Language Reference

CALL NLPQUA (rc, xr, quad, x0 <, opt> <, blc> <, tc> <, par> <, "ptit"> <, lin> ) ;

This subroutine is supported only by the IML procedure.

The NLPQUA subroutine computes an optimum value of a quadratic objective function.

See the section Nonlinear Optimization and Related Subroutines for a listing of all NLP subroutines. See Chapter 16 for a description of the arguments of NLP subroutines.

The NLPQUA subroutine uses a fast algorithm for maximizing or minimizing the quadratic objective function

subject to boundary constraints and general linear equality and inequality constraints. The algorithm is memory-consuming for problems with general linear constraints.

The matrix must be symmetric but not necessarily positive definite (or negative definite for maximization problems). The constant term c affects only the value of the objective function, not its derivatives or the optimal point .

The algorithm is an active-set method in which the update of active boundary and linear constraints is done separately. The decomposition of the matrix of active linear constraints is updated iteratively (Gill et al. 1984). If is the number of free parameters (that is, n minus the number of active boundary constraints) and is the number of active linear constraints, then is an orthogonal matrix that contains null space Z in its first columns and range space Y in its last columns. The matrix is an triangular matrix of the form for . The Cholesky factor of the projected Hessian matrix is updated simultaneously with the decomposition when the active set changes.

The objective function is specified by the input arguments quad and lin, as follows:

As in the other optimization subroutines, you can use the blc argument to specify boundary and general linear constraints, and you must provide a starting point x0 to determine the number of parameters. If x0 is not feasible, a feasible initial point is computed by linear programming, and the elements of x0 can be missing values.

Assuming nonnegativity constraints , the quadratic optimization problem is solved with the LCP call, which solves the linear complementarity problem.

Choosing a sparse (or dense) input form of the quad argument does not mean that the algorithm used in the NLPQUA subroutine is necessarily sparse (or dense). If the following conditions are satisfied, the NLPQUA algorithm stores and processes the matrix as sparse:

The sparse NLPQUA algorithm uses a modified form of minimum degree Cholesky factorization (George and Liu 1981).

In addition to the standard iteration history, the NLPNRA subroutine prints the following information:

The Betts problem (see the section Constrained Betts Function) can be expressed as a quadratic problem in the following way:

Then

The following statements use the NLPQUA subroutine to solve the Betts problem:

lin  = { 0. 0. -100};
quad = {  0.02   0.0 ,
          0.0    2.0 };
con  = {  2. -50.  .   .,
         50.  50.  .   .,
         10.  -1. 1. 10.};
x = { -1. -1.};
opt = {0 2};
call nlpqua(rc, xres, quad, x, opt, con, , , , lin);

The quad argument specifies the matrix, and the lin argument specifies the g vector with the value of c appended as the last element. The matrix con specifies the boundary constraints and the general linear constraint.

The iteration history follows.