Running Time Analysis of MOEA/D on Pseudo-Boolean Functions

Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving the multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present running time analysis of a simple MOEA with mutation and crossover based on the MOEA/D framework (MOEA/D-C) on five pseudo-Boolean functions. Our rigorous theoretical analysis shows that by properly setting the number of subproblems, the upper bounds of expected running time of MOEA/D-C obtaining a set of solutions to cover the Pareto fronts (PFs) of these problems are apparently lower than those of the one with mutation-only (MOEA/D-M). Moreover, to effectively obtain a set of solutions to cover the PFs of these problem, MOEA/D-C only needs to decompose these MOPs into several subproblems with a set of simple weight vectors while MOEA/D-M needs to find
$\Omega (n)$
optimally decomposed weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vectors for different problems and make the algorithm more efficient. This paper provides some insights into the working principles of MOEA/D and explains why some existing decomposition-based MOEAs work well in computational experiments.