TOMLAB Solver Reference: Difference between revisions

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{{Part Of Manual|title=the TOMLAB Manual|link=[[TOMLAB|TOMLAB Manual]]}}
[[Category:Manuals]]
[[Category:Manuals]]
{{cleanup|Clean this article.}}
{{Part Of Manual|title=the TOMLAB Manual|link=[[TOMLAB|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.
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.


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*[[nlpSolve]]
*[[nlpSolve]]


====pdcoTL====
==pdcoTL==
 
'''Purpose'''


''pdcoTL ''solves linearly constrained convex nonlinear optimization problems.
''pdcoTL ''solves linearly constrained convex nonlinear optimization problems.


*[[pdcoTL]]


<math>
==pdscoTL==
\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>


''pdscoTL'' solves linearly constrained convex nonlinear optimization problems.


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>.
*[[pdscoTL]]
 
'''Calling  Syntax'''


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


'''Description  of Inputs'''
Solve general quadratic programming problems.


{|
*[[qpSolve]]
|''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'''
==slsSolve==


{|
Find a Sparse Least Squares (sls) solution to a constrained least squares problem with the use of any suitable TOMLAB  NLP solver.
|''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'''
*[[slsSolve]]


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


The problem (19) is first reformulated into SOL PDCO form:
Solve optimization problems constrained by a convex feasible region.


*[[sTrustr]]


<math>
==Tfmin==
\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>


Minimize function of one variable. Find miniumum x in [x_L, x_U] for function Func within tolerance xTol.  Solves using Brents minimization algorithm.


The problem actually solved by ''pdco ''is
*[[Tfmin]]


==Tfzero==


<math>
Tfzero, TOMLAB fzero routine.
\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>


*[[Tfzero]]


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.
==ucSolve==


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


'''Examples'''
*[[ucSolve]]


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


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


''pdcoSet.m'', ''pdco.m'', ''Tlsqrmat.m''
*goalSolve - For sparse multi-objective goal attainment problems, with linear and nonlinear constraints.


'''See Also'''
*Tlsqr - Solves large, sparse linear least squares problem, as well as unsymmetric linear systems.


''pdscoTL.m''
*lsei - For linearly constrained least squares problem with both equality and inequality constraints.


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


''pdscoTL'' solves linearly constrained convex nonlinear optimization problems.
*qld - For convex quadratic programming problem.
 
*[[pdscoTL]]

Latest revision as of 13:30, 17 July 2011

Notice.png

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.

conSolve

Solve general constrained nonlinear optimization problems.

cutPlane

Solve mixed integer linear programming problems (MIP).

DualSolve

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

expSolve

Solve exponential fitting problems for given number of terms p.

glbDirect

Solve box-bounded global optimization problems.

glbSolve

Solve box-bounded global optimization problems.

glcCluster

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

glcDirect

Solve global mixed-integer nonlinear programming problems.

glcSolve

Solve general constrained mixed-integer global optimization problems.

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.

infSolve

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

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.

lpSimplex

Solve general linear programming problems.

L1Solve

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

MilpSolve

Solve mixed integer linear programming problems (MILP).

minlpSolve

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

mipSolve

Solve mixed integer linear programming problems (MIP).

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.

multiMINLP

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

nlpSolve

Solve general constrained nonlinear optimization problems.

pdcoTL

pdcoTL solves linearly constrained convex nonlinear optimization problems.

pdscoTL

pdscoTL solves linearly constrained convex nonlinear optimization problems.

qpSolve

Solve general quadratic programming problems.

slsSolve

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

sTrustr

Solve optimization problems constrained by a convex feasible region.

Tfmin

Minimize function of one variable. Find miniumum x in [x_L, x_U] for function Func within tolerance xTol. Solves using Brents minimization algorithm.

Tfzero

Tfzero, TOMLAB fzero routine.

ucSolve

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

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.
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