π€ AI Summary
The runtime behavior of the SMS-EMOA algorithm in multi-objective optimization lacks rigorous theoretical analysis.
Method: We introduce the analytically tractable *m*-objective multimodal benchmark *m*OJZJ, provably converging to the complete Pareto front; conduct theoretical analysis of crowding distanceβs detrimental impact on NSGA-II in multimodal settings; examine the acceleration effect of heavy-tailed mutation in multi-objective contexts; and formalize convergence under stochastic population updates.
Contribution/Results: We establish an expected convergence time of *O*(*M*Β²*n*α΅) for SMS-EMOA on *m*OJZJ; demonstrate competitive performance with GSEMO and NSGA-II on OneMinMax and LOTZ; and show limited acceleration from stochastic updates in high dimensions. This work provides the first rigorous analytical framework and scalable benchmark for runtime analysis of multi-objective evolutionary algorithms.
π 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/Ξ(min(Mk^(1/2)/2^(k/2),1)) speed-up, which is Ξ(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-Ξ²). 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.
