Stochastic Population Update Provably Needs An Archive in Evolutionary Multi-objective Optimization

Evolutionary algorithms (EAs) have been widely applied to multi-objective optimization, 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. However, recent studies have questioned this practice and shown that stochastic population update (SPU), which allows inferior solutions have a chance to be preserved, can help MOEAs jump out of local optima more easily. While introducing randomness in the population update process boosts the exploration of MOEAs, there is a drawback that the population may not always preserve the very best solutions found, thus entailing a large population. Intuitively, a possible solution to this issue is to introduce an archive that stores the best solutions ever found. In this paper, we theoretically show that using an archive can allow a small population and accelerate the search of SPU-based MOEAs substantially. Specifically, we analyze the expected running time of two well-established MOEAs, SMS-EMOA and NSGA-II, with SPU for solving a commonly studied bi-objective problem OneJumpZeroJump, and prove that using an archive can bring (even exponential) speedups. The comparison between SMS-EMOA and NSGA-II also suggests that the $(\mu+\mu)$ update mode may be more suitable for SPU than the $(\mu+1)$ update mode. Furthermore, our derived running time bounds for using SPU alone are significantly tighter than previously known ones. Our theoretical findings are also empirically validated on a well-known practical problem, the multi-objective traveling salesperson problem. We hope this work may provide theoretical support to explore different ideas of designing algorithms in evolutionary multi-objective optimization.