Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants

Abstract

We consider a global optimization problem for Lipschitz-continuous functions with an unknown Lipschitz constant. Our approach is based on the well-known DIRECT (DIviding RECTangles) algorithm and motivated by the diagonal partitioning strategy. One of the main advantages of the diagonal partitioning scheme is that the objective function is evaluated at two points at each hyper-rectangle and, therefore, more comprehensive information about the objective function is considered than using the central sampling strategy used in most DIRECT-type algorithms. In this paper, we introduce a new DIRECT-type algorithm, which we call BIRECT (BIsecting RECTangles). In this algorithm, a bisection is used instead of a trisection which is typical for diagonal-based and DIRECT-type algorithms. The bisection is preferable to the trisection because of the shapes of hyper-rectangles, but usual evaluation of the objective function at the center or at the endpoints of the diagonal are not favorable for bisection. In the proposed algorithm the objective function is evaluated at two points on the diagonal equidistant between themselves and the endpoints of a diagonal. This sampling strategy enables reuse of the sampling points in descendant hyper-rectangles. The developed algorithm gives very competitive numerical results compared to the DIRECT algorithm and its well know modifications.