Runtime Analysis of the SMS-EMOA for Many-Objective Optimization
Abstract
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/Θ(min(Mk^(1/2)/2^(k/2),1)) speed-up, which is Θ(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-β). 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.
BibTeX
@article{Zheng_Doerr_2024, title={Runtime Analysis of the SMS-EMOA for Many-Objective Optimization}, volume={38}, url={https://ojs.aaai.org/index.php/AAAI/article/view/30077}, DOI={10.1609/aaai.v38i18.30077}, abstractNote={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/Θ(min(Mk^(1/2)/2^(k/2),1)) speed-up, which is Θ(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-β). 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.}, number={18}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Zheng, Weijie and Doerr, Benjamin}, year={2024}, month={Mar.}, pages={20874-20882} }