Out of the Niche: Using Direct Search Methods to Find Multiple Global Optima

Multimodal optimization deals with problems where multiple feasible global solutions coexist. Despite sharing a common objective function value, some global optima may be preferred to others for various reasons. In such cases, it is paramount to devise methods that are able to find as many global optima as possible within an affordable computational budget. Niching strategies have received an overwhelming attention in recent years as the most suitable technique to tackle these kinds of problems. In this paper we explore a different approach, based on a systematic yet versatile use of traditional direct search methods. When tested over reference benchmark functions, our proposal, despite its apparent simplicity, noticeably resists the comparison with state-of-the-art niching methods in most cases, both in the number of global optima found and in the number of function evaluations required. However, rather than trying to outperform niching methods-far more elaborated-our aim is to enrich them with the knowledge gained from exploiting the distinctive features of direct search methods. To that end, we propose two new performance measures that can be used to evaluate, compare and monitor the progress of optimization algorithms of (possibly) very different nature in their effort to find as many global optima of a given multimodal objective function as possible. We believe that adopting these metrics as reference criteria could lead to more sophisticated and computationally-efficient algorithms, which could benefit from the brute force of derivative-free local search methods.