all variables are assumed non-integer. If islogical(IntVars) (=all el- ements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element >1, IntVars is the indices for integer variables.
	nMax	Number of integer combinations possible, if empty multiMINLP computes nMax.
	Rfac	Reduction factor for real variables in MINLP subproblem close to local multiMINLP minimum. Bounds set to x_L = max(x_L,x-Rfac\(x_U-x_L)) and x_U= min(x_U,x+Rfac\(x_U-x_L)). Default 0.25.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
		Result structure from the last good optimization step giving the best f(x) value, the possible global MINLP minimum.
		The following fields in Result are changed by multiMINLP before return:
	ExitFlag	= 0 normal output, of if fGoal set and found.
		= 1 A solution reaching the user defined fGoal was not found.
		= 2 Unbounded problem.
		=4 Infeasible problem.
		The Solver, SolverAlgorithm and ExitText fields are also reset.
		A special field in Result is also returned, Result.multiMINLP:
	xOpt	Prob.N x k matrix with k distinct local optima, the test being norm(x k- xOpt(:,i)) <= xEqTol\*max(1,norm(x k)) that if fulfilled assumes x_k to be to close to xOpt(:,i).
	fOpt	The k function values in the local optima xOpt(:,i),i=1,...,k.
	Inform	The Inform value returned by the local solver when finding each of the local optima xOpt(:,i); i=1,...,k. The Inform value can be used to judge the validity of the local minimum reported.
	localTry	Total number of local searches.
	Iter	Total number of iterations.
	FuncEv	Total number of function evaluations.
	GradEv	Total number of gradient evaluations.
	HessEv	Total number of Hessian evaluations.
	ConstrEv	Total number of constraint function evaluations.
	ExitText	Text string giving Inform information.

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 (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 = 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

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) \\ \mathrm{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 f(x)} is a convex nonlinear 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 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 (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 \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 (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) \\ \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 (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}{\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 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_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 D_2} are positive-definite diagonal matrices defined from 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_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 d_2} given in Prob.SOL.d1 and Prob.SOL.d2.

In particular, 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_2} indicates the accuracy required for satisfying each row 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 Ax=b} . See pdco for a detailed discussion 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 D_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 D_2} . Note that in pdco, 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.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

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) \\ \mathrm{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 f(x)} is a convex separable nonlinear 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 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 (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\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:

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}{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.

TOMLAB Solver Reference: Difference between revisions

Revision as of 05:47, 11 July 2011

Contents

clsSolve

conSolve

cutPlane

DualSolve

expSolve

glbDirect

glbSolve

glcCluster

glcDirect

glcSolve

infLinSolve

infSolve

linRatSolve

lpSimplex

L1Solve

MilpSolve

minlpSolve

mipSolve

multiMin

multiMINLP

nlpSolve

pdcoTL

pdscoTL

qpSolve

slsSolve

sTrustr

Tfmin

Tfzero

ucSolve

Additional solvers

Navigation menu

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@@ Line 99: / Line 99: @@
 *[[L1Solve]]
-====MILPSOLVE====
+==MilpSolve==
-'''Purpose'''
 Solve mixed integer linear programming problems (MILP).
-''MILPSOLVE  ''solves problems of the form
+*[[MilpSolve]]
-<math>
-\begin{array}{cccccc}
-\min\limits_{x} & f(x) & = & c^{T}x &  \\s/t & x_{L} & \leq  & x  & \leq & x_{U} \\& b_{L} & \leq  & Ax & \leq & b_{U} \\&       &       & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$}  \\
-\end{array}
-</math>
-where <math>c, x, x_L, x_U \in \MATHSET{R}^{n}</math> , <math>A\in\MATHSET{R}^{m\times n}</math> and <math>b_L, b_U \in MATHSET{R}^{m}</math>. The variables <math>x \in I</math> , the index subset of <math>1,...,n</math> are restricted to
-be integers.
-'''Calling  Syntax'''
-Result = tomRun('MILPSOLVE',Prob, 1); or
-Prob = ProbCheck(Prob, 'MILPSOLVE');
-Result = milpsolveTL(Prob);
-PrintResult(Result,1);
-'''Description  of Inputs'''
-{|
-|''Prob''||colspan="2"|Problem description structure. The following fields are used:
-|-
-|||''x_L, x_U''||Lower and upper bounds on variables. (Must be dense).
-|-
-|||''b_L, b_U''||Lower and upper bounds on linear constraints. (Must be dense).
-|-
-|||''A''||Linear constraint matrix. (Sparse or dense).
-|-
-|||''QP.c''||Linear objective function coefficients,  size ''nx''1.
-|-
-|||''BIG''||Definition of infinity.  Default is 1e30.
-|-valign="top"
-|||''LargeScale''||Defines if milpsolveTL will convert the A matrix to a sparse matrix or not.
-|-
-|||||Largescale ! = 0 - sparse
-|-
-|||||LargeScale = 0 - dense
-|-
-|||||Default is to use the A matrix just as it is defined.
-|-
-|||''PriLevOpt''||Specifies the printlevel that will be used by MILPSOLVE.
-|-
-|||||0 (NONE) No outputs
-|-
-|||||1 (NEUTRAL) Only some specific debug messages in debug print rou- tines are reported.
-|-valign="top"
-|||||2 (CRITICAL) Only critical  messages are  reported.  Hard errors like instability,  out of memory.
-|-
-|||||3 (SEVERE) Only severe messages are reported. Errors.
-|-
-|||||4 (IMPORTANT) Only important messages are reported. Warnings and Errors.
-|-
-|||||5 (NORMAL) Normal messages are reported.
-|-valign="top"
-|||||6 (DETAILED) Detailed messages are reported. Like model size, con- tinuing B&B improvements.
-|-
-|||||7 (FULL)  All messages are reported. Useful for debugging purposes and small models.
-|-valign="top"
-|||||Default print level is 0, no outputs. PriLevOpt ''< ''0 is interpreted as 0, and larger than 7 is interpreted as 7.
-|-valign="top"
-|||''MaxCPU''||Maximal CPU Time (in seconds) to be used by MILPSOLVE, stops with best point found.
-|-valign="top"
-|colspan="3"|Fields used in '''Prob.MILPSOLVE '''(Structure with MILPSOLVE specific parameters)
-|-valign="top"
-|||''ANTI_DEGEN''||Binary vector. If empty, no anti-degeneracy handling is applied. If the length (i) of the vector is less than 8 elements,only the i first modes are considered. Also if i is longer than 8 elements, the elements after element8 are ignored.
-|-
-|||||ANTI_DEGEN specifies if special handling must be done to reduce de- generacy/cycling while solving. Setting this flag can avoid cycling, but can also increase numerical instability.
-|-
-|||||ANTIDEGEN FIXEDVARS  ! = 0 Check if  there are equality slacks in the basis and try  to drive  them out in order to reduce chance of degeneracy in Phase 1.
-|-
-|||||ANTIDEGEN COLUMNCHECK  != 0
-|-
-|||||ANTIDEGEN STALLING  != 0
-|-
-|||||ANTIDEGEN NUMFAILURE != 0
-|-
-|||||ANTIDEGEN LOSTFEAS != 0
-|-
-|||||ANTIDEGEN INFEASIBLE  != 0
-|-
-|||||ANTIDEGEN DYNAMIC != 0
-|-
-|||||ANTIDEGEN DURINGBB != 0
-|-valign="top"
-|||''basis''||If empty or erroneous, default basis is used. Default start base is the all slack basis (the default simplex starting basis).
-|-
-|||||Prob.MILPSOLVE.basis stores the basic variables. If an element is less then zero then it means on lower bound, else on upper bound. Element 0 of the array is unused. The default initial basis is bascolumn\[x\] = -x. By MILPSOLVE  convention, a basic variable is always on its lower bound, meaning that basic variables is always represented with a minus sign.
-|-
-|||||When a restart is done, the basis vector must be assigned  a correct starting basis.
-|-valign="top"
-|||''BASIS_CRASH''||The set basiscrash  function  specifies  which basis crash mode  MILP- SOLVE will used.
-|-
-|||||When no base crash is done (the default), the initial  basis from which MILPSOLVE starts to solve the model is the basis containing all slack or artificial variables that is automatically associates with each constraint.
-|-
-|||||When base crash is enabled, a heuristic "crash procedure" is executed before the first simplex iteration to quickly choose a basis matrix that has fewer artificial  variables. This procedure tends to reduce the num- ber of iterations to optimality  since a number of iterations are skipped. MILPSOLVE  starts iterating from this basis until optimality.
-|-
-|||||BASIS_CRASH ! = 2 - No basis crash
-|-
-|||||BASIS_CRASH = 2 - Most feasible basis
-|-
-|||||Default is no basis crash.
-|-valign="top"
-|||''BB_DEPTH_LIMIT''||Sets the maximum branch-and-bound depth.  This value makes  sense only if there are integer, semi-continuous or SOS variables in the model so that the branch-and-bound algorithm is used to solve the model. The branch-and-bound algorithm will not go deeper than this level. When BB DEPTH LIMIT i set to 0 then there is no limit  to the depth. The default value is -50. A positive value means that the depth is absolute. A negative value means a relative B&B  depth.  The "order"  of a MIP problem is defined to be 2 times the number of binary variables plus the number of SC and SOS variables. A relative value of -x results in a maximum depth of x times the order of the MIP problem.
-|-
-|||''BB_FLOOR_FIRST''||Specifies which branch to take first in branch-and-bound algorithm. Default value is 1.
-|-
-|||||BB_FLOOR_FIRST = 0 (BRANCH CEILING) Take ceiling branch first
-|-
-|||||BB_FLOOR_FIRST = 1 (BRANCH FLOOR) Take floor branch first
-|-
-|||||BB_FLOOR_FIRST = 2 (BRANCH AUTOMATIC) MILPSOLVE decides which branch being taken first
-|-
-|||''BB_RULE''||Specifies the branch-and-bound rule. Default value is 0.
-|-
-|||||BB_RULE  = 0 (NODE FIRSTSELECT)  Select lowest  indexed non- integer column
-|-
-|||||BB_RULE = 1 (NODE GAPSELECT) Selection based on distance from the current bounds
-|-
-|||||BB_RULE = 2 (NODE RANGESELECT) Selection based on the largest current bound
-|-
-|||||BB_RULE = 3 (NODE FRACTIONSELECT) Selection based on largest fractional value
-|-
-|||||BB_RULE = 4 (NODE PSEUDOCOSTSELECT4) Simple, unweighted pseudo-cost of a variable
-|-
-|||||BB_RULE = 5 (NODE PSEUDONONINTSELECT)  This is an ex- tended pseudo-costing strategy based on minimizing the number of in- teger infeasibilities.
-|-
-|||||BB RULE = 6 (NODE PSEUDORATIOSELECT)  This is an extended pseudo-costing strategy based on maximizing the normal pseudo-cost divided by the number of infeasibilities. Effectively, it is similar to (the reciprocal of ) a cost/benefit ratio.
-|-
-|||||BB_RULE = 7 (NODE USERSELECT)
-|-valign="top"
-|||''BB_RULE_ADD''||Additional values for the BB RULE. BB RULE is a vector. If the length i of the vector is less than 10 elements, only the i first modes are considered. Also if i is longer than 10 elements, the elements after element 10 is ignored.
-|-
-|||||BB_RULE ADD(1) ! = 0 (NODE WEIGHTREVERSEMODE)
-|-
-|||||BB_RULE ADD(2)  ! = 0 (NODE BRANCHREVERSEMODE) In case when get bb floorfirst is BRANCH AUTOMATIC, select the opposite direction (lower/upper branch) that BRANCH AUTOMATIC had cho- sen.
-|-
-|||||BB_RULE_ADD(3) ! = 0 (NODE GREEDYMODE)
-|-
-|||||BB_RULE_ADD(4) ! = 0 (NODE PSEUDOCOSTMODE)
-|-
-|||||BB_RULE_ADD(5) ! = 0 (NODE DEPTHFIRSTMODE) Select  the node that has already been selected before the number of times
-|-
-|||||BB_RULE_ADD(6) ! = 0 (NODE RANDOMIZEMODE)
-|-
-|||||BB_RULE_ADD(7) !  = 0 (NODE DYNAMICMODE)    When NODE DEPTHFIRSTMODE is selected, switch off this mode when a first solution is found.
-|-
-|||||BB_RULE_ADD(8) ! = 0 (NODE RESTARTMODE)
-|-
-|||||BB_RULE_ADD(9) ! = 0 (NODE BREADTHFIRSTMODE) Select the node that has been selected before the fewest number of times or not at all BB RULE ADD(10) ! = 0 (NODE AUTOORDER)
-|-
-|||''BFP''||Defines which Basis Factorization Package that will be used by MILP- SOLVE.
-|-
-|||||BFP = 0 : LUSOL
-|-
-|||||BFP = 1 : built in etaPHI from MILPSOLVE  v3.2
-|-
-|||||BFP = 2 : Additional  etaPHI BFP = 3 : GLPK
-|-
-|||||Default BFP is LUSOL.
-|-valign="top"
-|||''BREAK_AT_FIRST''||Specifies if the branch-and-bound algorithm stops at the first found so- lution (BREAK_AT_FIRST != 0) or not (BREAK_AT_FIRST = 0). De- fault is not to stop at the first found solution.
-|-valign="top"
-|||''BREAK_AT_VALUE'||Specifies if the branch-and-bound algorithm stops when the object value is better than a given value. The default value is (-) infinity.
-|-valign="top"
-|||''EPAGAP''||Specifies the absolute MIP gap tolerance for the branch and bound algo- rithm.  This tolerance is the difference between the best-found solution yet and the current solution. If the difference is smaller than this toler- ance then the solution (and all the sub-solutions) is rejected. The default value is 1e-9.
-|-valign="top"
-|||''EPGAP''||Specifies the relative MIP gap tolerance for the branch and bound algo- rithm.  The default value is 1e-9.
-|-valign="top"
-|||''EPSB''||Specifies the value that is used as a tolerance for the Right Hand Side (RHS) to determine whether  a value should be considered  as 0.  The default epsb value is 1.0e-10
-|-valign="top"
-|||''EPSD''||Specifies the value that is used as a tolerance for reduced costs to deter- mine whether a value should be considered as 0. The default epsd value is 1e-9. If EPSD is empty, EPSD is read from ''Prob.optParam.eps f''.
-|-valign="top"
-|||''EPSEL''||Specifies the value that is used as a tolerance for rounding values to zero. The default epsel value is 1e-12.
-|-valign="top"
-|||''EPSINT''||Specifies  the  tolerance that  is used  to determine whether a floating- point number is in fact an integer. The default value for epsint is 1e-7. Changing this tolerance value can result in faster solving times, but the solution is less integer.
-|-valign="top"
-|||''EPSPERTURB''||Specifies the value that is used as perturbation scalar for degenerative problems. The default epsperturb value is 1e-5.
-|-valign="top"
-|||''EPSPIVOT''||Specifies the value that is used as a tolerance pivot element to determine whether a value should be considered as 0. The default epspivot value is 2e-7
-|-
-|||''IMPROVEMENT_LEVEL''||Specifies the iterative improvement level.
-|-
-|||||IMPROVEMENT LEVEL = 0 (IMPROVE  NONE) improve none
-|-
-|||||IMPROVEMENT LEVEL = 1 (IMPROVE  FTRAN)  improve FTRAN
-|-
-|||||IMPROVEMENT LEVEL = 2 (IMPROVE  BTRAN)  improve BTRAN
-|-
-|||||IMPROVEMENT LEVEL  = 3 (IMPROVE  SOLVE) improve FTRAN + BTRAN.
-|-
-|||||IMPROVEMENT LEVEL  = 4 (IMPROVE  INVERSE)  triggers automatic inverse accuracy control in the dual simplex, and when an error gap is exceeded the basis is reinverted
-|-
-|||||Choice 1,2,3 should not be used with MILPSOLVE  5.1.1.3,  because of problems with the solver. Default is 0.
-|-valign="top"
-|||''LoadFile''||File that  contains the  model. If LoadFile is a nonempty string which corresponds to actual file, then the model is read from this file rather than from the Prob struct.
-|-
-|||''LoadMode''||1 - LP - MILPSOLVE  LP format
-|-
-|||||2 - MPS - MPS format
-|-
-|||||3 - FMPS - Free MPS format
-|-
-|||||A default value for this field does not exist. Both LoadFile and Load- Mode must be set if a problem will be loaded.
-|-
-|||||If there is something wrong with LoadMode or LoadFile, an error mes- sage will be printed and MILPSOLVE  will be terminated. Leave Load- Mode and LoadFile empty if the problem not will be loaded from file.
-|-
-|||''LogFile''||Name of file to print MILPSOLVE  log on.
-|-
-|||''MAXIMIZE	''||If  MAXIMIZE ! = 0, MILPSOLVE  is set to maximize the objective function, default is to minimize.
-|-
-|||''MAX_PIVOT''||Sets the maximum number of pivots between a re-inversion of the matrix. Default is 42.
-|-
-|||''NEG_RANGE''||Specifies the negative value below which variables are split into a negative and a positive part.  This value must always be zero or negative. If a positive value is specified, then 0 is taken. The default value is -1e6.
-|-
-|||''PRESOLVE''||Vector containing possible presolve options. If the length i of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored.
-|-
-|||||PRESOLVE(1) ! = 0 (PRESOLVE ROWS) Presolve rows
-|-
-|||||PRESOLVE(2) ! = 0 (PRESOLVE COLS) Presolve columns
-|-
-|||||PRESOLVE(3) ! = 0 (PRESOLVE LINDEP)  Eliminate linearly depen- dent rows
-|-
-|||||PRESOLVE(4) ! = 0 (PRESOLVE SOS) Convert constraints to SOSes (only SOS1 handled)
-|-
-|||||PRESOLVE(5) ! = 0 (PRESOLVE REDUCEMIP)  If the phase 1 solu- tion process finds that a constraint is redundant then this constraint is deleted.
-|-
-|||||PRESOLVE(6) ! = 0 (PRESOLVE DUALS) Calculate duals PRESOLVE(7) ! = 0 (PRESOLVE SENSDUALS) Calculate sensitivity if there are integer variables
-|-
-|||||Default is not to do any presolve.
-|-
-|||''PRICING_RULE''||The pricing rule can be one of the following rules.
-|-
-|||||PRICING_RULE = 0 Select first (PRICER_FIRSTINDEX)
-|-
-|||||PRICING_RULE = 1 Select according to Dantzig (PRICER_DANTZIG)
-|-
-|||||PRICING_RULE = 2 Devex pricing from Paula Harris (PRICER_DEVEX)
-|-
-|||||PRICING_RULE = 3 Steepest Edge (PRICER STEEPESTEDGE)
-|-valign="top"
-|||''PRICING_MODE''||Additional pricing settings, any combination of the modes below. This is a binary vector. If the length i of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored.
-|-
-|||||PRICE_PRIMALFALLBACK ! = 0 In case of Steepest Edge, fall back to DEVEX in primal.
-|-
-|||||PRICE_MULTIPLE ! = 0 Preliminary implementation of the multiple pricing scheme.  This means that attractive candidate entering columns from one iteration may be used in the subsequent iteration, avoiding full updating of reduced costs. In the current implementation, MILPSOLVE only reuses the 2nd best entering column alternative.
-|-
-|||||PRICE_PARTIAL ! = 0 Enable partial pricing
-|-
-|||||PRICE_ADAPTIVE ! = 0 Temporarily use First Index if cycling is detected
-|-
-|||||PRICE_RANDOMIZE  ! = 0 Adds a small randomization effect to the selected pricer
-|-
-|||||PRICE_LOOPLEFT  ! = 0 Scan entering/leaving columns left rather than right
-|-
-|||||PRICE_LOOPALTERNATE ! = 0 Scan entering/leaving columns alternatingly left/right
-|-
-|||||Default basis is PRICER DEVEX combined with PRICE ADAPTIVE.
-|-
-|||''sa''||Struct  containing information  of the sensitivity analysis (SA) MILPSOLVE will perform.
-|-
-|||||sa.obj =! 0 Perform sensitivity analysis on the objective function
-|-
-|||||sa.obj = 0 Do not perform sensitivity analysis on the objective function
-|-
-|||||sa.rhs =! 0 Perform sensitivity analysis on the right hand sides.
-|-
-|||||sa.rhs = 0 Do not perform sensitivity analysis on the right hand sides.
-|-valign="top"
-|||''SaveFileAfter''||Name  of a file to save the MILPSOLVE object after presolve.The name must be of type string (char), Example: Prob.MILPSOLVE.SaveFileAfter = 'save2' If the type is not char SaveFileBefore is set to save2.\[file extension\].
-|-valign="top"
-|||''SaveFileBefore''||Name of a file to save the MILPSOLVE object before presolve. The name must be  of type string (char), Example: Prob.MILPSOLVE.SaveFileBefore = 'save1'. If the type  is not char SaveFileBefore is set to save1.\[file extension\].
-|-
-|||''SaveMode''||1 - LP - MILPSOLVE  LP format
-|-
-|||||2 - MPS - MPS format
-|-
-|||||3 - FMPS - Free MPS format
-|-
-|||||If empty, the default format LP is used.
-|-valign="top"
-|||''SCALE_LIMIT''||Sets the relative  scaling convergence  criterion  to  the  absolute value of SCALE_LIMIT for the active scaling mode. The integer part of SCALE_LIMIT specifies the maximum number of iterations. Default is 5.
-|-
-|||''SCALING_ALG''||Specifies which scaling algorithm will be used by MILPSOLVE.
-|-
-|||||0 No scaling algorithm
-|-
-|||||1 (SCALE EXTREME) Scale to convergence using largest absolute value
-|-
-|||||2 (SCALE RANGE) Scale based on the simple numerical range
-|-
-|||||3 (SCALE MEAN) Numerical range-based scaling
-|-
-|||||4 (SCALE GEOMETRIC) Geometric scaling
-|-
-|||||7 (SCALE CURTISREID) Curtis-reid scaling
-|-
-|||||Default is 0, no scaling algorithm.
-|-valign="top"
-|||''SCALING_ADD''||Vector containing possible additional scaling parameters. If the length (i) of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored. SCALING ADD ! = 0 (SCALE_QUADRATIC)
-|-
-|||||SCALING ADD ! = 0 (SCALE_LOGARITHMIC) Scale to convergence using logarithmic mean of all values
-|-
-|||||SCALING_ADD  !  = 0  (SCALE_USERWEIGHT) User can specify scalars
-|-
-|||||SCALING_ADD ! = 0 (SCALE_POWER2) also do Power scaling
-|-
-|||||SCALING_ADD  ! = 0 (SCALE_EQUILIBRATE) Make  sure that  no scaled number is above 1
-|-
-|||||SCALING_ADD ! = 0 (SCALE_INTEGERS) Also scaling integer vari- ables
-|-
-|||||SCALING_ADD ! = 0 (SCALE_DYNUPDATE) Dynamic update
-|-
-|||||Default is 0, no additional mode.
-|-
-|||||Settings SCALE_DYNUPDATE is a way to make sure that scaling factors are recomputed. In that case, the scaling factors are recomputed also when a restart is done.
-|-
-|||''SIMPLEX_TYPE''||Sets the desired combination of primal and dual simplex algorithms.
-|-
-|||||5 (SIMPLEX_PRIMAL_PRIMAL) Phase1 Primal, Phase2 Primal
-|-
-|||||6 (SIMPLEX_DUAL_PRIMAL) Phase1 Dual, Phase2 Primal
-|-
-|||||9 (SIMPLEX_PRIMAL_DUAL) Phase1 Primal, Phase2 Dual
-|-
-|||||10 (SIMPLEX_DUAL_DUAL) Phase1 Dual, Phase2 Dual
-|-
-|||||Default is SIMPLEX DUAL PRIMAL (6).
-|-valign="top"
-|||''SOLUTION_LIMIT''||Sets the solution number that will be returned. This value is only consid- ered if there are integer, semi-continuous or SOS variables in the model so that the branch-and-bound algorithm is used. If there are more solutions with the same objective value, then this number specifies which solution must be returned. Default is 1.
-|-
-|||''sos''||List of structs containing data about Special Ordered Sets (SOS). See below for further description.
-|-
-|colspan="3"|Fields used in '''Prob.MIP '''(Structure with MIP specific parameters)
-|-
-|||''IntVars''||Defines which variables are integers, of general type I or binary type B Variable indices should be in the range \[1,...,n\].
-|-
-|||||IntVars is a logical vector ==''> x''(''f ind''(''I ntV ars > ''0)) are integers
-|-
-|||||IntVars is a vector of indices ==''> x''(''I ntV ars'') are integers (if \[\], then no integers of type I or B are defined) variables with x L=0 and x U=1, is are set to binary. It is possible to combine integer and semi-continuous type to obtain the semi-integer type.
-|-valign="top"
-|||''fIP''||This parameter specifies the initial "at least better than" guess for objective function. This is only used in the branch-and-bound algorithm when integer variables exist in the model. All solutions with a worse objective value than this value are immediately rejected. The default is infinity.
-|-valign=top"
-|||''SC''||A vector with indices for variables of type semi-continuous (SC), a logical vector or a scalar (see MIP.IntVars). A semi-continuous variable i takes either the value 0 or some value in the range \[x L(i), x U(i)\].  It is possible to combine integer and semi-continuous type to obtain the semi-integer type.
-|-valign="top"
-|||''sos1''||List of structures defining the Special Ordered Sets of Order One (SOS1). For SOS1 set k, sos1(k).var is a vector of indices for variables of type SOS1 in set k, sos1(k).row is the priority  of SOS k in the set of SOS1 and sos1(k).weight is a vector of the same length as sos1(k).var and it describes the order MILPSOLVE  will weight the variables in SOS1 set k.
-|-
-|||||a low number of a row and a weight means high priority.
-|-
-|||''sos2''||List of n structures defining the Special Ordered Sets (SOS) of Order Two (SOS2). (see MIP.sos1)
-|}
-'''Description  of Outputs'''
-{|
-|''Result''||colspan="2"|Structure with result from optimization. The following fields are changed:
-|-
-|||''x_k''||Optimal  solution (or some other solution if optimum could not been found)
-|-
-|||''f_k''||Optimal objective value.
-|-
-|||''v_k''||\[rc; duals\]. If Reduced cost and dual variables are not available, then v_k is empty.
-|-
-|||''ExitFlag''||TOMLAB information parameter.
-|-
-|||||0 = Optimal solution found.
-|-
-|||||1 = Suboptimal solution or user abort.
-|-
-|||||2 = Unbounded solution.
-|-
-|||||3 = Numerical failure.
-|-
-|||||4 = Infeasible model.
-|-
-|||||10 = Out of memory.
-|-
-|||||11 = Branch and bound stopped.
-|-
-|||''ExitText''||Status text from MILPSOLVE.
-|-
-|||''Inform''||MILPSOLVE  information parameter.
-|-
-|||||-2 = Out of memory.
-|-
-|||||0 = Optimal solution found.
-|-
-|||||1 = Suboptimal solution.
-|-
-|||||2 = Infeasible model.
-|-
-|||||3 = Unbounded solution.
-|-
-|||||4 = Degenerate solution.
-|-
-|||||5 = Numerical failure.
-|-
-|||||6 = User abort.
-|-
-|||||7 = Timeout.
-|-
-|||||10 = Branch and bound failed.
-|-
-|||||11 = Branch and bound stopped.
-|-
-|||||12 = Feasible branch and bound solution.
-|-
-|||||13 = No feasible branch and bound solution. Other = Unknown status.
-|-valign="top"
-|||''Iter''||The total  number of nodes processed in the branch-and-bound algorithm. Is only applicable if the model contains integer variables. In the case of an LP model Result.Iter contains the number of iterations. This is however not documented.
-|-
-|||''MinorIter''||The total number of Branch-and-bound iterations. When the problem is LP, MinorIter equals Result.Iter
-|-
-|||''MILPSOLVE.basis''||Optimal basis, on the format described above under Prob.MILPSOLVE.basis.
-|-
-|||''MILPSOLVE.MaxLevel''||The deepest Branch-and-bound  level of the last solution. Is only applicable if the model contains integer variables.
-|-
-|||''MILPSOLVE.sa.objStatus''||1 successful
-|-
-|||||0 SA not requested
-|-
-|||||-1 Error: error from MILPSOLVE
-|-
-|||||-3 no SA available
-|-
-|||''MILPSOLVE.sa.ObjLower''||An array that  will  contain the values of the lower limits  on the objective function.
-|-
-|||''MILPSOLVE.sa.ObjUpper''||An array that will contain the values of the upper limits on the objective function.
-|-
-|||''MILPSOLVE.sa.RhsStatus''||see MILPSOLVE.sa.objStatus.
-|-
-|||''MILPSOLVE.sa.RhsLower''||An array that will contain the values of the lower limits on the RHS.
-|-
-|||''MILPSOLVE.sa.RhsUpper''||An array that will contain the values of the upper limits on the RHS.
-|-
-|||''xState''||State of each variable
-|-
-|||||0 - free variable,
-|-
-|||||1 - variable on lower bound,
-|-
-|||||2 - variable on upper bound,
-|-
-|||||3 - variable is fixed, lower bound = upper bound.
-|-
-|||''bState''||State of each linear constraint
-|-
-|||||0 - Inactive constraint,
-|-
-|||||1 - Linear constraint on lower bound,
-|-
-|||||2 - Linear constraint on upper bound,
-|-
-|||||3 - Linear equality constraint.
-|}
 ====minlpSolve====