Peeking beyond peaks: Challenges and research potentials of continuous multimodal multi-objective optimization

Christian Grimme, Pascal Kerschke, Pelin Aspar, Heike Trautmann, Mike Preuss, André H. Deutz, Hao Wang, Michael Emmerich

Abstract: Highlights • Multimodality in continuous multi-objective optimization (MOO) is defined very differently. • The notion of multimodality is rather transferred from the single-objective case than understood. • This work suggests a common terminology to the community. • It surveys current development in multimodal continuous MOO. • It suggests new directions of research in the context of multimodal continuous MOO. Abstract Multi-objective (MO) optimization, i.e., the simultaneous optimization of multiple conflicting objectives, is gaining more and more attention in various research areas, such as evolutionary computation, machine learning (e.g., (hyper-)parameter optimization), or logistics (e.g., vehicle routing). Many works in this domain mention the structural problem property of multimodality as a challenge from two classical perspectives: (1) finding all globally optimal solution sets, and (2) avoiding to get trapped in local optima. Interestingly, these streams seem to transfer many traditional concepts of single-objective (SO) optimization into claims, assumptions, or even terminology regarding the MO domain, but mostly neglect the understanding of the structural properties as well as the algorithmic search behavior on a problem’s landscape. However, some recent works counteract this trend, by investigating the fundamentals and characteristics of MO problems using new visualization techniques and gaining surprising insights. Using these visual insights, this work proposes a step towards a unified terminology to capture multimodality and locality in a broader way than it is usually done. This enables us to investigate current research activities in multimodal continuous MO optimization and to highlight new implications and promising research directions for the design of benchmark suites, the discovery of MO landscape features, the development of new MO (or even SO) optimization algorithms, and performance indicators. For all these topics, we provide a review of ideas and methods but also an outlook on future challenges, research potential and perspectives that result from recent developments.