Finding the Set of Nearly Optimal Solutions of a Multiobjective Optimization Problem

Oliver Schütze, Angel E. Rodriguez-Fernandez, Carlos Segura, Carlos Hernández

Abstract: Evolutionary multiobjective optimization (EMO) is a highly active research field that has attracted many researchers and practitioners over the past three decades. Surprisingly, until now, the goal of almost all EMO algorithms is to compute a suitable finite size representation of the Pareto set/front of a given multiobjective optimization problem or at least a part of it. In other words, the quest is restricted to optimal solutions. In this work, we argue that the entire set of nearly optimal solutions—which includes all optimal ones—is of potential interest for the decision maker as they contain in addition to the optimal solutions alternative realizations or backup solutions. We further make a first effort to reliably compute the set of nearly optimal solutions via EMO algorithms. To this end, we first propose a new set of interest, $N_{Q,\epsilon }$ , and analyze its topology. In a next step, we propose an unbounded archiver that aims to capture $N_{Q,\epsilon }$ and analyze it with respect to monotonicity and limit behavior. After this, we discuss the related subset selection problem which comes with unbounded archivers leading to four different algorithms. Finally, we numerically investigate the behavior of the archiver and the selection strategies, and present some results when using the archiver as external archiver to three widely used multiobjective evolutionary algorithms indicating the benefit of the new approach.