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

Weijie Zheng, Benjamin Doerr

TL;DR

It is shown 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 in subexponential time when the number of objectives is at least three.

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 sub-exponential 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 for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency For Many Objectives

TL;DR

It is shown that even on the simple -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 in subexponential time when the number of objectives is at least three.

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 -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 sub-exponential 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).
Paper Structure (18 sections, 8 theorems, 20 equations, 3 figures, 3 algorithms)

This paper contains 18 sections, 8 theorems, 20 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Preliminaries
  4. Many-Objective Optimization
  5. Algorithms
  6. Ineffectiveness for Four and More Objectives
  7. The $m$OneMinMax Benchmark
  8. Deficiencies of the Many-Objective Crowding Distance
  9. An Exponential Lower Bound for $m\ge 4$
  10. Also the Combined Parent and Offspring Population Does Not Cover the Pareto Front
  11. Synthetic MOEAs
  12. Ineffectiveness for Three Objectives
  13. 3-OneMinMax
  14. An Exponential Lower Bound for 3-OneMinMax
  15. Also the Combined Parent and Offspring Population Does Not Cover the Pareto Front
  16. ...and 3 more sections

Key Result

Lemma 3

Let $m\in \mathbb{N}$ with $m \ge 4$. Let $S$ be a set of pair-wise non-dominated individuals in $\{0,1\}^n$. Assume that we compute the crowding distance $\mathop{\mathrm{cDis}}\nolimits(S)$ with respect to the objective function $m$OneMinMax via Algorithm alg:cDis. Then at most $4n+2m$ individuals

Figures (3)

  • Figure 1: Size of the part of the Pareto front covered by the parent population $P_t$ and the combined parent and offspring population $R_t$ in the first 1000 iterations of exemplary runs of the NSGA-II on the $4$-OneMinMax problem with problem size $n=40$.
  • Figure 2: The size of the Pareto front covered by $P_t$ and $R_t$ (mean value $\pm$ standard deviation) in the $1,000$-th iteration for the NSGA-II (fair selection, standard bit-wise mutation) with population size $N=4M,16M,64M,256M$ on $m$OneMinMax with problem size $n=40$ and $4$ objectives, where $M=441$ is the Pareto front size (10 independent runs).
  • Figure 3: Covered $(f_2,f_4)$ objective values before (the combined parent population $R_t$ as well as the individuals in $R_t$ with positive crowding distance) and after (the next population $P_{t+1}$) the original survival selection of the NSGA-II (fair selection, standard bit-wise mutation) with population size $N=4M,16M,64M,256M$ on $m$OneMinMax with problem size $n=40$ and $4$ objectives where $M=441$ is the Pareto front size. Displayed is one typical run.

Theorems & Definitions (16)

  • Definition 1: $m$OneMinMax
  • Example 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Definition 6: Neighbors
  • proof : Proof of Theorem \ref{['lem:rtcover']}
  • Theorem 7
  • ...and 6 more