Lipschitz-inspired HALRECT Algorithm for Derivative-free Global Optimization

Abstract: This article considers a box-constrained global optimization problem for Lipschitz-continuous functions with an unknown Lipschitz constant. Motivated by the famous DIRECT (DIviding RECTangles), a new HALRECT (HALving RECTangles) algorithm is introduced. A new deterministic approach combines halving (bisection) with a new multi-point sampling scheme in contrast to trisection and midpoint sampling used in the most existing DIRECT-type algorithms. A new partitioning and sampling scheme utilizes more comprehensive information about the objective function. Four different strategies of selecting potentially optimal hyper-rectangles are introduced to exploit the information about the objective function effectively. The original HALRECT algorithm and other introduced HALRECT variations (twelve in total) are tested and compared with the other twelve recently introduced DIRECT-type algorithms on $96$ box-constrained benchmark functions from DIRECTGOLib v1.1, and 96 perturbed their versions. The extensive experimental results show a very promising performance compared to state-of-the-art DIRECT-type global optimization. New HALRECT approaches offers high robustness across problems of different degrees of complexity, varying from simple - uni-modal and low dimensional to complex - multi-modal and higher dimensionality.