%0 Journal Article
%T Equal Angle Distribution of Polling Directions in Direct-Search Methods
%A Benjamin Van Dyke
%J Journal of Optimization
%D 2014
%R 10.1155/2014/619249
%X The purpose of this paper is twofold: first, to introduce deterministic strategies for directional direct-search methods, including new instances of the mesh adaptive direct-search (MADS) and the generating set search (GSS) class of algorithms, which utilize a nice distribution of PoLL directions when compared to other strategies, and second, to introduce variants of each algorithm which utilize a minimal positive basis at each step. The strategies base their PoLL directions on the use of the QR decomposition to obtain an orthogonal set of directions or on using the equal angular directions from a regular simplex centered at the origin with vertices on the unit sphere. Test results are presented on a set of smooth, nonsmooth, unconstrained, and constrained problems that give comparisons between the various implementations of these directional direct-search methods. 1. Introduction In [1], Vicente and Cust¨Ždio introduced a framework for directional direct-search methods (DSM) encompassing both the MADS [2] class of algorithms (using an underlying mesh and simple decrease) and the Gss [3] class of algorithms (using sufficient decrease under a forcing function ) for black-box optimization problems of the form where the objective function is nonsmooth, possibly discontinuous and is the set of feasible points. The algorithms under this framework are shown to have first order convergence results in the Rockafellar sense [4] for discontinuous functions. The framework is shown in Algorithm 1. Algorithm 1: The directional direct-search method of [ 1]. While both algorithms fall under this framework and are shown to have the same convergence results, there are tradeoffs for either choice. The MADS class of algorithms not only requires a bit more complexity in choosing polling directions, but also requires simple decrease in order to accept an iteration as successful. Under the given framework, any instance of Gss requires sufficient decrease using a forcing function that must be chosen under unknown criteria. The sufficient decrease condition, however, can be implemented in such a way that it is mostly negligible and the method for choosing polling directions is simpler because of the absence of a mesh. Each iteration of this DSM framework is characterized by two steps, a SEARCH and a POLL. The POLL step may be opportunistic (the POLL stops if a successful point has been found) and is governed by a POLL size parameter (MADS denotes this ) such that under certain assumptions. These assumptions include the requirement that at each iteration either the set of SEARCH
%U http://www.hindawi.com/journals/jopti/2014/619249/