SOL SQOPT

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 #Using TOMLAB for information on how to use SQOPT with TOMLAB.

Purpose

sqopt solves dense quadratic optimization problems defined as

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

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

Calling Syntax

The full input matrix A has two parts A = [A; d']; The position of the row d' is iObj. iObj=0 means no linear part in A.

NOTE: There are two ways to give the linear objective: either explicit as vector c or as part of the sparse matrix A, as d (or both ways).

[xs, hs, pi, rc, Inform, nS, nInf, sInf, Obj, iwCount] = sqopt( A, bl, bu, H, c, hElast, 
iObj, optPar, Warm, hs, xs, nS, SpecsFile, PrintFile,  SummFile, ObjAdd, moremem, ProbName, 
Prob);

Description of Inputs

The following fields are used:
A	Constraint matrix, m x n (SPARSE).
bl	Lower bounds on (x,Ax,d').
bu	Upper bounds on (x,Ax,d').
H	Quadratic matrix, n x n, SPARSE or DENSE, empty if LP problem. If H is a string, H should be the name of a function routine, e.g if H = 'HxComp' then the function routine: function Hx = HxComp(x, nState, Prob) should compute H * x. The user must define this routine. nState == 1 if calling for the first time, otherwise 0. Third argument, the Prob structure, should only be used if calling SQOPT with the additional input parameter Prob, see below. Tomlab implements this callback to the predefined Matlab function HxFunc.m, using the call if Prob.SOL.callback == 1.
c	Linear objective.
hElast	Defines which bounds are elastic in elastic mode. hElast(j): 0 = variable j cannot be infeasible. 1 = variable j can violate its lower bound. 2 = variable j can violate its upper bound. 3 = variable j can violate either its lower or upper bound.
iObj	Says which row of A is a free row containing a linear objective vector d. If there is no such vector, iObj = 0.
optPar	Vector with optimization parameters overriding defaults and the optionally specified SPECS file. Set empty if only using default parameters.
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 variables: 0=nonbasic (on bl), 1=nonbasic (on bu), 2=superbasic (between bounds), 3=basic (between bounds).
xs	Initial x vector (nx1), optionally including m slacks at the end. If Warm start, full n+m vector xs must be supplied.
nS	# of superbasics. Used if a Warm Start, otherwise set to 0.
SpecsFile	Name of the SPECS input parameter file.
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 characters = 500.
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 to 0.
ProbName	Name of the problem. ¡=100 characters are used in the MEX interface. In the SQOPT solver the first 8 characters are used in the printed solution and in some routines that output BASIS files. Blank is OK.
Prob	Sending the Prob structure is optional, only of use if sending H as a function string, see input H.

Description of Outputs

The following fields are used:
xs	Solution vector (n+m by 1) with n decision variable values together with the m slack variables.
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 value of the optional parameter 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 xs(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).
pi	Lagrangian multipliers (dual solution vector) (m x 1 vector).
rc	A vector of reduced costs, g - ( A - I )T p, where g is the gradient of the objective if xs is feasible (or the gradient of the Phase-1 objective otherwise). The last m entries are p.
Inform	Result of SQOPT run. 0 finished successfully 1 optimality conditions satisfied 2 feasible point found 4 weak QP minimizer 10 the problem appears to be infeasible 11 infeasible linear constraints 12 infeasible linear equalities 14 infeasibilities minimized 20 the problem appears to be unbounded 21 unbounded objective 30 resource limit error 31 iteration limit reached 33 the superbasics limit is too small 40 terminated after numerical difficulties 42 singular basis 43 cannot satisfy the general constraints 44 ill-conditioned null-space basis 50 error in the user-supplied functions 53 the QP Hessian is indefinite 70 user requested termination 73 terminated during QP objective evaluation 74 terminated from monitor routine 80 insufficient storage allocated 81 work arrays must have at least 500 elements 82 not enough character storage 83 not enough integer storage 84 not enough real storage 90 input arguments out of range 91 invalid input argument 92 basis file dimensions do not match this problem 93 the QP Hessian is indefinite 140 system error 141 wrong number of basic variables 142 error in basis package
nS	# of superbasics.
nInf	Number of infeasibilities.
sInf	Sum of infeasibilities.
Obj	Objective function value at optimum.
iwCount	Number of QP iterations in iwCount(1), number of Hx products.

Using TOMLAB

Purpose

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

Calling Syntax

Using the driver routine tomRun :

Prob = ''o''Assign( ... );
Result = tomRun('sqopt', Prob ... );
or
Prob = ProbCheck( ... ); 
Result = sqoptTL(Prob);

Call Prob = oAssign( ... ) 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.callback	If 1, use a callback to Matlab to compute QP.F * x for different x. Faster when F is very large and almost dense, avoiding copying of F from Matlab to MEX.
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.hElastic	Defines which variables are elastic in elastic mode. hElastic(j): 0 = variable j is non-elastic and cannot be infeasible. 1 = variable j can violate its lower bound. 2 = variable j can violate its upper bound. 3 = variable j can violate either its lower or upper bound.
SOL.moremem	Add more memory if SQOPT stops with not enough storage message. 1E6 is 10MB of memory. Default 0.
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 of the function.
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 sqopt.m (similar to TOMLAB).
Inform	Result of SQOPT run. 0 finished successfully 1 optimality conditions satisfied 2 feasible point found 4 weak QP minimizer 10 the problem appears to be infeasible 11 infeasible linear constraints 12 infeasible linear equalities 14 infeasibilities minimized 20 the problem appears to be unbounded 21 unbounded objective 30 resource limit error 31 iteration limit reached 33 the superbasics limit is too small 40 terminated after numerical difficulties 42 singular basis 43 cannot satisfy the general constraints 44 ill-conditioned null-space basis 50 error in the user-supplied functions 53 the QP Hessian is indefinite 70 user requested termination 73 terminated during QP objective evaluation 74 terminated from monitor routine 80 insufficient storage allocated 81 work arrays must have at least 500 elements 82 not enough character storage 83 not enough integer storage 84 not enough real storage 90 input arguments out of range 91 invalid input argument 92 basis file dimensions do not match this problem 93 the QP Hessian is indefinite 140 system error 141 wrong number of basic variables 142 error in basis package
rc	A vector of reduced costs, g - ( A - I )T p, where g is the gradient of the objective if xs is feasible (or the gradient of the Phase-1 objective otherwise). The last m entries are p.
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.
Solver	Name of the solver (sqopt).
SolverAlgorithm	Description of the solver.
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 value of the optional parameter 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 xs(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.hs	State for variables and slacks in xs.
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 SQOPT, unless the value is altered in the SPECS file (input SpecsFile).

See TOMLAB User's Guide for the SPECS keywords and description.

Description of Inputs

#	SPECS keyword text	Lower	Default	Upper	Comment
LP/QP Parameters I - Printing
2.	PRINT LEVEL	0	0	10	0, 1 or 10
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
LP/QP Parameters II - Convergence Tolerances
11.	FEASIBILITY TOLERANCE	> 0	1E-6
12.	OPTIMALITY TOLERANCE	> 0	1E-6
Scaling
18.	SCALE OPTION	0	2	2
19.	SCALE TOLERANCE	> 0	0.9	< 1
20.	SCALE PRINT	0	0	1	1 = True
21.	CRASH TOLERANCE	0	0.1	< 1
LU I
23.	LU FACTOR TOLERANCE	1	100
24.	LU UPDATE TOLERANCE	1	10
25.	LU SWAP TOLERANCE	> 0	1.22E-4	eps1/4
26.	LU SINGULARITY TOLERANCE	> 0	3.25E-11	eps0.67
LP/QP Parameters III
27.	PIVOT TOLERANCE	>' 0	3.25E-11	eps0.67
28.	CRASH OPTION	0	0	3	0,1,2,3
29.	ELASTIC WEIGHT	0	1
30.	ITERATIONS LIMIT	0	10000
31.	PARTIAL PRICE	1	10
32.	MAXIMIZE	0	0	1	1=maximize
QP Objective
45.	UNBOUNDED STEP SIZE
48.	SUPERBASICS LIMIT	> 0	> 0	1E20 min(500,1+nnObj)
LP/QP Parameters IV
49.	ELASTIC MODE	0	1		0,1,2
50.	ELASTIC OBJECTIVE	0	2		0,1,2
Frequencies II
51.	CHECK FREQUENCY	> 0	60
52.	EXPAND FREQUENCY	> 0	10000
53.	FACTORIZATION FREQUENCY	> 0	50
LU II
63.	LU COMPLETE PIVOTING or LU PARTIAL PIVOTING	0	0	1	1=complete, 0=partial