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## NLPNRR Call

nonlinear optimization by Newton-Raphson ridge method

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

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 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 and the Hessian matrix ,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 may 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 n 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 may 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 , 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 "Constrained Betts Function" ). The iteration history is shown in Figure 17.8.

   proc iml;
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

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
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

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
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

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


Figure 17.8: Iteration History for the NLPNRR Subroutine

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