TOMLAB Solver Reference: Difference between revisions

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*[[minlpSolve]]
*[[minlpSolve]]


====mipSolve====
==mipSolve==
 
'''Purpose'''


Solve mixed integer linear programming problems (MIP).
Solve mixed integer linear programming problems (MIP).


''mipSolve ''solves problems of the form
*[[mipSolve]]
 
 
<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 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''||colspan="2"|Problem description structure. The following fields are used:
|-
|||''c''||The vector ''c ''in <math>c^{T}x</math>.
|-
|||''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.
|-valign="top"
|||''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.
|-valign="top"
|||''SolverLP''||Name of solver used for initial  LP subproblem. Default solver is used if empty, see ''GetSolver.m ''and ''tomSolve.m''.
|-valign="top"
|||''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'''
 
{|
|-valign"top"
|''Result''||colspan="2"|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: Illegal''x_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====
====multiMin====

Revision as of 05:55, 11 July 2011


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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

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 , the index subset of 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 , the index subset of 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



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 (unknown function "\MATHSET"): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (unknown function "\MATHSET"): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} .

Calling Syntax

Result = 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



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

Calling Syntax

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

Description of Inputs

Prob Problem description structure. The following fields are used:
x_0 Initial x vector, used if non-empty.
A The linear constraint matrix.
b_L,b_U Lower and upper bounds for the linear constraints.
PriLevOpt Print level in pdsco solver. If > 0: prints summary information.
SOL Structure with SOL special parameters:
pdco Options structure with fields as defined by pdcoSet.
d1 Primal regularization vector. Must be a positive vector (length n) or scalar, in which case D1 = diag(d1) is used. Default: 10-4 .
d2 Dual regularization vector. Must be a positive vector (length m) or a scalar value, in which case D2 = diag(d2) is used. Default: 10-4 .
y0 Initial dual parameters for linear constraints (default 0)
z0 Initial dual parameters for simple bounds (default 1/N )

xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method.

xsize Estimate of the biggest x at the solution. (default 1/N )
zsize Estimate of the biggest z at the solution. (default 1/N )
optParam Structure with optimization parameters. The following fields are used:
MaxIter Maximum number of iterations. (Prob.SOL.pdco.MaxIter).
MinorIter Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter).
eps_x Accuracy for satisfying x1 . * z1 = 0, x2 .z1 = 0, where z = z1 - z2 and z1 , z2 > 0.(Prob.SOL.pdco.OptTol)
bTol Accuracy for satisfying Ax + D2 r = b, AT y + z = ?f (x) and x - x1 = bL , x +x2 = bU , where x1 , x2 > 0 (Prob.SOL.pdco.FeaTol)
wait 0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait)

Description of Outputs

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

Description

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

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


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


The problem actually solved by pdco is


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


where and are positive-definite diagonal matrices defined from , given in Prob.SOL.d1 and Prob.SOL.d2.

In particular, indicates the accuracy required for satisfying each row of . See pdco for a detailed discussion of and . Note that in pdco, the objective is denoted , and .

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


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

Calling Syntax

Result=tomRun('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:



The problem actually solved by pdsco is


Failed to parse (unknown function "\multicolumn"): {\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 is the primal regularization parameter, typically small but 0 is allowed. Furthermore, is the dual regularization parameter, typically small or 1; must be strictly greater than zero.

With positive the primal-dual solution is bounded and unique.

See pdsco for a detailed discussion of </math>\gamma</math> and . Note that in pdsco, the objective is denoted , and .

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



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



where Failed to parse (unknown function "\Rdim"): {\displaystyle x,x_L,x_U \in \Rdim{n}} , Failed to parse (unknown function "\Rdim"): {\displaystyle r(x) \in \Rdim{m}} , Failed to parse (unknown function "\Rdim"): {\displaystyle A \in\Rdim{m_1,n}} , Failed to parse (unknown function "\Rdim"): {\displaystyle b_L,b_U \in \Rdim{m_1}} and Failed to parse (unknown function "\Rdim"): {\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 are added at the end of the vector of unknowns. The reformulated problem


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



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

Calling Syntax

Result = 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 in the interval . Tfzero searches for a zero of a function between the given scalar values and until the width of the interval (xLow, xUpp) has collapsed to within a tolerance specified by the stopping criterion, . 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



where Failed to parse (unknown function "\MATHSET"): {\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.
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