Language Reference: NLPQUA Call :: SAS/IML(R) 9.2 User's Guide

Language Reference

NLPQUA Call

nonlinear optimization by quadratic method

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

See the section "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11 for a description of the inputs to and outputs of all NLP subroutines.

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

$\frac{1}2 x^tgx + g^tx + {con}$

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

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 (refer to Gill et al. 1984). If

is the number of free parameters (that is,

minus the number of active boundary constraints), and

is the number of active linear constraints, then

is an

orthogonal matrix containing null space

in its first

columns and range space

in its last

columns. The matrix

is an

triangular matrix of the form $t_{ij}=0$ for $i \lt n-j$ . 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:

The quad argument specifies the symmetric Hessian matrix, , of the quadratic term. The input can be in dense or sparse form. In dense form, all entries of the quad matrix must be specified. If $n\leq 3$ , the dense specification must be used. The sparse specification can be useful when has many zero elements. You can specify an matrix in which each row represents one of the nonzero elements of . The first column specifies the row location in , the second column specifies the column location, and the third column specifies the value of the nonzero element.
The lin argument specifies the linear part of the quadratic optimization problem. It must be a vector of length or . If lin is a vector of length , it specifies the vector of the linear term, and the constant term is considered zero. If lin is a vector of length , then the first elements of the argument specify the vector and the last element specifies the constant term of the objective function.

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 $x \geq 0$ , the quadratic optimization problem solved with the LCP call, which solves the linear complementarity problem. Refer to SAS/IML Software: Usage and Reference, Version 6, First Edition for details.

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 will store and process the matrix

as sparse:

No general linear constraints are specified.
The memory needed for the sparse storage of is less than 80% of the memory needed for dense storage.
is not a diagonal matrix. If is diagonal, it is stored and processed as a diagonal matrix.

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 heading alpha is the step size, $\alpha$ , computed with the line-search algorithm.
The heading slope refers to , the slope of the search direction at the current parameter iterate $x^{(k)}$ . For minimization, this value should be significantly smaller than zero. Otherwise, the line-search algorithm has difficulty reducing the function value sufficiently.

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

$x = [ x_1 \ x_2 ], g = [ 0.02 & 0 \ 0 & 2 ], g = [ 0 \ 0 ], {con} = -100$

Then

$\frac{1}2 x^tgx - g^tx + {con} = 0.5[0.02x_1^2 + 2x_2^2] - 100 = 0.01x_1^2 + x_2^2 - 100$

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 }; 
    c    = {  2. -50.  .   ., 
             50.  50.  .   ., 
             10.  -1. 1. 10.}; 
    x = { -1. -1.}; 
    optn = {0 2}; 
    CALL NLPQUA(rc,xres,quad,x,optn,c,,,,lin);

The quad argument specifies the

matrix, and the lin argument specifies the

vector with the value of con appended as the last element. The matrix C specifies the boundary constraints and the general linear constraint.

The iteration history follows.

  
  
                          Optimization Start 
                          Parameter Estimates 
                                       Gradient           Lower           Upper 
                                       Objective           Bound           Bound 
   N Parameter         Estimate        Function      Constraint      Constraint 
  
   1 X1                6.800000        0.136000        2.000000       50.000000 
   2 X2               -1.000000       -2.000000      -50.000000       50.000000 
  
           Value of Objective Function = -98.5376 
  
  
  
                       Linear Constraints 
  
   1   59.00000 :  10.0000 <= + 10.0000 * X1 - 1.0000 * X2 
  
  
                  Null Space Method of Quadratic Problem 
  
                   Parameter Estimates                2 
                   Lower Bounds                       2 
                   Upper Bounds                       2 
                   Linear Constraints                 1 
                   Using Sparse Hessian               _ 
  
                    Optimization Start 
  
    Active Constraints        0  Objective Function  -98.5376 
    Max Abs Gradient Element  2

  
  
                     Function        Active       Objective 
  Iter    Restarts      Calls   Constraints        Function 
  
     1           0          2             1       -99.87349 
     2           0          3             1       -99.96000 
  
         Objective    Max Abs              Slope of 
          Function   Gradient      Step      Search 
  Iter      Change    Element      Size   Direction 
  
     1      1.3359     0.5882     0.706      -2.925 
     2      0.0865          0     1.000      -0.173 
  
  
                             Optimization Results 
  
 Iterations                             2  Function Calls            4 
 Gradient Calls                         3  Active Constraints        1 
 Objective Function                -99.96  Max Abs Gradient Element  0 
 Slope of Search Direction   -0.173010381 
  
    ABSGCONV convergence criterion satisfied. 
  
  
  
  
  
  
  
  
  
                      Optimization Results 
                      Parameter Estimates 
  
                                       Gradient    Active 
                                       Objective    Bound 
   N Parameter         Estimate        Function    Constraint 
  
   1 X1                2.000000        0.040000     Lower BC 
   2 X2                       0               0 
  
              Value of Objective Function = -99.96 
  
  
          Linear Constraints Evaluated at Solution 
  
  1    10.00000 = -10.0000 + 10.0000 * X1 - 1.0000 * X2

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