Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency for Many Objectives

Abstract

The NSGA-II is one of the most prominent algorithms to solve multi-objective optimization problems. Despite numerous successful applications, several studies have shown that the NSGA-II is less effective for larger numbers of objectives. In this work, we use mathematical runtime analyses to rigorously demonstrate and quantify this phenomenon. We show that even on the simple m-objective generalization of the discrete OneMinMax benchmark, where every solution is Pareto optimal, the NSGA-II also with large population sizes cannot compute the full Pareto front (objective vectors of all Pareto optima) in subexponential time when the number of objectives is at least three. The reason for this unexpected behavior lies in the fact that in the computation of the crowding distance, the different objectives are regarded independently. This is not a problem for two objectives, where any sorting of a pair-wise incomparable set of solutions according to one objective is also such a sorting according to the other objective (in the inverse order).

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

“…Again, its special case k = 1 will be (essentially) equal to the mONEMINMAX problem. With this, our results are comparable both the ones in (Zheng and Doerr 2023b) and (Bian et al 2023).…”

“…This work quickly inspired many interesting follow-up results in bi-objective optimization Bian and Qian 2022;Doerr and Qu 2023a,b,c;Dang et al 2023a,b;Cerf et al 2023). In contrast to these positive results for two objectives, Zheng and Doerr (2023b) proved that for m ≥ 3 objectives the NSGA-II needs at least exponential time (in expectation and with high probability) to cover the full Pareto front of the m-objective ONEMINMAX benchmark, a simple manyobjective version of the basic ONEMAX problem where all search points are Pareto optimal. They claimed that the main reason for this low efficiency is the independent computation of the crowding distance in each objective.…”

“…As mentioned earlier, the NSGA-II was proven to have enormous difficulties in optimizing many-objective problems (Zheng and Doerr 2023b). In that paper, the mobjective counterpart mONEMINMAX of the bi-objective ONEMINMAX benchmark was proposed and analyzed.…”

“…We note that this general block construction goes back to the seminal paper of (Laumanns, Thiele, and Zitzler 2004). Definition 1 (Zheng and Doerr (2023b)). Let m be the number of objectives and be even, and the problem size n be a multiple of m/2.…”

Approximation Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

“…A runtime analysis in the presence of noise (together with the independent parallel work [16] the first mathematical runtime analysis of a MOEA in a noisy environment) was conducted in [41]. That the NSGA-II can have difficulties with more than two objectives was shown for ONEMINMAX in [42], whereas the NSGA-III was proven to be efficient on the 3-objective ONEMINMAX problem in [43], and the efficiency of the SMS-EMOA, a variant of the steady-state NSGA-II, on the many-objective ONEJUMPZEROJUMP problem was proven in [44].…”