The NLPNRR subroutine computes an optimum value of a function by using a Newton-Raphson ridge method.

See the section Nonlinear Optimization and Related Subroutines for a listing of all NLP subroutines. See Chapter 14 for a description of the arguments of 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 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 , 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);