TOMLAB OPTIMIZATION ENVIRONMENT: slsSolve

slsSolve

Purpose

For finding a sparse least squares solution to a constrained least squares problem.

Description

Finds a Sparse Least Squares (sls) solution to a constrained least squares problem with many variables, with the use of any suitable NLP TOMLAB solver

The use is mainly for sparse problems, where the structure in both the Jacobian of the residuals and the Jacobian of the nonlinear constraints are utilized by a sparse NLP solver, e.g. SNOPT.

See slsDemo.m in tomlab\examples for an example

Problem

        min  0.5*r'r, where r(x) is in R^m
         x
        s/t   x_L <=   x  <= x_U, x is in R^n
              b_L <= A x  <= b_U
              c_L <= c(x) <= c_U

The constrained least squares (L2) problem is solved in slsSolve by rewriting the problem as a general constrained optimization problem. A set of m additional variables z , stored as x(n+1:m), is added

        min    0.5*z'z, 
         x
        s/t   x_L <=   x              <= x_U
              b_L <= A x              <= b_U
              c_L <= c(x)             <= c_U
                0 <= r(x) - z         <= 0

Input Parameters

   Prob       Structure Prob. Prob must be defined!!! 
              Best is to use Prob = clsAssign(.....), if using the TQ format.
              The problem should be created in the Tomlab constrained 
              nonlinear least squares format (cls)
              This format is aimed for a vector valued function, with a
              Jacobian matrix as the derivative.

   PriLev     The second input argument. Default == 2.
              If PriLev == 0, slsSolve is silent, except for error messages.
              If > 0, slsSolve prints summary output about problem
              transformation
              slsSolve calls PrintResult(Result,PriLev), i.e. printing in
              PrintResult is made if PriLev > 0. 
              PriLev == 2 displays standard output in PrintResult. 

Use Prob = probInit('name of file',problem_number'); if solving a predefined problem in the Init File (IF) format.

Extra fields used in Prob:

   SolverL2    Name of the Tomlab solver. Valid names are:
               conSolve, nlpSolve, (sTrustr, clsSolve)
               If Tomlab /SOL is installed: minos, snopt, npopt

   L2Type     =1 standard constrained formulation

The rest of the fields in Prob should be defined as wanted by the selected solver. See the help for the solver.

In particular:

   x_0:     Starting point
   x_L:     Lower bounds for x
   x_U:     Upper bounds for x
   b_L:     Lower bounds for linear constraints
   b_U:     Upper bounds for linear constraints
   A:       Linear constraint matrix
   c_L:     Lower bounds for nonlinear constraints
   c_U:     Upper bounds for nonlinear constraints

   ConsPattern The pattern of the constraint Jacobian
   JacPattern  The pattern of the residual Jacobian

Note that Prob.LS.y must have the correct residual length, if JacPattern is empty (but not ConsPattern).

slsSolve will create the new Prob.ConsPattern to be used by the solver using the information in ConsPattern and JacPattern.

Output Parameters

 Result Structure with results from optimization. See help for the used solver
        The output in Result, i.e. fields Result.x_k, Result.r_k, Result.J_k,
        Result.c_k, Result.cJac, Result.x_0, Result.xState, Result.cState,
        Result.v_k, is transformed back to the original problem. 
        Result.g_k is Result.J_k'*Result.r_k.
        The output in Result.Prob is the result after slsSolve
        transformed the problem, i.e. the altered Prob structure

qpSolve

sTrustr