Running Time Analysis of Multi-objective Evolutionary Algorithms on a Simple Discrete Optimization Problem

Marco Laumanns, Lothar Thiele, Eckart Zitzler, Emo Welzl, Kalyanmoy Deb

Abstract: For the first time, a running time analysis of populationbased multi-objective evolutionary algorithms for a discrete optimization problem is given. To this end, we define a simple pseudo-Boolean bi-objective problem (Lotz: leading ones– trailing zeroes) and investigate time required to find the entire set of Pareto-optimal solutions. It is shown that different multi-objective generalizations of a (1+1) evolutionary algorithm (EA) as well as a simple population-based evolutionary multi-objective optimizer (SEMO) need on average at least Θ(n 3) steps to optimize this function. We propose the fair evolutionary multi- objective optimizer (FEMO) and prove that this algorithm performs a black box optimization in Θ(n 2 log n) function evaluations where n is the number of binary decision variables.