TOMLAB Solver Reference: Difference between revisions

Revision as of 06:49, 11 July 2011

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

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

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

clsSolve

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

clsSolve

conSolve

Solve general constrained nonlinear optimization problems.

conSolve

cutPlane

Solve mixed integer linear programming problems (MIP).

cutPlane

DualSolve

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

dualSolve

expSolve

Solve exponential fitting problems for given number of terms p.

expSolve

glbDirect

Solve box-bounded global optimization problems.

glbDirect

glbSolve

Solve box-bounded global optimization problems.

glbSolve

glcCluster

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

glcCluster

glcDirect

Solve global mixed-integer nonlinear programming problems.

glcDirect

glcSolve

Solve general constrained mixed-integer global optimization problems.

glcSolve

infLinSolve

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

infLinSolve

infSolve

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

infSolve

linRatSolve

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

linRatSolve

lpSimplex

Solve general linear programming problems.

lpSimplex

L1Solve

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

L1Solve

MilpSolve

Solve mixed integer linear programming problems (MILP).

MilpSolve

minlpSolve

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

minlpSolve

mipSolve

Solve mixed integer linear programming problems (MIP).

mipSolve

multiMin

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

multiMin

multiMINLP

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

multiMINLP

nlpSolve

Solve general constrained nonlinear optimization problems.

nlpSolve

pdcoTL

Purpose

pdcoTL solves linearly constrained convex nonlinear optimization problems.

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

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

Calling Syntax

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

Description of Inputs

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

Description of Outputs

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

Description

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

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

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

The problem actually solved by pdco is

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

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

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

Examples

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

M-files Used

pdcoSet.m, pdco.m, Tlsqrmat.m

See Also

pdscoTL.m

pdscoTL

pdscoTL solves linearly constrained convex nonlinear optimization problems.

pdscoTL

@@ Line 135: / Line 135: @@
 *[[nlpSolve]]
-==pdcoTL==
+====pdcoTL====
+'''Purpose'''
 ''pdcoTL ''solves linearly constrained convex nonlinear optimization problems.
-*[[pdcoTL]]
+<math>
+\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}
+</math>
+where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \RR^n</math> , <math>A\in \RR^{m \times n}</math> and <math>b_L, b_U \in \RR^m</math>.
+'''Calling  Syntax'''
+Result=tomRun('pdco',Prob,...);
+'''Description  of Inputs'''
+{|
+|''Prob''||colspan="2"|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''.
+|-valign="top"
+|||''d1''||Primal regularization vector. Must be a positive vector (length ''n'') or scalar, in which case ''D''1 = ''diag''(''d''1) is used. Default: 10''-''4 .
+|-valign="top"
+|||''d2''||Dual regularization vector.  Must be a positive vector (length ''m'') or a scalar value, in which case ''D''2 = ''diag''(''d''2) is used. Default: 10''-''4 .
+|-
+|||''y0''||Initial  dual parameters for linear constraints (default 0)
+|-valign="top"
+|||''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'').
+|-valign="top"
+|||''MinorIter''||Maximum number of iterations in ''LSQR'' (''Prob.SOL.pdco.LSQRMaxIter'').
+|-
+|||''eps_x''||Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')
+|-valign="top"
+|||''bTol''||Accuracy for satisfying ''Ax ''+ ''D''2 ''r ''= ''b'', ''AT y ''+ ''z ''= ''?f ''(''x'') and ''x - x''1  = ''bL , x ''+''x''2 = ''bU '', where ''x''1 '', x''2  ''> ''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''||colspan="2"|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;
+|-valign="top"
+|||''bState''||State of linear constraints.  Free == 0; Lower == 1; Upper == 2; Equality == 3;
+|-
+|||''v_k''||Lagrangian multipliers (orignal bounds + constraints )
+|-valign="top"
+|||''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''||''pdco''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 ('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:
+<math>
+\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}
+</math>
+The problem actually solved by ''pdco ''is
+<math>
+\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}
+</math>
+where <math>D_1</math> and <math>D_2</math> are positive-definite diagonal matrices defined from <math>d_1</math>, <math>d_2</math> given in Prob.SOL.d1 and Prob.SOL.d2.
+In particular, <math>d_2</math> indicates the accuracy required for satisfying each row of <math>Ax=b</math>. See ''pdco'' for a detailed discussion of <math>D_1</math> and <math>D_2</math>. Note that in ''pdco'', the objective <math>f(x)</math> is denoted <math>\phi(x)</math>, <math>bl == x\_L</math> and <math>bu == x\_U</math>.
+'''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==

TOMLAB Solver Reference: Difference between revisions

Revision as of 06:49, 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

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