Speeding Up the NSGA-II With a Simple Tie-Breaking Rule

TL;DR

The paper tackles two weaknesses of NSGA-II in many-objective optimization: difficulty with three or more objectives and high sensitivity to population size. It proposes Balanced NSGA-II, a simple tie-breaking rule that distributes selections evenly across objective values within the critical tie-break group, backed by rigorous runtime analyses on OMM, LOTZ, and OJZJ_k and by bi-objective results. Theoretical bounds show polynomial-time performance for constant objective counts, with substantial improvements over classic NSGA-II when the population size is not carefully tuned, complemented by empirical evidence of speedups and robustness. The work suggests that modest, structure-aware tie-breaking can unlock scalable performance for MOEAs and invites extension to other algorithms and more objectives.

Abstract

The non-dominated sorting genetic algorithm~II (NSGA-II) is the most popular multi-objective optimization heuristic. Recent mathematical runtime analyses have detected two shortcomings in discrete search spaces, namely, that the NSGA-II has difficulties with more than two objectives and that it is very sensitive to the choice of the population size. To overcome these difficulties, we analyze a simple tie-breaking rule in the selection of the next population. Similar rules have been proposed before, but have found only little acceptance. We prove the effectiveness of our tie-breaking rule via mathematical runtime analyses on the classic OneMinMax, LeadingOnesTrailingZeros, and OneJumpZeroJump benchmarks. We prove that this modified NSGA-II can optimize the three benchmarks efficiently also for many objectives, in contrast to the exponential lower runtime bound previously shown for OneMinMax with three or more objectives. For the bi-objective problems, we show runtime guarantees that do not increase when moderately increasing the population size over the minimum admissible size. For example, for the OneJumpZeroJump problem with representation length $n$ and gap parameter $k$, we show a runtime guarantee of $O(\max\{n^{k+1},Nn\})$ function evaluations when the population size is at least four times the size of the Pareto front. For population sizes larger than the minimal choice $N = Θ(n)$, this result improves considerably over the $Θ(Nn^k)$ runtime of the classic NSGA-II.

Figures (4)

• Figure 1: The average number of function evaluations of the classic (dashed lines) and the balanced (solid lines) NSGA/̄II optimizing OneMinMax, for the shown population sizes $N$ and problem sizes $n$. The value $M$ denotes the size of the Pareto front, i.e., $M = n + 1$. Each point is the average of $50$ independent runs.
• Figure 2: The average number of function evaluations of the classic (dashed lines) and the balanced (solid lines) NSGA/̄II optimizing LeadingOnesTrailingZeros, for the shown population sizes $N$ and problem sizes $n$. The value $M$ denotes the size of the Pareto front, i.e., $M = n + 1$. Each point is the average of $50$ independent runs.
• Figure 3: The average number of function evaluations of the classic (dashed lines) and the balanced (solid lines) NSGA/̄II optimizing OneJumpZeroJump with $k = 3$, for the shown population sizes $N$ and problem sizes $n$. The value $M$ denotes the size of the Pareto front, i.e., $M = n - 2k + 3 = n - 3$. Each point is the average of $50$ independent runs.
• Figure 4: Average runtimes of the balanced NSGA/̄II with different population sizes $N$ on the 4-objective OMM problem. The value $M$ denotes the size of the Pareto front, i.e., $M = (n/2 + 1)^2$. Each point is the average of $50$ independent runs.

Theorems & Definitions (32)

• Lemma 1
• Lemma 2
• Theorem 3
• Lemma 4
• Theorem 5
• Lemma 6
• Lemma 7
• Theorem 8
• Corollary 9
• Lemma 10