Runtime Analysis of the SMS-EMOA for Many-Objective Optimization

The widely used multiobjective optimizer NSGA-II was recently proven to have considerable difficulties in many-objective optimization. In contrast, experimental results in the literature show a good performance of the SMS-EMOA, which can be seen as a steady-state NSGA-II that uses the hypervolume contribution instead of the crowding distance as the second selection criterion. This paper conducts the first rigorous runtime analysis of the SMS-EMOA for many-objective optimization. To this aim, we first propose a many-objective counterpart, the m-objective mOJZJ problem, of the bi-objective OJZJ benchmark, which is the first many-objective multimodal benchmark used in a mathematical runtime analysis. We prove that SMS-EMOA computes the full Pareto front of this benchmark in an expected number of $O(M^2 n^k)$ iterations, where $n$ denotes the problem size (length of the bit-string representation), $k$ the gap size (a difficulty parameter of the problem), and $M=(2n/m-2k+3)^{m/2}$ the size of the Pareto front. This result together with the existing negative result on the original NSGA-II shows that in principle, the general approach of the NSGA-II is suitable for many-objective optimization, but the crowding distance as tie-breaker has deficiencies. We obtain three additional insights on the SMS-EMOA. Different from a recent result for the bi-objective OJZJ benchmark, the stochastic population update often does not help for mOJZJ. It results in a $1/\Theta(\min\{Mk^{1/2}/2^{k/2},1\})$ speed-up, which is $\Theta(1)$ for large $m$ such as $m>k$. On the positive side, we prove that heavy-tailed mutation still results in a speed-up of order $k^{0.5+k-\beta}$. Finally, we conduct the first runtime analyses of the SMS-EMOA on the bi-objective OneMinMax and LOTZ benchmarks and show that it has a performance comparable to the GSEMO and the NSGA-II.