Uncrowded Hypervolume-Based Multiobjective Optimization with Gene-Pool Optimal Mixing

Domination-based multiobjective (MO) evolutionary algorithms (EAs) are today arguably the most frequently used type of MOEA. These methods, however, stagnate when the majority of the population becomes nondominated, preventing further convergence to the Pareto set. Hypervolume-based MO optimization has shown promising results to overcome this. Direct use of the hypervolume, however, results in no selection pressure for dominated solutions. The recently introduced Sofomore framework overcomes this by solving multiple interleaved single-objective dynamic problems that iteratively improve a single approximation set, based on the uncrowded hypervolume improvement (UHVI). It thereby however loses many advantages of population-based MO optimization, such as handling multimodality. Here, we reformulate the UHVI as a quality measure for approximation sets, called the uncrowded hypervolume (UHV), which can be used to directly solve MO optimization problems with a single-objective optimizer. We use the state-of-the-art gene-pool optimal mixing evolutionary algorithm (GOMEA) that is capable of efficiently exploiting the intrinsically available grey-box properties of this problem. The resulting algorithm, UHV-GOMEA, is compared with Sofomore equipped with GOMEA, and the domination-based MO-GOMEA. In doing so, we investigate in which scenarios either domination-based or hypervolume-based methods are preferred. Finally, we construct a simple hybrid approach that combines MO-GOMEA with UHV-GOMEA and outperforms both.