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

Language Reference

NLPNRA Call

nonlinear optimization by Newton-Raphson method

CALL NLPNRA( rc, xr, "fun", x0 <,opt, blc, tc, par, "ptit", "grd", "hes">);

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 NLPNRA algorithm uses a pure Newton step at each iteration when both the Hessian is positive definite and the Newton step successfully reduces the value of the objective function. Otherwise, it performs a combination of ridging and line-search to compute successful steps. If the Hessian is not positive definite, a multiple of the identity matrix is added to the Hessian matrix to make it positive definite (refer to Eskow & Schnabel 1991).

The subroutine uses the gradient $g^{(k)} = \nabla f(x^{(k)})$ and the Hessian matrix

$g^{(k)} = \nabla^2 f(x^{(k)})$ , and it requires continuous first- and second-order derivatives of the objective function inside the feasible region. If second-order derivatives are computed efficiently and precisely, the NLPNRA method does not need many function, gradient, and Hessian calls, and it can perform well for medium to large problems.

Note that using only function calls to compute finite difference approximations for second-order derivatives can be computationally very expensive and can contain significant rounding errors. If you use the "grd" input argument to specify a module that computes first-order derivatives analytically, you can reduce drastically the computation time for numerical second-order derivatives. The computation of the finite difference approximation for the Hessian matrix generally uses only calls of the module that specifies the gradient.

In each iteration, a line search is done along the search direction to find an approximate optimum of the objective function. The default line-search method uses quadratic interpolation and cubic extrapolation. You can specify other line-search algorithms with the fifth element of the opt argument. See the section "Options Vector" for details.

In unconstrained and boundary constrained cases, the NLPNRA algorithm can take advantage of diagonal or sparse Hessian matrices that are specified by the input argument "hes". To use sparse Hessian storage, the value of the ninth element of the opt argument must specify the number of nonzero Hessian elements returned by the Hessian module. See the section "Objective Function and Derivatives" for more details.

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 following statements invoke the NLPNRA subroutine to solve the constrained Betts optimization problem (see the section "Constrained Betts Function"). The iteration history follows.

  
    start F_BETTS(x); 
       f = .01 * x[1] * x[1] + x[2] * x[2] - 100.; 
       return(f); 
    finish F_BETTS; 
  
    con = {  2. -50.  .   ., 
            50.  50.  .   ., 
            10.  -1. 1. 10.}; 
    x = {-1. -1.}; 
    optn = {0 2}; 
    call nlpnra(rc,xres,"F_BETTS",x,optn,con); 
    quit;

  
  
                          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 
  
            Newton-Raphson Optimization with Line Search 
  
                     Without Parameter Scaling 
               Gradient Computed by Finite Differences 
          CRP Jacobian Computed by Finite Differences 
                   Parameter Estimates               2 
                   Lower Bounds                      2 
                   Upper Bounds                      2 
                   Linear Constraints                1 
  
                    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             0       -98.81551 
     2*          0          3             0       -99.40840 
     3*          0          4             1       -99.87504 
     4           0          5             1       -99.96000 
     5           0          6             1       -99.96000 
  
  
         Objective    Max Abs              Slope of 
          Function   Gradient      Step      Search 
  Iter      Change    Element      Size   Direction 
  
     1      0.2779     1.8000     0.100      -2.925 
     2*     0.5929     1.2713     0.294      -2.365 
     3*     0.4666     0.5829     0.542      -1.181 
     4      0.0850   0.000039     1.000      -0.170 
     5     3.9E-10   9.537E-7     1.000     -76E-11 
  
                      Optimization Results 
  
    Iterations                           5  Function Calls            7 
    Hessian Calls                        6  Active Constraints        1 
    Objective Function              -99.96  Max Abs Gradient Element  0 
    Slope of Search Direction -7.64376E-10  Ridge                     0 
  
    GCONV 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            -0.000000196               0 
  
               Value of Objective Function = -99.96 
  
  
  
         Linear Constraints Evaluated at Solution 
  
  1     10.00000  =  -10.0000 + 10.0000 * X1 - 1.0000 * X2

Top of Page