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Runtime Analyses of NSGA-III on Many-Objective Problems

Andre Opris, Duc-Cuong Dang, Frank Neumann, Dirk Sudholt

TL;DR

The paper provides the first runtime analyses of NSGA-III on constant-m many-objective benchmarks (m-LOTZ, m-OMM, m-COCZ), extending beyond the previously analyzed 3-objective case. It develops a toolkit of geometric and probabilistic methods to reason about reference-point survival and derives explicit parameter guidelines for the number of reference points and population size that guarantee good performance. The main results establish polynomial-time running times in terms of the problem size for each benchmark (e.g., $O(n^2)$ generations for $m$-LOTZ and $O(n\log n)$ for $m$-OMM and $m$-COCZ) with corresponding population sizes and reference-point counts, showing how these scale with $n$, $m$, and $f_{\max}$. Together, these findings justify NSGA-III’s applicability to many-objective problems and provide actionable guidance for parameter settings in practice, while laying groundwork for further analyses of NSGA-III on more complex or combinatorial problems.

Abstract

NSGA-II and NSGA-III are two of the most popular evolutionary multi-objective algorithms used in practice. While NSGA-II is used for few objectives such as 2 and 3, NSGA-III is designed to deal with a larger number of objectives. In a recent breakthrough, Wietheger and Doerr (IJCAI 2023) gave the first runtime analysis for NSGA-III on the 3-objective OneMinMax problem, showing that this state-of-the-art algorithm can be analyzed rigorously. We advance this new line of research by presenting the first runtime analyses of NSGA-III on the popular many-objective benchmark problems mLOTZ, mOMM, and mCOCZ, for an arbitrary constant number $m$ of objectives. Our analysis provides ways to set the important parameters of the algorithm: the number of reference points and the population size, so that a good performance can be guaranteed. We show how these parameters should be scaled with the problem dimension, the number of objectives and the fitness range. To our knowledge, these are the first runtime analyses for NSGA-III for more than 3 objectives.

Runtime Analyses of NSGA-III on Many-Objective Problems

TL;DR

The paper provides the first runtime analyses of NSGA-III on constant-m many-objective benchmarks (m-LOTZ, m-OMM, m-COCZ), extending beyond the previously analyzed 3-objective case. It develops a toolkit of geometric and probabilistic methods to reason about reference-point survival and derives explicit parameter guidelines for the number of reference points and population size that guarantee good performance. The main results establish polynomial-time running times in terms of the problem size for each benchmark (e.g., generations for -LOTZ and for -OMM and -COCZ) with corresponding population sizes and reference-point counts, showing how these scale with , , and . Together, these findings justify NSGA-III’s applicability to many-objective problems and provide actionable guidance for parameter settings in practice, while laying groundwork for further analyses of NSGA-III on more complex or combinatorial problems.

Abstract

NSGA-II and NSGA-III are two of the most popular evolutionary multi-objective algorithms used in practice. While NSGA-II is used for few objectives such as 2 and 3, NSGA-III is designed to deal with a larger number of objectives. In a recent breakthrough, Wietheger and Doerr (IJCAI 2023) gave the first runtime analysis for NSGA-III on the 3-objective OneMinMax problem, showing that this state-of-the-art algorithm can be analyzed rigorously. We advance this new line of research by presenting the first runtime analyses of NSGA-III on the popular many-objective benchmark problems mLOTZ, mOMM, and mCOCZ, for an arbitrary constant number of objectives. Our analysis provides ways to set the important parameters of the algorithm: the number of reference points and the population size, so that a good performance can be guaranteed. We show how these parameters should be scaled with the problem dimension, the number of objectives and the fitness range. To our knowledge, these are the first runtime analyses for NSGA-III for more than 3 objectives.
Paper Structure (10 sections, 9 theorems, 26 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous related work:
  3. Our contribution:
  4. Preliminaries
  5. Sufficiently Many Reference Points Protect Good Solutions
  6. $m$-LOTZ
  7. $m$-OMM
  8. $m$-COCZ
  9. Conclusions
  10. Useful results

Key Result

Lemma 3

Consider NSGA-III on a multiobjective function $f:\{0,1\}^n \to \mathbb{N}_0^m$and suppose that $\varepsilon_{\mathrm{nad}} \geq f_{\max}$. In every generation, after running Algorithm alg:normalization, for every $x \in R_t$ and every objective $j$ we have $0 \leq f_j^n(x) \leq 1$ and $y_j^{\mathrm

Figures (1)

  • Figure 1: Sketch of the way search points (orange) are associated with reference points (blue dots connected by dashed blue lines from the origin) for $m=2$ dimensions. The axes show the normalized fitness and all points lie in the unit cube $[0, 1]^m$ (blue shading). In the example, $\varphi_1 < \varphi_2$ and via $\sin(\varphi_1) = d_1/\vert{v}\vert$ and $\sin(\varphi_2) = d_2/\vert{v}\vert$ we conclude $d_1 < d_2$; the search point $v$ is associated with the leftmost reference point.

Theorems & Definitions (23)

  • Definition 1
  • Example 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 13 more