Experimental Study of Excessive Local Refinement Reduction Techniques for Global Optimization DIRECT-Type Algorithms

Abstract

This article considers a box-constrained global optimization problem for Lipschitz continuous functions with an unknown Lipschitz constant. The well-known derivative-free global search algorithm DIRECT (DIvide RECTangle) is a promising approach for such problems. Several studies have shown that recent two-step (global and local) Pareto selection-based algorithms are very efficient among all DIRECT-type approaches. However, despite its encouraging performance, it was also observed that the candidate selection procedure has two possible shortcomings. First, there is no limit on how small the size of selected candidates can be. Secondly, a balancing strategy between global and local candidate selection is missing. Therefore, it may waste function evaluations by over-exploring the current local minimum and delaying finding the global one. This paper reviews and employs different strategies in a two-step Pareto selection framework (1-DTC-GL) to overcome these limitations. A detailed experimental study has revealed that existing strategies do not always improve and sometimes even worsen results. Since 1-DTC-GL is a DIRECT-type algorithm, the results of this paper provide general guidance for all DIRECT-type algorithms on how to deal with excessive local refinement more efficiently.

Diagonal Partitioning Strategy Using Bisection of Rectangles and a Novel Sampling Scheme

“…One of these possible improvements is to evaluate the objective function only once at each vertex of each hyperrectangle, where the objective function values at vertices could be stored in a special vertex database, thus avoiding re-evaluation of the objective function at certain shared vertices in adjacent hyper-rectangles. Another feature, as shown during the previous test process, is to find a specific rule about how the change in the original optimization domain should be applied in order to improve the performance of the BIRECT-V algorithm (see [48,46,45,54]).…”

A Novel Approach to Enhance DIRECT-Type Algorithms for Hyper-Rectangle Identification

“…Another study provided a comprehensive exploration of hyper-rectangle splitting in the DIRECT algorithm, revealing an improved strategy for identifying potentially optimal hyper-rectangles, with the full text offering detailed insights into the algorithmic enhancements [25]. Further contributing to DIRECT optimization, a study proposed a new strategy for selecting potentially optimal hyper-rectangles, thereby enriching our understanding of optimization algorithms [23].Finally, an experimental study highlighted the effectiveness of the well-known derivative-free global search algorithm DIRECT, giving an overview of its performance and its practical benefits in solving optimization problems [34]. Together, these works represent a variety of advances, refining and extending the capabilities of optimization algorithms for hyper-rectangle identification and global search tasks.…”

Review and Computational Study on Practicality of Derivative-Free DIRECT-Type Methods

“…To overcome these shortcomings, various techniques (see, e.g. Paulavičius et al, 2020;Stripinis and Paulavičius, 2022a, 2022c, 2023a, including hybrid ones that combine the DIRECT algorithm with other optimization techniques, have been widely adopted in practical applications (Liuzzi et al, 2010(Liuzzi et al, , 2016. By integrating DIRECT with complementary methods, such as local search or metaheuristics, these hybrid approaches aim to mitigate the weaknesses of the algorithm and improve its overall performance.…”