Matlab input problem for optimization

A Prob - the Input Problem Structure

The Input Problem Structure, here referred to as Prob , is one of the most central aspects of working with TOMLAB . It contains numerous fields and substructures that influence the behaviour and performance of the solvers.

There are default values for everything that is possible to set defaults for, and all routines are written in a way that makes it possible for the user to just set an input argument empty ([ ]) and get the default.

TOMLAB is using the structure variable optParam , see Table A, for the optimization parameters controlling the execution of the optimization solvers.

Subproblems

Many algorithms need sub-problems solved as part of the main algorithm. For example, in SQP algorithms for general nonlinear programs, QP problems are solved as sub-problems in each iteration. As QP solver any solver, even a general NLP solver, may be used. To send parameter information to the QP subsolver, the fields Prob.optParam , Prob.Solver and Prob.SOL could be put as subfields to the Prob.QP field (see Table 16), i.e. as fields Prob.QP.optParam , Prob.QP.Solver . The field Prob.QP.optParam need not have all subfields, the missing ones are filled with default values for the particular QP solver.

The same Prob.QP subfield is used for the other types of subproblems recognized, i.e. Phase 1 feasibility problems, LP and dual LP problems. Note that the fields Prob.SolverQP , Prob.SolverFP Prob.SolverLP and Prob.SolverDLP are set to the name of the solver that should solve the subproblem. If the field is left empty, a suitable default solver is used, dependent on the license for TOMLAB .

Information stored in the problem structure Prob . Fields defining sub-structures are defined in Table 15

Field

Description

Tomlab

TOMLAB Version number.

Matrix with linear constraints, one constraint per row (dense or sparse).

ADObj

Automatic differentiation flag. If 1, -1 MAD is used to obtain gradient and Hessian, respectively.

ADCons

Automatic differentiation flag. If 1, -1 MAD is used to obtain the constraint Jacobian and nonlinear constraint Hessian, respectively.

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.

CHECK

If true, no more check is done by ProbCheck. Set to true (=1) after first call to ProbCheck.

CheckNaN

If Prob.CheckNaN =0, nlp_d2c , nlp_H , nlp_d2r checks for NaN elements and estimates the corresponding derivatives numerically. If Prob.CheckNaN >0, the same applies for nlp_dc , nlp_g , nlp_J . Off-diagonal elements in symmetric Hessians should both be set as NaN. fdng, fdng2, fdng3, only estimate NaN elements in gradient, if gradient vector is input.

cName

Name of each general constraint.

cols

The columns in the user computed matrix that will be accessed, and needs to be set.

ConIx

A vector with the sequence of calls required to compute the numerical constraint Jacobian efficiently. See findpatt for more information.

ConsDiff

Numerical approximation of the constraint derivatives. If set to 1, the classical approach with forward or backward differences together with automatic step selection will be used. If set to 2, 3 or 4 the spline routines csapi , csaps or spaps in the SPLINE Toolbox will be used. If set to 5, derivatives will be estimated by use of complex variables. For the SOL solvers, the value 6 gives the internal derivative approximation.

ConsIdx

Internally used to speed up the SOL solver computations. Used to send linear index from multiple subscripts for nonzero pattern in constraint Jacobian.

ConsPattern

Matrix with non-zero pattern in the constraint gradient matrix.

d2LPattern

Sparsity pattern of the Hessian of the Lagrangian function.

f_Low

Lower bound on optimal function value.

f_opt

Objective function value f(x^*) corresponding to the points given in x_opt .

GradTolg

Size of step length to estimate first order derivatives in the gradient vector. If this field is empty, optParam.DiffInt is used instead.

GradTolH

Size of step length to estimate the Hessian matrix. If this field is empty , optParam.DiffInt is used instead.

GradTolJ

Size of step length to estimate the Jacobian matrix or the constraint gradient matrix. If this field is empty, optParam.DiffInt is used instead.

HessIx

A vector with the sequence of calls required to compute the numerical Jacobian efficiently. See findpatt for more information.

HessPattern

Matrix with non-zero pattern in the Hessian matrix.

JacIx

A vector with the sequence of calls required to compute the numerical Jacobian efficiently. See findpatt for more information.

JacPattern

Matrix with non-zero pattern in the Jacobian matrix.

LargeScale

Flag if the problem is large scale. If this flag is set no collection of search steps are made. Also, for some solvers, LargeScale chooses between dense (=0) or sparse (=1) versions of the solver. This flag also controls several other features in TOMLAB such as estimation of patterns.

MaxCPU

Maximum execution time in seconds for the solver. The feature is available for a limited number of solvers.

MENU

Flag used to tell if a menu, the GUI or a driver is calling, to avoid unnecessary checks of the fields in Prob (0).

Mode

Indicates whether the user should return function values and/or derivatives. Is best used when the user computes both function values and derivatives at the same time to save CPU time. 0= Assign function values. 1= Assign known derivatives, unknown derivative values set as -11111 (=missing value). 2= Assign function and known derivatives, unknown derivatives set as -11111.

nState

Indicates the first and last calls to the user routines to compute function and derivatives. Used by the SOL solvers.

1	First call.
0	Other calls before the last call.
2+Inform	Last call, see the Inform parameter for the solver used.

Problem dimension (number of variables).

mLin

Number of linear constraints.

mNonlin

Number of nonlinear constraints.

Name

Problem name.

NumDiff

Numerical approximation of the derivatives of the objective function. If set to 1, the classical approach with forward or backward differences together with automatic step selection will be used. If set to 2, 3 or 4 either the standard Matlab spline function or the spline routines csapi , csaps or spaps in the SPLINE Toolbox will be used. If set to 5, derivatives will be estimated by use of complex variables. For the some solvers, the value 6 gives the internal derivative approximation.

Problem number (1).

plotLine

Flag if to do a plot of the line search problem.

PriLev

Print level in the driver routines (0).

PriLevOpt

Print level in the TOM solver and the Matlab part of the solver interface ( 0).

PrintLM

Flag: controls whether or not PrintResult should calculate Lagrange multipliers and reduced (projected) gradient after a solver has been run.

Since PrintResult operates on a Result structure, and not on Prob , this flag is accessed as Result.Prob.PrintLM , but of course it is possible to set it before solving the problem.

probFile

Name of m-file in which the problems are defined.

probType

TOMLAB problem type, see Table 2.2.

rows

The rows in the user computed vector/matrix that will be accessed, and needs to be set.

simType

A flag indicating when the TOMLAB simulation format is used. The objective and constraints are calculated at the same function. The gradient and Jacobian are also calculated in the same function.

smallA

If 1 then small elements in the linear constraints are removed. The elements have to be smaller than eps*max(max(abs(Prob.A))).

SolverDLP

Name of the solver that should solve dual LP sub-problems.

* SolverLP

Name of the Solver that should solve LP sub-problems.

* SolverFP

Name of the solver that should solve a phase one LP sub-problem, i.e. finding a feasible point to a convex set.

* SolverQP

Name of the solver that should solve QP sub problems.

User supplied parameters for the problem (Init File Format). Use Prob.user for extra parameters.

uPName

Problem name (Prob.Name ) connected to the user supplied parameters in ( uP).

user

User supplied parameters for the problem. Should be stored in subfields, for example Prob.user.aParam .

WarmStart

For solver with support for warmstarts, WarmStart > 0 indicates that the solver should do a warm start.

x_0

Starting point.

x_L

Lower bounds on the variables x.

x_U

Upper bounds on the variables x.

x_min

Lower bounds on plot region.

x_max

Upper bounds on plot region.

x_opt

Stationary points x^*, one per row (if known). It is possible to define an extra column, in which a zero (0) indicates a minimum point, a one (1) a saddle point, and a two (2) a maximum. As default, minimum points are assumed. The corresponding function values for each row in x_opt should be given in Prob.f_opt .

xName

Name of each decision variable in x.

Warning

If 1 (default), some warnings may be issued.

Table 15: The fields defining sub-structures in the problem structure Prob . Default values are in all tables given in parenthesis at the end of each item.

Field Description

QP Structure with special fields for linear and quadratic problems, see Table 16.

LS Structure with special fields for least squares problems, see Table 17.

MIP Structure with special fields for mixed-integer programming, see Table 18.

GO Structure with special fields for global optimization.

CGO Structure with special fields for costly global optimization.

ExpFit Structure with special fields for exponential fitting problems, see Table A.

NTS Structure with special fields for nonlinear time series.

LineParam Structure with special fields for line search optimization parameters, see Table 19.

optParam Structure with special fields for optimization parameters, see Table A.

PartSep Structure with special fields for partially separable functions, see Table 20.

SOL Structure with special fields for SOL (Stanford Optimization Laboratory) solvers, see Table 21.

Solver Structure with fields Name , Alg and Method . Name is the name of the solver. Alg is the solver algorithm to be used. Method is the solver sub-method technique. See the solver descriptions Section 11.

FUNCS Structure with user defined names of the m-files computing the objective, gradient, Hessian etc. See Table 22. These routines are called from the corresponding gateway routine

DUNDEE Structure with special fields for the MINLP solvers (University of Dundee).

PENOPT Structure with special fields for the PENOPT solvers.

Table 16: Information stored in the structure Prob.QP . The three last sub-fields, always part of the Prob subfields, could optionally be put here to give information to a subproblem QP, LP, dual LP or feasible point (Phase 1) solver.

Field Description

F Constant matrix F in 1/2 x'Fx+c'x

c Cost vector c in 1/2 x'Fx+c'x

B Logical vector of the same length as the number of variables. A zero corresponds to a variable in the basis.

y Dual parameters y.

Q Orthogonal matrix Q in QR-decomposition.

R Upper triangular matrix R in QR-decomposition.

E Pivoting matrix E in QR-decomposition. Stored sparse.

Ascale Flag if to scale the A matrix, the linear constraints.

DualLimit Stop limit on the dual objective.

UseHot True if to use a crash basis (hot start).

HotFile If nonempty and UseCrash true, read basis from this file.

HotFreq How often to save a crash basis.

HotN The number of crash basis files.

optParam Structure with special fields for optimization parameters, see Table A.

SOL Structure with special fields for SOL (Stanford Optimizaton Laboratory) solvers, see Table 21.

Solver Structure with fields Name , Alg and Method . Name is the name of the solver. Alg is the solver algorithm to be used. Method is the solver sub-method technique. See the solver descriptions Section 11.

Table 17: Information stored in the structure Prob.LS

Field Description

weightType Weighting type:

0 No weighting.

1 Weight with data in y . If y(t)=0, the weighting is 0, i.e. deleting this residual element.

2 Weight with weight vector or matrix in weightY . If weightY is a vector then weighting by weigthY.*r (element wise multiplication). If weightY is a matrix then weighting by weigthY*r (matrix multiplication).

3 nlp_r calls the routine weightY (must be a string with the routine name) to compute the residuals.

weightY Either empty, a vector, a matrix or a string, see weightType .

t Time vector t.

y Vector or matrix with observations y(t).

E Fixed matrix, in linear least squares problem E*x − y.

yUse If yUse =0 compute residual as f(x,t)−y(t) (default), otherwise y(t) should be treated separately by the solver and the residual routines just return f(x,t).

SepAlg If SepAlg=1, use separable non linear least squares formulation (default 0).

Table 18: Information stored in the structure Prob.MIP

Field Description

IntVars Which variables are integer valued

VarWeight Priority vector for each variable.

fIP Function value for point defined in xIP . Gives an upper bound on the IP value wanted. Makes it possible to cut branches and avoid node computations

xIP The point x giving the function value fIP .

PI See the Tomlab /Xpress User's Guide.

SC See the Tomlab /Xpress User's Guide.

SI See the Tomlab /Xpress User's Guide.

sos1 See the Tomlab /Xpress User's Guide.

sos2 See the Tomlab /Xpress User's Guide.

xpcontrol See the Tomlab/ Xpress User's Guide.

callback See the Tomlab /Xpress User's Guide.

KNAPSACK See the Tomlab /Xpress User's Guide.

captionInformation stored in the structure Prob.ExpFit . Default values in parenthesis.

Field Description

p Number of exponential terms (2).

wType Weighting type (1).

eType Type of exponential terms (1).

infCR Information criteria for selection of best number of terms (0).

dType Differentiation formula (0).

geoType Type of equation (0).

qType Length q of partial sums (0).

sigType Sign to use in (P ± √Q)/D in exp_geo for p=3,4 (0).

lambda Vector of dimension p, intensities.

alpha Vector of dimension p, weights.

beta Vector of dimension p, weights in generalized exponential models.

x0Type Type of starting value algorithm.

sumType Type of exponential sum.

Table 19: Information stored in the structure Prob.LineParam

Field Description

LineAlg Line search algorithm. 0= quadratic interpolation, 1= cubic interpolation, 2= curvilinear quadratic interpolation (not robust), 3= curvilinear cubic interpolation (not robust) ( LineAlg=1).

sigma Line search accuracy; 0<sigma<1. sigma=0.9 inaccurate line search. sigma=0.1 accurate line search (0.9).

InitStepLength Initial length of step (1.0).

MaxIter Maximum number of line search iterations.

fLowBnd Lower bound on optimal function value. Used in the line search by Fletcher, m-file LineSearch (=−realmax).

rho Determines the ρ line (0.01).

tau1 Determines how fast step grows in phase 1 (9).

tau2 How near end point of [a,b] (0.1).

tau3 Choice in [a,b] phase 2 (0.5).

eps1 Minimal length for interval [a,b] (10⁻⁷).

eps2 Minimal reduction (100× є).

Information stored in the structure Prob.optParam . Default values in parenthesis. Items marked (SOL-n) are corresponding to SOL parameters, and are given different initial values dependent on which solver is used. The number n gives the element number used in the optParam parameter vector in Prob.SOL .

Field

Description

PriLev

Solver major print level in file output (SOL-1).

PriFreq

Print frequency (SOL-5) in optimization solver.

SummFreq

Summary frequency (SOL-6) in optimization solver.

MinorPriLev

Minor print level in file output (SOL-2) in SOL sub-problem solver.

IterPrint

Flag for one-row-per-iteration printout during optimization (0). If SOL-1 not explicitely set, setting this flag will set SOL-1 to 1 for the SOL solvers.

wait

Flag, if true use pause statements after output in each iteration (0).

MaxFunc

Maximal number of function evaluations.

* MaxIter

Maximum number of iterations (SOL-30).

* MajorIter

Maximum number of iterations in major problem (SOL-35).

* MinorIter

Maximum number of iterations in minor problem (SOL-36).

eps_f

Relative convergence tolerance in f (10⁻⁸).

eps_absf

Absolute convergence tolerance for the function value ( −realmax).

eps_x

Relative convergence tolerance in parameter solution x (SOL-10).

eps_dirg

Convergence tolerance for the directed derivative (10⁻⁸).

eps_c

Feasibility tolerance for nonlinear constraints (SOL-9).

eps_g

Gradient (or reduced gradient) convergence tolerance (10⁻⁷).

eps_Rank

Rank test tolerance (SOL-27).

EpsGlob

Global/local weight parameter in global optimization (10⁻⁴).

fTol

Relative accuracy in the computation of the function value (SOL-41).

* xTol

If x

[x_L,x_L+bTol] or [x_U−bTol,x_U ], fix x on bound (100*є = 2.2204 10⁻¹³).

bTol

Feasibility tolerance for linear constraints (SOL-11).

* cTol

Feasibility tolerance for nonlinear constraints (SOL-9).

MinorTolX

Relative convergence tolerance in parameters x in sub-problem (SOL-12).

size_x

Size at optimum for the variables x, used in the convergence tests ( 1). Only changed if scale very different, x >> 1.

* size_f

Size at optimum for the function f, used in the convergence tests ( 1). Only changed if scale very different, f >> 1.

* size_c

Size at optimum for the constraints c, used in the convergence tests ( 1). Only changed if scale very different, c >> 1.

PreSolve

Flag if presolve analysis is to be applied on linear constraints ( 0).

DerLevel

Derivative Level, knowledge about nonlinear derivatives: 0= Some components of the objective gradient are unknown and some components of the constraint gradient are unknown, 1= The objective gradient is known but some or all components of the constraint gradient are unknown, 2= All constraint gradients are known but some or all components of the objective gradient are unknown, 3= All objective and constraint gradients are known (3,SOL-39).

GradCheck

0,1,2,3 gives increasing level of user-supplied gradient checks (SOL-13).

DiffInt

Difference interval in derivative estimates (SOL-42).

* CentralDiff

Central difference interval in derivative estimates (SOL-43).

QN_InitMatrix

Initial matrix for Quasi-Newton, may be set by the user. When QN_InitMatrix is empty, the identity matrix is used.

splineSmooth

Smoothness parameter sent to the SPLINE Toolbox routine csaps.m when computing numerical approximations of the derivatives (−1 default). 0 means least squares straight line fit. 1 means natural (variational) cubic spline interpolant. The transition range in [0,1] is small, suggested tries are 0.2 or 0.4. <0 lets the routine csaps make the choice (default)

* splineTol

Tolerance parameter sent to the SPLINE Toolbox routine spaps.m when computing numerical approximations of the derivatives (10⁻³). Should be set in the order of the noise level.

BigStep

Unbounded step size. Used to detect unbounded nonlinear problems. (SOL-45).

* BigObj

Unbounded objective value. Used to detect unbounded nonlinear problems. (SOL-46).

CHECK

If true, no more check is done on the structure. Set to true (=1) after first call to optParamSet .

Table 20: Information stored in the structure Prob.PartSep

Field Description

pSepFunc Number of partially separable functions.

index Index for the partially separable function to compute, i.e. if i=index, compute f_i(x). If index=0, compute the sum of all, i.e. f(x)=Σ_i=1^Mf_i(x).

Table 21: Information stored in the structure Prob.SOL

Field Description

SpecsFile If nonempty gives the name of a file which is written in the SOL SPECS file format.

PrintFile If nonempty gives the name of a file which the SOL solver should print information on. The amount printed is dependent on the print level in Prob.SOL.optPar(1) .

SummFile If nonempty gives the name of a file which the SOL solver prints summary information on.

xs Vector with solution x and slack variables s. Used for warm starts.

hs Vector with basis information in the SOL sparse solver format. Used for warm starts.

nS Number of superbasics. Used for warm starts.

hElastic Elastic variable information in SQOPT .

iState Vector with basis information in the SOL dense solver format. Used for warm starts.

cLamda Vector with Lagrange multiplier information in the SOL dense solver format. Used for warm starts.

H Cholesky factor of Hessian Approximation. Either in natural order (Hessian Yes) or reordered (Hessian No). Used for warm starts using natural order with NPSOL and NLSSOL .

callback For large dense or nearly dense quadratic problems (probType == qp) it is more efficient to use a callback function from the MEX routine to compute the matrix-vector product F x. Then Prob.QP.F is never copied into the MEX solver. This option applies to SQOPT only.

optPar Vector with optParN elements with parameter information for SOL solvers. Initialized to missing value, −999. The elements used are described in the help for each solver. If running TOMLAB format, also see the help of the TOMLAB solver interface routine whos name always has the letters TL added, e.g. minosTL .

optParN Number of elements in optPar , defined as 62.

Table 22: Information stored in the structure Prob.FUNCS

Field Description

f Name of m-file computing the objective function f(x).

g Name of m-file computing the gradient vector g(x). If Prob.FUNCS.g is empty then numerical derivatives will be used.

H Name of m-file computing the Hessian matrix H(x).

c Name of m-file computing the vector of constraint functions c(x).

dc Name of m-file computing the matrix of constraint normals ∂ c(x)/dx.

d2c Name of m-file computing the 2nd part of 2nd derivative matrix of the Lagrangian function, Σ_i λ_i ∂²c(x)/dx².

r Name of m-file computing the residual vector r(x).

J Name of m-file computing the Jacobian matrix J(x).

d2r Name of m-file computing the 2nd part of the Hessian for nonlinear least squares problem, i.e. Σ_i=1^m r_i(x) ∂²r_i(x) / ∂ x_j ∂ x_k.

Information stored in the structure Prob.DUNDEE

Field

Description

callback

If 1, use a callback to Matlab to compute QP.F x in BQPD and miqpBB . Faster when F is large and nearly dense. Avoids copying the matrix to the MEX solvers.

kmax

Maximum dimension of the reduced space (k ), default equal to dimension of problem. Set to 0 if solving an LP problem.

mlp

Maximum number of levels of recursion.

mode

Mode of operation, default set as 2 * Prob.WarmStart.

Solution (warmstart).

Dimension of reduced space (warmstart).

Steepest-edge normalization coefficient (warmstart).

Indices of active constraints, first n−k. (warmstart).

List of pointers to recursion information in ls (warmstart).

peq

Pointer to end of equality constraint indices in ls (warmstart).

PrintFile

Name of print file. Amount/print type is determined by Prob.DUNDEE.optPar(1) .

optPar

Vector with optimization parameters. Described in Table 23.

Table 23: Prob.DUNDEE.optPar values used by TOMLAB /MINLP solvers . −999 in any element gives default value.

Used by:

Index Name Default Description BQPD miqpBB filterSQP minlpBB

1 iprint 0 Print level in DUNDEE solvers. O O O O

2 tol 10⁻¹⁰ Relative accuracy in BQPD solution. O O O O

3 emin 1 1/0: Use/do not use constraint scaling in BQPD O O O O

4 sgnf 5 10⁻⁴ Maximum relative error in two numbers equal in exact arithmetic. O O O O

5 nrep 2 Maximum number of refinement steps. O O O O

6 npiv 3 No repeat of more than npiv steps were taken. O O O O

7 nres 2 Maximum number of restarts if unsuccessful. O O O O

8 nfreq 500 Maximum interval between refactorizations. O O O O

9 ubd 100 Constraint violation parameter I - - O O

10 tt 0.125 Constraint violation parameter II - - O O

11 NLP_eps 10⁻⁶ Relative tolerance for NLP solutions - - O O

12 epsilon 10⁻⁶ Accuracy for x tests O O O O

13 MIopttol 10⁻⁴ Accuracy for f tests - O - O

14 fIP 10²⁰ Upper bound on the IP value wanted. - O - O

15 timing 0 1/0: Use/do not use timing - O - -

16 max_time 4000 Maximum time (sec's) allowed for the run - O - -

17 branchtype 1 Branching strategy (1,2,3) - O - -

18 ifsFirst 0 Exit when first IP solution is found - O - -

19 infty 10²⁰ Large value used to represent infinity O O O O

20 Nonlin 0 Treat all constraints as nonlinear if 1 - - O O

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