Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like stochastic domination and coupling or Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results presented here have appeared previously, some, however, only in recent conference publications. While the focus is on collecting tools for the analysis of randomized search heuristics, many of these may be useful as well in the analysis of classic randomized algorithms or discrete random structures.Comment: 91 page

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 multi-objective 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 bi-objective 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. © 2013 IEEE