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nonlinear optimization by quadratic method

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

See "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11, "Nonlinear Optimization Examples," 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
(1/2) xTGx + gTx + 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 G must be symmetric but not necessarily positive definite (or negative definite for maximization problems). The constant term con affects only the value of the objective function, not its derivatives or the optimal point x*.

The algorithm is an active-set method in which the update of active boundary and linear constraints is done separately. The QT decomposition of the matrix Ak of active linear constraints is updated iteratively (refer to Gill, Murray, Saunders, and Wright, 1984). If nf is the number of free parameters (that is, n minus the number of active boundary constraints), and na is the number of active linear constraints, then Q is an nf ×nf orthogonal matrix containing null space Z in its first nf-na columns and range space Y in its last na columns. The matrix T is an na ×na triangular matrix of the form tij=0 for i < n-j. The Cholesky factor of the projected Hessian matrix ZTkGZk is updated simultaneously with the QT 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 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 G 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 "Constrained Betts Function" ) can be expressed as a quadratic problem in the following way:
x = [ x_1 \ 
G = [ 0.02 & 0 \ 
 0 & 2 
g = [ 0 \ 
 {con} = -100
(1/2) xTGx - gTx + con = 0.5[0.02x12 + 2x22] - 100 = 0.01x12 + x22 - 100
The following statements use the NLPQUA subroutine to solve the Betts problem:
   proc iml;
      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 G matrix, and the lin argument specifies the g 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 is shown in Figure 17.10.

                         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

Figure 17.10: Iteration History for the NLPQUA Subroutine

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.