Robustness in Hypervolume-based Multiobjective Search

The use of quality indicators within the search has become a popular approach in the field of evolutionary multiobjective optimization. It relies on the concept to transform the original multiobjective problem into a set problem that involves a single objective function only, namely a quality indicator, reflecting the quality of a Pareto set approximation. Especially the hypervolume indicator has gained a lot of attention in this context since it is the only set quality measure known that guarantees strict monotonicity. Accordingly, various hypervolume-based search algorithms for approximating the Pareto set have been proposed, including sampling-based methods that circumvent the problem that the hypervolume is in general hard to compute. Despite these advances, there are several open research issues in indicator-based multiobjective search when considering real-world applications—the issue of robustness is one of them. For instance with mechanical manufacturing processes, there exist unavoidable inaccuracies that prevent a desired solution to be realized with perfect precision; therefore, a solution in terms of a concrete decision vector is not associated with just one one fixed vector of objective values, but rather with a range of objective values that reflect the variance when slightly changing the decision variables. As a consequence, the optimization model needs to account for such uncertainties and the search method is required to explicitly integrate robustness considerations. While in single-objective optimization, there are various studies dealing with the robustness issue, there are considerably fewer in the multiobjective optimization literature and none in the context of hypervolume-based multiobjective search. This study is set in the latter context and addresses the question of how to incorporate robustness when using the hypervolume indicator within an evolutionary algorithm. To this end, three common robustness concepts are translated to and tested for hypervolume-based search on the one hand, and an extension of the hypervolume indicator is proposed on the other hand that not only unifies those three concepts, but also enables to realize much more general trade-offs between objective values and robustness of a solution. Finally, the approaches are compared on two test problem suites as well as on a newly proposed real-world bridge construction problem.