SOL MINOS

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

Direct Solver Call

A direct solver call is not recommended unless the user is 100 % sure that no other solvers will be used for the problem. Please refer to subsection Using TOMLAB for information on how to use MINOS with TOMLAB.

Purpose

minos solves nonlinear optimization problems defined as

${\begin{array}{ll}\min \limits _{x}&f(x)\\&\\s/t&{\begin{array}{lcccl}&&x&&,\\b_{L}&\leq &Ax&\leq &b_{U}\\&&c(x)&&\\\end{array}}\end{array}}$

where $x\in \mathbb {R} ^{n}$ , $f(x)\in \mathbb {R}$ , $A\in \mathbb {R} ^{m_{1}\times n}$ , $b_{L},b_{U}\in \mathbb {R} ^{n+m_{1}+m_{2}}$ and $c(x)\in \mathbb {R} ^{m_{2}}$ .

or quadratic optimization problems defined as

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x\\&\\s/t&{\begin{array}{lcccl}&&x&&,\\b_{L}&\leq &g(x)&\leq &b_{U}\\&&Ax&&\\\end{array}}\end{array}}$

where $c,x\in \mathbb {R} ^{n}$ , $F\in \mathbb {R} ^{n\times n}$ , $A\in \mathbb {R} ^{m_{1}\times n}$ , and $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$ .

The full input matrix A has three parts A = [d/dx g(x); A; c'];

Calling Syntax

The file 'funfdf.m' must be defined and contain: function [mode, f, g] = funfdf(x, Prob, mode, nstate) to compute the objective function f and the gradient g at the point x.

The file 'funcdc.m' must be defined and contain: function [mode ,c ,dcS] = funcdc(x, Prob, mode, nstate) to compute the nonlinear constraint value c and the constraint Jacobian dcS for the nonlinear constraints at the point x.

NOTE: The matrix dcS MUST be a SPARSE MATLAB matrix. Do dcS = sparse(dcS); after dcS has been computed.

[hs, xs, pi, rc, Inform, nS, nInf, sInf, Obj, iwCount, gObj, fCon, gCon] = minos(H, A, bl, bu, nnCon, nnObj, 
nnJac, Prob, iObj,  optPar, Warm, hs, xs, pi, nS, SpecsFile, PrintFile,  SummFile, PriLev, ObjAdd,  moremem, 
ProbName );

Description of Inputs

The following fields are used:

Input	Description
H	Matrix n x n in a quadratic programming (QP) problem. DENSE or SPARSE. Leave empty if LP, or NLP problem.
A	Constraint matrix, m x n SPARSE (nonlinear, linear and objective) m > 0 always!!! Define dummy constraint for unconstrained problems.
bl	Lower bounds on (x,g(x),Ax,c').
bu	Upper bounds on (x,g(x),Ax,c'). NOTE! The bl and bu values for the last nonlinear constraint c must have reverse signs and be put in each other places: If cL <= c(x) <= cU , then bl = -cU and bu = -cL . This is because the bounds acts as the constraints on the slack variables for the nonlinear constraints.
nnCon	Number of nonlinear constraints.
nnObj	Number of nonlinear objective variables.
nnJac	Number of nonlinear Jacobian variables.
Prob	Must be a structure. No check is made in the MEX interface. If TOMLAB calls minos, then Prob is the standard TOMLAB problem structure, otherwise the user should set: Prob.P = ProblemNumber, where ProblemNumber is some integer. If the problem is a LP or QP problem (H defined), the user does not have to specify anything else in the structure. For a general nonlinear objective or nonlinear constraints names of two user written routines must be given: funfdf, actual name stored in Prob.FUNCS.fg, with syntax [mode, f, g] = funfdf(x, Prob, mode, nstate). funcdc, actual name stored in Prob.FUNCS.cdc, with syntax [mode, c, dcS] = funcdc(x, Prob, mode, nstate). MINOS is calling the TOMLAB routines nlp fg.m and nlp cdcS.m in the call- back, and they call funfdf and funcdc, respectively. If these fields in Prob are empty (Prob.FUNCS.fg, Prob.FUNCS.cdc), the TOMLAB callback routines calls the usual function routines. Then the Prob struct should be normally defined, and the fields Prob.FUNCS.f, Prob.FUNCS.g, Prob.FUNCS.c, Prob.FUNCS.dc be set in the normal way (e.g. by the routine mFiles.m, or one of the Assign-routines like conAssign.m). If the mode parameter is 0, funfdf should return f, otherwise both f and the gradient vector g. If the mode parameter is 0, funcdc should return c, otherwise both c and dcS. Note that each row in dcS corresponds to a constraint, and that dcS must be a SPARSE matrix. The user could also write his own versions of the routines nlp fg.m and nlp cdcS.m and put them before in the path.
iObj	Says which row of A is a free row containing a linear objective vector c. If there is no such vector, iObj = 0. Otherwise, this row must come after any nonlinear rows, so that nnCon <= iObj <= m.
optPar	Vector with optimization parameters overriding defaults and the optionally specified SPECS file. If using only default options, set optPar as an empty matrix.
Warm	Flag, if true: warm start. Default cold start (if empty). If 'Warm Start' xs, nS and hs must be supplied with correct values.
hs	Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds), 3=basic (between bounds).
xs	Initial vector, optionally including m slacks at the end. If warm start, full xs must be supplied.
pi	Lagrangian multipliers for the nnCon nonlinear constraints. If empty, set as 0.
nS	# of superbasics. Only used if calling again with a Warm Start.
SpecsFile	Name of the SPECS input parameter file, see TOMLAB.
PrintFile	Name of the Print file. Name includes the path, maximal number of characters = 500.
SummFile	Name of the Summary file. Name includes the path, maximal number of char- acters = 500.
PriLev	Printing level in the minos m-file and minos MEX-interface. = 0 Silent = 1 Summary information = 2 More detailed information
ObjAdd	Constant added to the objective for printing purposes, typically 0.
moremem	Add extra memory for the sparse LU, might speed up the optimization. 1E6 is 10MB of memory. If empty, set as 0.
ProbName	Name of the problem. ¡=100 characters are used in the MEX interface. In the MINOS solver the first 8 characters are used in the printed solution and in some routines that output BASIS files. Blank is OK.

Description of Outputs

The following fields are used:

Output	Description
hs	Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds), 3=basic (between bounds). Basic and superbasic variables may be outside their bounds by as much as the Feasibility tolerance. Note that if scaling is specified, the Feasibility tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as 0.1 outside their bounds, but this is unlikely unless the problem is very badly scaled. Check the "Primal infeasibility" printed after the EXIT message. Very occasionally some nonbasic variables may be outside their bounds by as much as the Feasibility tolerance, and there may be some nonbasics for which xn(j) lies strictly between its bounds. If ninf > 0, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used).
xs	Solution vector (n+m by 1) with n decision variable values together with the m slack variables.
pi	Lagrangian multipliers (dual solution vector) (m x 1 vector)
rc	Vector of reduced costs, g - ( A I )Tp, where g is the gradient of the objective function if xn is feasible, or the gradient of the Phase-1 objective otherwise. If ninf = 0, the last m entries are -p. Reduced costs vector is of n+m length.
Inform	Result of MINOS run. 0 Optimal solution found. 1 The problem is infeasible. 2 The problem is unbounded (or badly scaled). 3 Too many iterations. 4 Apparent stall. The solution has not changed for a large number of iterations (e.g. 1000). 5 The Superbasics limit is too small. 6 User requested termination (by returning bad value). 7 Gradient seems to be giving incorrect derivatives. 8 Jacobian seems to be giving incorrect derivatives. 9 The current point cannot be improved. 10 Numerical error in trying to satisfy the linear constraints (or the linearized nonlinear constraints). The basis is very ill-conditioned. 11 Cannot find a superbasic to replace a basic variable. 12 Basis factorization requested twice in a row. Should probably be treated as inform = 9. 13 Near-optimal solution found. Should probably be treated as inform = 9. 20 Not enough storage for the basis factorization. 21 Error in basis package. 22 The basis is singular after several attempts to factorize it (and add slacks where necessary). 30 An OLD BASIS file had dimensions that did not match the current problem. 32 System error. Wrong number of basic variables. 40 Fatal errors in the MPS file. 41 Not enough storage to read the MPS file. 42 Not enough storage to solve the problem.
nS	# of superbasics.
nInf	Number of infeasibilities.
sInf	Sum of infeasibilities.
Obj	Objective function value at optimum.
iwCount	Number of iterations (major and minor), function and constraint calls.
gObj	Gradient of the nonlinear objective.
fCon	Nonlinear constraint vector.
gCon	Gradient vector (non-zeros) of the nonlinear constraint vector.

Using TOMLAB

Purpose

minosTL solves nonlinear optimization problems defined as

${\begin{array}{ll}\min \limits _{x}&f(x)\\&\\s/t&{\begin{array}{lcccl}x_{L}&\leq &x&\leq &x_{U},\\b_{L}&\leq &Ax&\leq &b_{U}\\c_{L}&\leq &c(x)&\leq &c_{U}\\\end{array}}\end{array}}$

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$ , $f(x)\in \mathbb {R}$ , $A\in \mathbb {R} ^{m_{1}\times n}$ , $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$ and $c_{L},c(x),c_{U}\in \mathbb {R} ^{m_{2}}$ .

Calling Syntax

Using the driver routine tomRun :

Prob = ''o''Assign( ... );
Result = tomRun('minos', Prob, 1);

Description of Inputs

Prob, The following fields are used:
x_L, x_U	Bounds on variables.
b_L, b_U	Bounds on linear constraints.
c_L, c_U	Bounds on nonlinear constraints.
A	Linear constraint matrix.
QP.c	Linear coefficients in objective function.
PriLevOpt	Print level.
WarmStart	If true, use warm start, otherwise cold start.
SOL.xs	Solution and slacks from previous run.
SOL.hs	State for solution and slacks from previous run.
SOL.nS	Number of superbasics from previous run.
SOL.moremem	Add more memory if MINOS stops with not enough storage message. 1E6 is 10MB of memory. Default n+m1+m2 (number of variables + linear + nonlin- ear constraints).
SOL.SpecsFile	Name of user defined SPECS file, read BEFORE optPar() is used.
SOL.PrintFile	Name of SOL Print file. Amount and type of printing determined by SPECS parameters or optPar parameters.
SOL.SummFile	Name of SOL Summary File.
SOL.optPar	Elements > -999 takes precedence over corresponding TOMLAB params. See Table in SOL Using the SOL Solvers in TOMLAB.

Description of Outputs

Result, The following fields are used:
Result	The structure with results (see ResultDef.m).
f_k	Function value at optimum.
x_k	Solution vector.
x_0	Initial solution vector.
g_k	Gradient of the function.
c_k	Nonlinear constraint residuals.
cJac	Nonlinear constraint gradients.
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;
cState	State of nonlinear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
v_k	Lagrangian multipliers (for bounds + dual solution vector).
ExitFlag	Exit status from minos.m (similar to TOMLAB).
Inform	Result of MINOS run. 0 Optimal solution found. 1 The problem is infeasible. 2 The problem is unbounded (or badly scaled). 3 Too many iterations. 4 Apparent stall. The solution has not changed for a large number of iterations (e.g. 1000). 5 The Superbasics limit is too small. 6 User requested termination (by returning bad value). 7 Gradient seems to be giving incorrect derivatives. 8 Jacobian seems to be giving incorrect derivatives. 9 The current point cannot be improved. 10 Numerical error in trying to satisfy the linear constraints (or the linearized nonlinear constraints). The basis is very ill-conditioned. 11 Cannot find a superbasic to replace a basic variable. 12 Basis factorization requested twice in a row. Should probably be treated as inform = 9. 13 Near-optimal solution found. Should probably be treated as inform = 9. 20 Not enough storage for the basis factorization. 21 Error in basis package. 22 The basis is singular after several attempts to factorize it (and add slacks where necessary). 30 An OLD BASIS file had dimensions that did not match the current problem. 32 System error. Wrong number of basic variables. 40 Fatal errors in the MPS file. 41 Not enough storage to read the MPS file. 42 Not enough storage to solve the problem.
rc	Vector of reduced costs, g-(AI)Tp, where g is the gradient of the objective function if xn is feasible, or the gradient of the Phase-1 objective otherwise. If ninf = 0, the last m entries are -p. Reduced costs vector is of n+m length.
Iter	Number of iterations.
FuncEv	Number of function evaluations. GradEv Number of gradient evaluations.
ConstrEv	Number of constraint evaluations.
QP.B	Basis vector in TOMLAB QP standard.
MinorIter	Number of minor iterations.
Solver	Name of the solver (minos).
SolverAlgorithm	Description of the solver.
SOL.xs	Solution and slack variables.
SOL.hs	Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds), 3=basic (between bounds). Basic and superbasic variables may be outside their bounds by as much as the Feasibility tolerance. Note that if scaling is specified, the Feasibility tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as 0.1 outside their bounds, but this is unlikely unless the problem is very badly scaled. Check the "Primal infeasibility" printed after the EXIT message. Very occasionally some nonbasic variables may be outside their bounds by as much as the Feasibility tolerance, and there may be some nonbasics for which xn(j) lies strictly between its bounds. If ninf > 0, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used).
SOL.nS	# of superbasics.
SOL.nInf	# of infeasibilities.
SOL.sInf	Sum of infeasibilities.

optPar

Description

Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by MINOS, unless the value is altered in the SPECS file.

Definition: nnL = max(nnObj,nnJac))

Description of Inputs

Table: The following fields are used:

#	SPECS keyword text	Lower	Default	Upper	Comment
Printing
1.	PRINT LEVEL	0	0	11111	JFLXB: Jac, fCon, lambda, x, B=LU stats
Frequencies I
5.	PRINT FREQUENCY	0	100
6.	SUMMARY FREQUENCY	0	100
7.	SOLUTION YES/NO	0	1	1	1 = YES; 0 = NO
8.	SUPPRESS PARAMETERS	0	0	1	1 = True
Convergence Tolerances
9.	ROW TOLERANCE	> 0	1E-6
10.	OPTIMALITY TOLERANCE	> 0	max(1E - 6, (10eps_R )^0.5 ) = 1.73E-6
11.	FEASIBILITY TOLERANCE	> 0	1E-6
Derivative checking
13.	VERIFY LEVEL	-1	-1	3	-1,0,1,2,3
14.	START OBJECTIVE CHECK AT COL	0	1	nnObj
15.	STOP OBJECTIVE CHECK AT COL	0	nnObj	nnObj
16.	START CONSTRAINT CHECK AT COL	0	1	nnJac
17.	STOP CONSTRAINT CHECK AT COL	0	nnJac	nnJac
Scaling
18.	SCALE OPTION See Section 6.2 for more information.	0	1 or 2	2	2 if LP,1 if NLP
19.	SCALE TOLERANCE	> 0	0.9	< 1
20.	SCALE PRINT	0	0	1	1 = True
21.	CRASH TOLERANCE	0	0.1	< 1
Other Tolerances
22.	LINESEARCH TOLERANCE	> 0	0.1	< 1
LU I
23.	LU FACTORIZATION TOLERANCE	1	100 or 5	100 if LP
24.	LU UPDATE TOLERANCE	1	10 or 5	10 if LP
25.	LU SWAP TOLERANCE	> 0	1.22E-4	eps^1/4
26.	LU SINGULARITY TOLERANCE	> 0	3.25E-11	eps^0.67
LP or LC subproblems
27.	PIVOT TOLERANCE	> 0	3.25E-11	eps^0.67
28.	CRASH OPTION	0	3	3	0,1,2,3
29.	WEIGHT ON LINEAR OBJECTIVE	0.0	0.0	during Phase 1
30.	ITERATIONS LIMIT m3=1 if length(Prob.QP.c) > 0, otherwise m3=0. TOMLAB default: max(10000,3(m+m3) + 10nnL).	0	3(m+m3) + 10nnL
SLC method
32.	MAXIMIZE	0	0	1	1=maximize
33.	LAGRANGIAN	0	1	1	1=YES, 0=NO
34.	PENALTY PARAMETER	0.0	1.0
35.	MAJOR ITERATIONS LIMIT	> 0	50
36.	MINOR ITERATIONS LIMIT	> 0	40
37.	MAJOR DAMPING PARAMETER	> 0	2.0
38.	MINOR DAMPING PARAMETER	> 0	2.0
39.	DERIVATIVE LEVEL Is always set by minosTL dependent on Prob.ConsDiff, Prob.NumDiff.	0	3	3	0,1,2,3
40.	RADIUS OF CONVERGENCE	0.0	0.01
41.	FUNCTION PRECISION	> 0	3.0E-13	eps0.8 = eps_R
42.	DIFFERENCE INTERVAL	> 0	5.48E-7	eps0.4
43.	CENTRAL DIFFERENCE INTERVAL	> 0	6.69E-5	eps0.8/3
44.	COMPLETION	0	1 LC, 0 1 NC	0=PARTIAL 1=FULL
45.	UNBOUNDED STEP SIZE	> 0	1E10
46.	UNBOUNDED OBJECTIVE	> 0	1E20
Hessian approximation
47.	HESSIAN DIMENSION	1	50	1+nnL
48.	SUPERBASICS LIMIT	1	50	1+nnL TOMLAB default (to avoid termination with Superbasics Limit too small): If n <= 5000: max(50, n + 1) If n > 5000: max(500, n + 200 - size(A, 1) - length(cL )) Avoid setting REDUCED HESSIAN (number of columns in reduced Hessian). It will then be set to the same value as the SUPERBASICS LIMIT by MINOS. Frequencies II
51.	CHECK FREQUENCY	> 0	60
52.	EXPAND FREQUENCY	> 0	10000
53.	FACTORIZATION FREQUENCY	> 0	50
LU II
63.	LU PARTIAL PIVOTING or LU COMPLETE PIVOTING or LU ROOK PIVOTING	0	0	3	0=partial 1=complete 2=rook
Additional parameters
67.	AIJ TOLERANCE Elements \|a(i, j)\| < AI J TOLERANCE are set as 0	0	1E-10
70.	SUBSPACE Convergence tolerance in current subspace before consider moving off another constraint.	> 0	0.5	1	Subspace tolerance

SOL MINOS

Contents

Direct Solver Call

Purpose

Calling Syntax

Description of Inputs

Description of Outputs

Using TOMLAB

Purpose

Calling Syntax

Description of Inputs

Description of Outputs

optPar

Description

Description of Inputs

Table: The following fields are used:

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