The NLPDD subroutine solves a nonlinear optimization problem by using the double dogleg 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 double dogleg optimization method combines the ideas of the quasi-Newton and trust-region methods. In each iteration, the algorithm computes the step, , as a linear combination of the steepest descent or ascent search direction, , and a quasi-Newton search direction, , as follows:

The step must remain within a specified trust-region radius (see Fletcher (1987)). Hence, the NLPDD subroutine uses the dual quasi-Newton update but does not perform a line search. You can specify one of two update formulas with the fourth element of the opt input argument.

The double dogleg optimization technique works well for medium to moderately large optimization problems, in which the objective function and the gradient are much faster to compute than the Hessian. The implementation is based on Dennis and Mei (1979) and Gay (1983), but it is extended for boundary and linear constraints. The NLPDD subroutine generally needs more iterations than the techniques that require second-order derivatives (NLPTR, NLPNRA, and NLPNRR), but each of the NLPDD iterations is computationally inexpensive. Furthermore, the NLPDD subroutine needs only gradient calls to update the Cholesky factor of an approximate Hessian.

In addition to the standard iteration history, the NLPDD routine prints the following information:

• The heading lambda refers to the parameter of the double dogleg step. A value of 0 corresponds to the full (quasi-) Newton step.

• The heading slope refers to , the slope of the search direction at the current parameter iterate . For minimization, this value should be significantly smaller than zero.

The following statements invoke the NLPDD subroutine to solve the constrained Betts optimization problem (see the section Constrained Betts Function).

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 1};
call nlpdd(rc, xres, "F_BETTS", x, optn, con);

The preceding statements produce the following iteration history. The optimization converged after six iterations.