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A Crowding Distance That Provably Solves the Difficulties of the NSGA-II in Many-Objective Optimization

Weijie Zheng, Yao Gao, Benjamin Doerr

TL;DR

The paper tackles the known failure of NSGA-II in many-objective optimization by introducing a truthful crowding distance that uses a normalized $L_1$ distance across objectives and correlated tie-breaking to better reflect solution isolation. It proves that NSGA-II with this truthful distance (NSGA-II-T), including a sequential variant, preserves Pareto-front values under reasonable population sizes and achieves polynomial runtimes on standard many-objective benchmarks, matching the performance of NSGA-III and SMS-EMOA. For bi-objective problems, NSGA-II-T retains the original NSGA-II performance while requiring smaller population sizes, and it offers slightly better MEI-based approximation in limited-population regimes. Together with experimental results, the work shows that the truthful distance preserves the practical strengths of NSGA-II in two objectives and dramatically improves its behavior in higher dimensions, offering a practical alternative to switching to NSGA-III or SMS-EMOA. $\overline{M}$, $N$, and $tCD$ feature in the formal guarantees and runtime analyses, underscoring the method’s theoretical grounding and potential impact on MOEA practice.

Abstract

Recent theoretical works have shown that the NSGA-II can have enormous difficulties to solve problems with more than two objectives. In contrast, algorithms like the NSGA-III or SMS-EMOA, differing from the NSGA-II only in the secondary selection criterion, provably perform well in these situations. To remedy this shortcoming of the NSGA-II, but at the same time keep the advantages of the widely accepted crowding distance, we use the insights of these previous work to define a variant of the crowding distance, called truthful crowding distance. Different from the classic crowding distance, it has for any number of objectives the desirable property that a small crowding distance value indicates that some other solution has a similar objective vector. Building on this property, we conduct mathematical runtime analyses for the NSGA-II with truthful crowding distance. We show that this algorithm can solve the many-objective versions of the OneMinMax, COCZ, LOTZ, and OJZJ$_k$ problems in the same (polynomial) asymptotic runtimes as the NSGA-III and the SMS-EMOA. This contrasts the exponential lower bounds previously shown for the classic NSGA-II. For the bi-objective versions of these problems, our NSGA-II has a similar performance as the classic NSGA-II, gaining however from smaller admissible population sizes. For the bi-objective OneMinMax problem, we also observe a (minimally) better performance in approximating the Pareto front. These results suggest that our truthful version of the NSGA-II has the same good performance as the classic NSGA-II in two objectives, but can resolve the drastic problems in more than two objectives.

A Crowding Distance That Provably Solves the Difficulties of the NSGA-II in Many-Objective Optimization

TL;DR

The paper tackles the known failure of NSGA-II in many-objective optimization by introducing a truthful crowding distance that uses a normalized distance across objectives and correlated tie-breaking to better reflect solution isolation. It proves that NSGA-II with this truthful distance (NSGA-II-T), including a sequential variant, preserves Pareto-front values under reasonable population sizes and achieves polynomial runtimes on standard many-objective benchmarks, matching the performance of NSGA-III and SMS-EMOA. For bi-objective problems, NSGA-II-T retains the original NSGA-II performance while requiring smaller population sizes, and it offers slightly better MEI-based approximation in limited-population regimes. Together with experimental results, the work shows that the truthful distance preserves the practical strengths of NSGA-II in two objectives and dramatically improves its behavior in higher dimensions, offering a practical alternative to switching to NSGA-III or SMS-EMOA. , , and feature in the formal guarantees and runtime analyses, underscoring the method’s theoretical grounding and potential impact on MOEA practice.

Abstract

Recent theoretical works have shown that the NSGA-II can have enormous difficulties to solve problems with more than two objectives. In contrast, algorithms like the NSGA-III or SMS-EMOA, differing from the NSGA-II only in the secondary selection criterion, provably perform well in these situations. To remedy this shortcoming of the NSGA-II, but at the same time keep the advantages of the widely accepted crowding distance, we use the insights of these previous work to define a variant of the crowding distance, called truthful crowding distance. Different from the classic crowding distance, it has for any number of objectives the desirable property that a small crowding distance value indicates that some other solution has a similar objective vector. Building on this property, we conduct mathematical runtime analyses for the NSGA-II with truthful crowding distance. We show that this algorithm can solve the many-objective versions of the OneMinMax, COCZ, LOTZ, and OJZJ problems in the same (polynomial) asymptotic runtimes as the NSGA-III and the SMS-EMOA. This contrasts the exponential lower bounds previously shown for the classic NSGA-II. For the bi-objective versions of these problems, our NSGA-II has a similar performance as the classic NSGA-II, gaining however from smaller admissible population sizes. For the bi-objective OneMinMax problem, we also observe a (minimally) better performance in approximating the Pareto front. These results suggest that our truthful version of the NSGA-II has the same good performance as the classic NSGA-II in two objectives, but can resolve the drastic problems in more than two objectives.
Paper Structure (12 sections, 13 theorems, 6 equations, 3 figures, 2 algorithms)

This paper contains 12 sections, 13 theorems, 6 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classic and Truthful Crowding Distance
  4. Original Crowding Distance
  5. Truthful Crowding Distance
  6. Runtime Analysis: Computing the Pareto Front
  7. Not Losing Pareto Front Points
  8. Runtime Results for Many Objectives
  9. Runtime Results for Two Objectives
  10. Approximation Ability and Runtime
  11. Experiments
  12. Conclusion and Future Work

Key Result

Lemma 1

Let $m\in\mathbb{N}$ be the number of objectives of the discussed function $f=(f_1,\dots,f_m)$. Let $S$ be a population of individuals in $\{0,1\}^n$. Assume that we compute the truthful crowding distance $\mathop{\mathrm{\mathop{\mathrm{tCD}}\nolimits}}\nolimits(S)$ via Algorithm alg:dcDis. Then fo

Figures (3)

  • Figure 1: Median number (with $1$st and $3$rd quartiles, in 20 runs) of function evaluations to compute the full Pareto front of the $4$-objective OneMinMax problem.
  • Figure 2: The number of function evaluations to cover the full Pareto front for OneMinMax.
  • Figure 3: The $\textsc{MEI}\xspace$ for generations $[1..100]$ and $[3001..3100]$ after the two extreme points were found (one run).

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:pfkept']}
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 7 more