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**CALL NLPTR(***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 NLPTR subroutine is a trust-region method that uses the gradient and Hessian matrix . It requires that the objective function

The

Note that finite difference approximations for
second-order derivatives using only function
calls are computationally very expensive.
If you specify first-order derivatives analytically with
the *"grd"* module argument, you can drastically reduce
the computation time for numerical second-order derivatives.
Computing the finite difference approximation for the
Hessian matrix **G** generally uses only *n* calls
of the module that computes the gradient analytically.

The NLPTR method performs well for small- to medium-sized problems
and does not need many function, gradient, and Hessian calls.
However, if the gradient is not specified analytically by using
the *"grd"* argument or if the computation of the Hessian
module, as specified by the *"hes"* module 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 NLPTR subroutine prints the following information:

- Under the heading
*Iter*, an asterisk (*) printed after the iteration number indicates that the computed Hessian approximation was singular and had to be ridged with a positive value. - The heading
*lambda*represents the Lagrange multiplier, . This has a value of zero when the optimum of the quadratic function approximation is inside the trust region, in which case a trust-region-scaled Newton step is performed. It is greater than zero when the optimum is at the boundary of the trust region, in which case the scaled Newton step is too long to fit in the trust region and a quadratically-constrained optimization is done. Large values indicate optimization difficulties, and as in Gay (1983), a negative value indicates the special case of an indefinite Hessian matrix. - The heading
*radius*refers to , the radius of the trust region. Small values of the radius combined with large values of in subsequent iterations indicate optimization problems.

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