Better Running Time of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) by Using Stochastic Tournament Selection

Chao Bian, Chao Qian

Abstract: Evolutionary algorithms (EAs) have been widely used to solve multi-objective optimization problems, and have become the most popular tool. However, the theoretical foundation of multi-objective EAs (MOEAs), especially the essential theoretical aspect, i.e., running time analysis, is still largely underdeveloped. The few existing theoretical works mainly considered simple MOEAs, while the non-dominated sorting genetic algorithm II (NSGA-II), probably the most influential MOEA, has not been analyzed except for a very recent work considering a simplified variant without crossover. In this paper, we present a running time analysis of the standard NSGA-II for solving LOTZ, the commonly used bi-objective optimization problem. Specifically, we prove that the expected running time (i.e., number of fitness evaluations) is $$O(n^3)$$ for LOTZ, which is the same as that of the previously analyzed simple MOEAs, GSEMO and SEMO, as well as the NSGA-II without crossover. Next, we introduce a new parent selection strategy, stochastic tournament selection (i.e., k tournament selection where k is uniformly sampled at random), to replace the binary tournament selection strategy of NSGA-II, decreasing the upper bound on the required expected running time to $$O(n^2)$$ . Experiments are also conducted, suggesting that the derived running time upper bounds are tight. We also empirically compare the performance of the NSGA-II using the two selection strategies on the widely used benchmark problem ZDT1, and the results show that stochastic tournament selection can help the NSGA-II converge faster.