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

Language Reference

NLPNRR Call

nonlinear optimization by Newton-Raphson ridge method

CALL NLPNRR( 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 NLPNRR algorithm uses a pure Newton step when both the Hessian is positive definite and the Newton step successfully reduces the value of the objective function. Otherwise, a multiple of the identity matrix is added to the Hessian matrix.

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.

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.

The NLPNRR method performs well for small to medium-sized problems, and it does not need many function, gradient, and Hessian calls. However, if the gradient is not specified analytically by using the "grd" module argument, or if the computation of the Hessian module specified with the "hes" argument is computationally expensive, one of the (dual) quasi-Newton or conjugate gradient algorithms might be more efficient.

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

The heading ridge refers to the value of the nonnegative ridge parameter. A value of zero indicates that a Newton step is performed. A value greater than zero indicates either that the Hessian approximation is zero or that the Newton step fails to reduce the optimization criterion. A large value can indicate optimization difficulties.
The heading rho refers to $\rho$ , the ratio of the achieved difference in function values and the predicted difference, based on the quadratic function approximation. A value that is much smaller than one indicates possible optimization difficulties.

The following statements invoke the NLPNRR 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 nlpnrr(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 Ridge Optimization 
  
                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             1       -99.87337 
     2           0          3             1       -99.96000 
     3           0          4             1       -99.96000 
  
  
                                             Ratio 
                                            Actual 
         Objective    Max Abs                  and 
          Function   Gradient            Predicted 
  Iter      Change    Element    Ridge      Change 
  
     1      1.3358     0.5887        0       0.706 
     2      0.0866   0.000040        0       1.000 
     3    4.07E-10          0        0       1.014 
  
  
  
                   Optimization Results 
  
    Iterations               3  Function Calls                       5 
    Hessian Calls            4  Active Constraints                   1 
    Objective Function  -99.96  Max Abs Gradient Element             0 
    Ridge                    0  Actual Over Pred Change   1.0135158294 
  
  
    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             0.000000134               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