SOL QP-MINOS

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

Using TOMLAB

Purpose

minosQPTL solves quadratic optimization problems defined as

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

where 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, 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 A \in \MATHSET{R}^{m_1 \times n}} , and Failed to parse (unknown function "\MATHSET"): {\displaystyle b_L,b_U \in \MATHSET{R}^{m_1}} .

Calling Syntax

Using the driver routine tomRun :

Prob = qpAssign( ... );
Result = tomRun('qp-minos', Prob ... );
or
Prob = ProbCheck( ... ); 
Result = minosQPTL(Prob);

Call Prob = qpAssign( ... ) or Prob=ProbDef to define the Prob for the second option.

Description of Inputs

Prob, The following fields are used:
x_L, x_U	Bounds on variables.
b_L, b_U	Bounds on linear constraints.
A	Linear constraint matrix.
QP.c	Linear coefficients in objective function.
QP.F	Quadratic matrix of size nnObj x nnObj. nnObj < n is OK.
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+m (number of variables + linear 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 50.

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 c (linear objective).
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 (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 - ( 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.
Iter	Number of iterations.
FuncEv	Number of function evaluations. Set to Iter.
GradEv	Number of gradient evaluations. Set to Iter.
ConstrEv	Number of constraint evaluations. Set to 0.
QP.B	Basis vector in TOMLAB QP standard.
MinorIter	Number of minor iterations. NOT SET.
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 variables: 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.

Description of Inputs

The following fields are used:

#	SPECS keyword text	Lower	Default	Upper	Comment
Printing
1.	PRINT LEVEL	0	0	1	0=brief 1=LU stats
Frequencies I
5.	PRINT FREQUENCY	0	100
6.	SUMMARY FREQUENCY	0	1
7.	SOLUTION YES/NO	0	1	1	1 = YES; 0 = NO
8.	SUPPRESS PARAMETERS	0	0	1	1 = True
Convergence Tolerances
10.	OPTIMALITY TOLERANCE	> 0	max(1E - 6, (10epsR )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
Scaling
18.	SCALE OPTION See Section 6.2 for more information.	0	1	2
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 FACTOR TOLERANCE	1	5.0
24.	LU UPDATE TOLERANCE	1	5.0
25.	LU SWAP TOLERANCE	>0	1.22E-4	eps1/4
26.	LU SINGULARITY TOLERANCE	>0	3.25E-11	eps0.67
LP parameters
27.	PIVOT TOLERANCE	> 0 3.25E-11	eps0.67
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
31.	PARTIAL PRICE	1	1
32.	MAXIMIZE	0	0	1	1=maximize
Reduced-gradient method
39.	DERIVATIVE LEVEL Is always set by minosqpTL to 3.	0	3	3	0,1,2,3
41.	FUNCTION PRECISION	> 0	3.0E-13	eps0.8 = epsR
42.	DIFFERENCE INTERVAL	> 0	5.48E-8	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 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.	1	50	1+nnL
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	0	1E-10 Elements |a(i, j)| < AI J TOLERANCE are set as 0
70.	SUBSPACE Convergence tolerance in current subspace before consider moving off another constraint.	> 0	0.5	1	Subspace tolerance