Multi-Objective Archiving

Most multiobjective optimization algorithms maintain an archive explicitly or implicitly during their search. Such an archive can be solely used to store high-quality solutions presented to the decision maker, but in many cases may participate in the search process (e.g., as the population in evolutionary computation). Over the last two decades, archiving, the process of comparing new solutions with previous ones and deciding how to update the archive/population, stands as an important issue in evolutionary multiobjective optimization (EMO). This is evidenced by constant efforts from the community on developing various effective archiving methods, ranging from conventional Pareto-based methods to more recent indicator-based and decomposition-based ones. However, the focus of these efforts is on empirical performance comparison in terms of specific quality indicators; there is lack of systematic study of archiving methods from a general theoretical perspective. In this article, we attempt to conduct a systematic overview of multiobjective archiving, in the hope of paving the way to understand archiving algorithms from a holistic perspective of theory and practice, and more importantly providing a guidance on how to design theoretically desirable and practically useful archiving algorithms. In doing so, we also present that archiving algorithms based on weakly Pareto-compliant indicators (e.g., $\epsilon $ -indicator and IGD+), as long as designed properly, can achieve the same theoretical desirables as archivers based on Pareto-compliant indicators (e.g., hypervolume indicator). Such desirables include the property limit-optimal, the limit form of the possible optimal property that a bounded archiving algorithm can have with respect to the most general form of superiority between solution sets.

I. Introduction

Multiobjective optimization refers to an optimization scenario where several conflicting objectives are optimized simultaneously. A prominent feature of a multiobjective optimization problem (MOP) is that, in contrast to its single-objective counterpart, it does not have a single optimal solution, but rather a set of tradeoff solutions, called Pareto-optimal solutions or the Pareto front in the objective space, whose size is usually prohibitively large or even infinite.