Convergence of Stochastic Search Algorithms to Finite Size Pareto Set Approximations

In this work we study the convergence of generic stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of $\epsilon$-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We investigate two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation. Finally, we demonstrate the potential for a possible hybridization of a given stochastic search algorithm with a particular local search strategy -- multi-objective continuation methods -- by showing that the concept of $\epsilon$-dominance can be integrated into this approach in a suitable way.