Lower bounds for the runtime of a global multi-objective evolutionary algorithm

While for single-objective evolutionary algorithms many sharp run-time analyses exist, there are only few for multiobjective evolutionary algorithms (MOEAs), and even fewer for global MOEAs, that is, MOEAs using standard bit mutation (instead of 1-bit mutation, which is easier to analyze, but less common in practice). For example, there is not a single lower bound result for the runtime of the classic “global simple evolutionary multiobjective optimizer” (GSEMO) on the biobjective test function LeadingOnesTrailingZeros (LOTZ). An upper bound of O(n²/p), where p ≤ 1/n is the mutation probability, for this runtime was proven ten years ago by Giel (CEC 2003). In this work, we show that this bound is sharp for small values of p, namely p <; n^-7/4.