Theoretical Advantage of Multiobjective Evolutionary Algorithms for Problems with Different Degrees of Conflict

Abstract

The field of multiobjective evolutionary algorithms (MOEAs) often emphasizes its popularity for optimization problems with conflicting objectives. However, it is still theoretically unknown how MOEAs perform for different degrees of conflict, even for no conflicts, compared with typical approaches outside this field. As the first step to tackle this question, we propose the OneMaxMin$_k$ benchmark class with the degree of the conflict $k\in[0..n]$, a generalized variant of COCZ and OneMinMax. Two typical non-MOEA approaches, scalarization (weighted-sum approach) and $\epsilon$-constraint approach, are considered. We prove that for any set of weights, the set of optima found by scalarization approach cannot cover the full Pareto front. Although the set of the optima of constrained problems constructed via $\epsilon$-constraint approach can cover the full Pareto front, the general used ways (via exterior or nonparameter penalty functions) to solve such constrained problems encountered difficulties. The nonparameter penalty function way cannot construct the set of optima whose function values are the Pareto front, and the exterior way helps (with expected runtime of $O(n\ln n)$ for the randomized local search algorithm for reaching any Pareto front point) but with careful settings of $\epsilon$ and $r$ ($r>1/(\epsilon+1-\lceil \epsilon \rceil)$). In constrast, the generally analyzed MOEAs can efficiently solve OneMaxMin$_k$ without above careful designs. We prove that (G)SEMO, MOEA/D, NSGA-II, and SMS-EMOA can cover the full Pareto front in $O(\max\{k,1\}n\ln n)$ expected number of function evaluations, which is the same asymptotic runtime as the exterior way in $\epsilon$-constraint approach with careful settings. As a side result, our results also give the performance analysis of solving a constrained problem via multiobjective way.