SOL LSSOL

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 LSSOL with TOMLAB.

Purpose

lssol solves least squares optimization problems defined as

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

where $x\in \mathbb {R} ^{n}$, $y\in \mathbb {R} ^{M}$, $H\in \mathbb {R} ^{M\times n}$, $A\in \mathbb {R} ^{m_{1}\times n}$, $b_{L},b_{U}\in \mathbb {R} ^{n+m_{1}}$.

Dependent on the input of H, y, c and Upper, any of the following ten problems are solved. If m == length(y) > 0, then a LS problem is assumed Upper is > 0 ==> H is upper trapezodial.

Problems solved by LSSOL.

H	y	c	Upper	Description
empty	empty	empty	-	Feasible point problem	(FP)
empty	empty	n x 1	-	Linear programming problem	(LP)
n x n	empty	empty	0	Quadratic programming problem	(QP1)
n x n	empty	n x 1	0	Quadratic programming problem	(QP2)
n x n	empty	empty	1	Quadratic programming problem	(QP3)
n x n	empty	n x 1	1	Quadratic programming problem	(QP4)
m x n	m	empty	0	Least squares problem	(LS1)
m x n	m	n x 1	0	Least squares problem	(LS2)
m x n	m	empty	1	Least squares problem	(LS3)
m x n	m	n x 1	1	Least squares problem	(LS4)

Calling Syntax

[x, Inform,  iState, cLamda, Iter,  fObj,  r, kx] = lssol(A, bl, bu, c, x, optPar, H, y, 
Warm, iState, Upper, kx, SpecsFile, PrintFile,  SummFile, PriLev, ProbName );

Description of Inputs

The following fields are used:
A	Constraint matrix, nclin x n (DENSE).
bl	Lower bounds on (x,Ax), nclin+n x 1 vector (DENSE).
bu	Upper bounds on (x,Ax), nclin+n x 1 vector (DENSE).
c	Linear objective function cost coeffs, n x 1 (DENSE). If isempty(c), setting c(1:n)=0;.
x	Initial estimate of solution vector x. If isempty(x), x(1:n)=0.
optPar	Vector with optimization parameters overriding defaults and the optionally specified SPECS file. If length(optPar) ¡ 62, lssol sets the rest of the values to missing value (-999).
H	Matrix H in quadratic part of objective function (DENSE). H is either empty (FP, LP), or quadratic n x n (QP), or m x n (LS).
y	Data vector of length m for LS problem, otherwise empty.
Warm	If Warm > 0, then warm start, otherwise cold Start. Default 0. If warm start, then x and iState must be set properly. Normally the values from last call to lssol are used.
iState	Working set (if Warm start) (nclin+m) x 1 (DENSE). If length(iState) ¡nclin+m, setting iState(1:n+m)=0 & Warm=0. iState(i)=0: Corresponding constraint not in the initial working set. iState(i)=1: Inequality constraint at its lower bound in working set. iState(i)=2: Inequality constraint at its upper bound in working set. iState(i)=3: Equality constraint in the initial working set, bl(i)==bu(i).
Upper	If > 0, then H must be an upper-trapezoidal matrix.
kx	Order of the n columns in A for the QP3, QP4, LS3 or LS4 problem.
SpecsFile	Name of the OPTIONS File, see TOMLAB Guide.
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.
PriLev	Print level in the lssol MEX-interface.. = 0 Silent = 10 Dimensions are printed if isempty(PriLev), set as 0.

Description of Outputs

The following fields are used:
x	Solution vector with decision variable values (n x 1 vector).
Inform	Result of LSSOL run. 0 = Optimal solution with unique minimizer found. 1 = Weak local solution (nonunique) was reached. 2 = The solution appears to be unbounded. 3 = The constraints could not be satisfied. The problem has no feasible solu- tion. 4 = Too many iterations, in either phase. 5 = 50 changes of working set without change in x, cycling? 6 = An input parameter was invalid. Other = UNKNOWN LSSOL Inform value.
iState	Status of working set, se input description of iState.
cLamda	Lagrangian multipliers (dual solution vector) (m x 1 vector).
Iter	Number of iterations.
Ax	A*x.
fObj	Value of objective function if feasible, or sum of infeasibilities.
r	Residual for LS problem.

Using TOMLAB

Purpose

lssolTL solves least squares optimization problems defined as

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}||Cx-d||\\&\\s/t&{\begin{array}{lcccl}x_{L}&\leq &x&\leq &x_{U},\\b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}\end{array}}$

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $d\in \mathbb {R} ^{M}$, $C\in \mathbb {R} ^{M\times n}$, $A\in \mathbb {R} ^{m_{1}\times n}$, $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$.

Calling Syntax

Using the driver routine tomRun :

Prob = ''o''Assign( ... );
Result = tomRun('lssol', Prob ... );
or
Prob = ProbCheck( ... ); 
Result = lssolTL(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.
PriLevOpt	Print level.
Warm	If true, use warm start, otherwise cold start.
SOL.xs	Solution (and slacks not used) from previous run (Warm start).
SOL.iState	Working set (if Warm start) (n+m) x 1 (DENSE) If length(iState) ¡ n+m, setting iState(1:n+m)=0; iState(i)=0: Corresponding constraint not in the initial QP working set. iState(i)=1: Inequality constraint at its lower bound in QP working set. iState(i)=2: Inequality constraint at its upper bound in QP working set. iState(i)=3: Equality constraint in the initial QP working set, bl(i)==bu(i).
SOL.cLamda	Lagrangian multipliers from previous run (Warm start).
SpecsFile	Name of user defined SPECS file, read BEFORE optPar() is used.
PrintFile	Name of SOL Print file. Amount and type of printing determined by SPECS parameters or optPar parameters.
SummFile	Name of SOL Summary File.
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	Exact gradient computed at optimum.
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 lssol.m (similar to TOMLAB).
Inform	LSSOL information parameter. 0 = Optimal solution with unique minimizer found. 1 = Weak local solution (nonunique) was reached. 2 = The solution appears to be unbounded. 3 = The constraints could not be satisfied. The problem has no feasible solu- tion. 4 = Too many iterations, in either phase. 5 = 50 changes of working set without change in x, cycling? 6 = An input parameter was invalid. Other = UNKNOWN LSSOL Inform value.
rc	Reduced costs. If ninf=0, last m == -v k.
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 (LSSOL).
SolverAlgorithm	Description of the solver.
SOL.xs	Solution and slack variables.
SOL.cLamda	Lagrangian multipliers.
SOL.iState	State for variables and constraints in iState.

optPar

Description

Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by LSSOL, 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	10	0,1,5,10,20,30
Convergence Tolerances
10.	OPTIMALITY TOLERANCE	> 0	1.1E-8	sqrt(eps)
11.	FEASIBILITY TOLERANCE	> 0	1.1E-8	sqrt(eps)
Other Tolerances
21.	CRASH TOLERANCE	> 0	0.01	< 1
27.	RANK TOLERANCE	> 0	1.1E-14	100*eps
30.	ITERATIONS LIMIT	> 0	max(50,5(n+m))
36.	FEASIBILITY PHASE ITERATIONS	> 0	max(50,5(n+m))
45.	INFINITE STEP SIZE	> 0	1E20
46.	INFINITE BOUND SIZE	> 0	1E20