Stochastic Population Update Can Provably Be Helpful in Multi-Objective Evolutionary Algorithms

TL;DR

It is proved that the expected running time of a well-established MOEA for solving a commonly studied bi-objective problem, OneJumpZeroJump, can be exponentially decreased if replacing its deterministic population update mechanism by a stochastic one.

Abstract

Evolutionary algorithms (EAs) have been widely and successfully applied to solve multi-objective optimization problems, due to their nature of population-based search. Population update, a key component in multi-objective EAs (MOEAs), is usually performed in a greedy, deterministic manner. That is, the next-generation population is formed by selecting the best solutions from the current population and newly-generated solutions (irrespective of the selection criteria used such as Pareto dominance, crowdedness and indicators). In this paper, we analytically present that stochastic population update can be beneficial for the search of MOEAs. Specifically, we prove that the expected running time of two well-established MOEAs, SMS-EMOA and NSGA-II, for solving two bi-objective problems, OneJumpZeroJump and bi-objective RealRoyalRoad, can be exponentially decreased if replacing its deterministic population update mechanism by a stochastic one. Empirical studies also verify the effectiveness of the proposed population update method. This work is an attempt to show the benefit of introducing randomness into the population update of MOEAs. Its positive results, which might hold more generally, should encourage the exploration of developing new MOEAs in the area.

Figures (5)

• Figure 1: Illustration of the OneJumpZeroJump problem when $n=20$ and $k=5$. The left subfigure: the function values vs. the number of 1-bits of a solution; the right subfigure: the second function value vs. the first function value, where the set of green points are the Pareto front.
• Figure 2: The objective vectors of the bi-objective RealRoyalRoad problem when $n=5$, where the set of green points are the Pareto front.
• Figure 3: Estimated number of generations of SMS-EMOA/NSGA-II using the deterministic population update divided by that using the stochastic population update for solving the OneJumpZeroJump problem with $k=2$.
• Figure 4: Estimated number of generations of SMS-EMOA/NSGA-II using the deterministic population update divided by that using the stochastic population update for solving the OneJumpZeroJump problem with $k=3$.
• Figure 5: Estimated number of generations of SMS-EMOA/NSGA-II using the deterministic population update divided by that using the stochastic population update for solving the bi-objective RealRoyalRoad problem.

Theorems & Definitions (22)

• Definition 1: Multi-objective Optimization
• Definition 2: Domination
• Lemma 1
• Definition 3: doerr2021ojzj
• Definition 4: dang2023crossover
• Lemma 2
• Lemma 3: robbins1955remark
• Theorem 1
• Lemma 4
• Theorem 2