On the globalization of Wilson-type optimization methods by means of generalized reduced gradient methods

Abstract

--Zusammenfassung On the Globalization of Wilson-type Optimization Methods by Means of Generalized Reduced GradientMethods. For solving nonlinear optimization problems, i.e. for the determination of Kuhn-Tucker points a numerical method is proposed. The considerations continue investigations of Best/ Br~iuninger/Ritter/Robinson and Kleinmichel/Richter/Sch6nefeld. In these papers (published in this journal) different local methods are combined with a penalty method in such a way that global convergence can be guaranteed. In order to show that the basic principle of coupling is applicable to a number of further globally convergent methods a local Wilson-type method is now initialized by a feasible direction method that uses reduced gradients. In both phases of the method similar subproblems (special quadratic programs) occur. Therefore, in contrast to the papers mentioned above systems of linear equations have to be solved exclusively. Under usual assumptions the algorithm is shown to be globally and superlinearly convergent.AMS Subject Classifications: 90C30, 65K05.

A class of nonmonotone stabilization methods in unconstrained optimization

“…In this paper we consider a general scheme which combines the nonmonotone line search technique of [-9] with the watchdog technique of [-6], and makes use of a steplength acceptability criterion of the kind employed in [7] and [8]. In particular, we discuss some aspects of the convergence properties of Newton-type methods.…”

“…However, in this case the sequence may leave the region of interest, with the possible occurrence of overflows in the evaluation of the objective function or its derivatives. A better criterion could be that of controlling the stepsize by means of some heuristic such as that proposed in [-7] and [8]. In any case, it is necessary to perform a periodic control of the function values by using a line search whenever a suitable reduction of these values is not achieved.…”

“…Noticeable exceptions to this point of view are some nonmonotone algorithms proposed in the field of nonsmooth optimization (see, e.g. [4,5]), the watchdog technique of [6], the steplength acceptability criterion employed in [7] and [8] in the context of constrained optimization and trajectory methods based on controlling the integration error at each step rather than the function values.…”

A note on a globalization of Wilson-type optimization methods

“…subject to (1 a) gi(x)<_O (i~I) (lb) gi(x)=O (i~J), (1 c) f, g~ (is I w J) I Rn--' R1 are twice continuously differentiable and t I: = { 1, ..., m}, J: = {m + 1 ..... p j, p > m. The investigations we are going to describe base on a hybrid method consisting of a feasible direction method in a first phase and a Wilson-type method in a second phase. This algorithm possessing the following structure (2) was proposed in [1] to solve problems (1) in that case that the equations (1 c) do not occur. In general, it can't be expected that the iterates generated by any first phase feasible direction method are strictly feasible if the nonlinear equations (1 c) have to be satisfied additionally.…”

“…phase feasible directi~ t ~ me t hod Wilson-type ~ me t hod ~test~2 phase 2 (2) To overcome these difficulties an approach proposed in [2] is used in the following. We apply the first phase method considered in [1] to inequality constrained problems of the form F (x, b) = rain ! subject to x E G with…”