TOMLAB OPTIMIZATION ENVIRONMENT: TlsqrTL

TlsqrTL

Purpose

TOMLAB LSQR Least Squares Interface

Syntax

   Result = TlsqrTL(Prob);

Description

LSQR finds a solution x to the following problems:

• Unsymmetric equations - solve A*x = b
• Linear least squares - solve A*x = b in the least-squares sense
• Damped least squares - solve

                  (   A    )*x = ( b )
                  ( damp*I )     ( 0 )
  

  in least-squares sense

where A is a matrix with m rows and n columns, b is an m-vector, and damp is a scalar (All quantities are real). The matrix A is intended to be a large and sparse array, but both full and sparse MATLAB arrays are handled.

Do help Tlsqr to see a discussion of the numerical aspects of using LSQR.

For a problem description, see Tlsqr.

Input Parameters

 Prob   Problem structure in TOMLAB format

Fields used in input structure Prob (call Prob=ProbDef; to define Prob)

 x_L, x_U  Bounds on variables. 
 b_L, b_U  Bounds on linear constraints (not used).
 A         Linear constraint matrix (not used).
 LS.C      Linear matrix m x n.
 LS.y      Data vector m x 1.
 LS.damp   Damping parameter.
 LS.condLim An upper limit on cond(Abar), where Abar = [A;damp * I].
 PriLevOpt Print Level.
 optParam.MaxIter - MaxIter in Tlsqr
 optParam.xTol    - aTol in Tlsqr
 optParam.bTol    - bTol in Tlsqr

 x_0       Initial solution vector. Normally LSQR starts with x=0, but
           if x_0 is given (n-vector), LSQR tries a warm start with x_0.

 D         Preconditioning. Only n diagonal elements of D are given.

 Alloc     Advanced memory handling if Alloc > 0 (default empty (or 0))
           See help Tlsqr.m , input parameter m, for a description of
           how to use Alloc.

Output Parameters

 Result   Structure with results (see ResultDef.m):
 r_k      Residual vector.
 J_k      Jacobian, is just the Prob.LS.C matrix.
 f_k      Function value at optimum.
 x_k      Solution vector.
 x_0      Initial  solution vector, empty if not given as a n-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  (similar to TOMLAB).
 Inform   LSQR information parameter.
 Iter     Number of iterations, set to -1.
 FuncEv   Number of function evaluations. Set to Iter.
 GradEv   Number of gradient evaluations. Set to Iter.
 ConstrEv Number of constraint evaluations. Set to 0.
 Solver   Name of the solver (Tlsqr).
 SolverAlgorithm  Description of the solver.

Tlsqr