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Stagnation in Evolutionary Algorithms: Convergence $\neq$ Optimality

Xiaojun Zhou

TL;DR

The paper challenges the view that stagnation hinders convergence and that convergence equates to optimality in evolutionary algorithms. Through theoretical analysis of simple pairwise dynamics and acceptance-based convergence, it shows that individual stagnation can promote population convergence while convergence need not coincide with optimality. Counterexamples and experiments on Zhou1–3 benchmarks across multiple EA families reveal that standard algorithms often converge to non-optimal points, even when stagnation is present. These findings urge a shift away from using convergence speed as the sole success metric and highlight the risk of overfitting to benchmarks in EA design.

Abstract

In the evolutionary computation community, it is widely believed that stagnation impedes convergence in evolutionary algorithms, and that convergence inherently indicates optimality. However, this perspective is misleading. In this study, it is the first to highlight that the stagnation of an individual can actually facilitate the convergence of the entire population, and convergence does not necessarily imply optimality, not even local optimality. Convergence alone is insufficient to ensure the effectiveness of evolutionary algorithms. Several counterexamples are provided to illustrate this argument.

Stagnation in Evolutionary Algorithms: Convergence $\neq$ Optimality

TL;DR

The paper challenges the view that stagnation hinders convergence and that convergence equates to optimality in evolutionary algorithms. Through theoretical analysis of simple pairwise dynamics and acceptance-based convergence, it shows that individual stagnation can promote population convergence while convergence need not coincide with optimality. Counterexamples and experiments on Zhou1–3 benchmarks across multiple EA families reveal that standard algorithms often converge to non-optimal points, even when stagnation is present. These findings urge a shift away from using convergence speed as the sole success metric and highlight the risk of overfitting to benchmarks in EA design.

Abstract

In the evolutionary computation community, it is widely believed that stagnation impedes convergence in evolutionary algorithms, and that convergence inherently indicates optimality. However, this perspective is misleading. In this study, it is the first to highlight that the stagnation of an individual can actually facilitate the convergence of the entire population, and convergence does not necessarily imply optimality, not even local optimality. Convergence alone is insufficient to ensure the effectiveness of evolutionary algorithms. Several counterexamples are provided to illustrate this argument.
Paper Structure (8 sections, 2 theorems, 16 equations, 3 figures, 1 table)

This paper contains 8 sections, 2 theorems, 16 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analysis of convergence and stagnation
  3. Analysis of optimality
  4. Experimental results and analysis
  5. Benchmark Functions
  6. Experiment settings
  7. Experimental results and analysis
  8. Conclusion

Key Result

Theorem 1

Considering a population with any two individuals $i$ and $j$ with the dynamics described by Eq. (2), if the parameter $\alpha$ satisfies $0 < \alpha < 1$, the population will converge, i.e.$\bm x_{i} (k) = \bm x_{j} (k), k \rightarrow \infty$.

Figures (3)

  • Figure 1: Convergence curves for $f_1$ with $T = 1000$
  • Figure 2: Convergence curves for $f_2$ with $T = 1000$
  • Figure 3: Convergence curves for $f_3$ with $T = 1000$

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Definition 1
  • Definition 2
  • Remark 4