Simplicial Lipschitz optimization without the Lipschitz constant

Abstract

In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant. Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Almost all previous modifications of Direct algorithm use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. Simplicial partitions may be preferable when the initial feasible region is either already a simplex or may be covered by one or a manageable number of simplices. Therefore in this paper we propose and investigate simplicial versions of the partition-based algorithm. In the case of simplicial partitions, definition of potentially optimal subregion cannot be the same as in the rectangular version. In this paper we propose and investigate two definitions of potentially optimal simplices: one involves function values at the vertices of the simplex and another uses function value at the centroid of the simplex. We use experimental investigation to compare performance of the algorithms with different definitions of potentially optimal partitions. The experimental investigation shows, that proposed simplicial algorithm gives very competitive results to Direct algorithm using standard test problems and performs particularly well when the search space and the numbers of local and global optimizers may be reduced by taking into account symmetries of the objective function.