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Analyzing the Runtime of the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) on the Concatenated Trap Function

Yukai Qiao, Marcus Gallagher

TL;DR

This work provides the first runtime analysis of GOMEA on the concatenated trap function, a modular, deceptive optimization benchmark. By proving that a truthful FOS enables efficient recombination of optimal subsolutions, the authors derive an upper bound of $O(m^{3}2^{k})$ on the expected runtime, achieving a substantial speedup over the $(1+1)$ EA’s $O(\ln{m}(mk)^{k})$. They validate the theory with experiments showing that, for a suitably large population $\mu=cm2^{k}$, GOMEA reliably finds the global optimum with high probability, and that the Optimal Mixing operator is key to preserving building blocks. The continuation work extends these results to generalized trap functions, highlighting the role of the optimal-region probability $p_{*}$ and offering bounds that scale as $O(\frac{1}{p_{*}}m^{3})$, with empirical evidence supporting robustness across parameter choices. Overall, the paper advances theoretical understanding of linkage learning and informed variation in modular, deceptive optimization problems, with implications for complex real-world domains that exhibit subfunction structure.

Abstract

The Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) is a state of the art evolutionary algorithm that leverages linkage learning to efficiently exploit problem structure. By identifying and preserving important building blocks during variation, GOMEA has shown promising performance on various optimization problems. In this paper, we provide the first runtime analysis of GOMEA on the concatenated trap function, a challenging benchmark problem that consists of multiple deceptive subfunctions. We derived an upper bound on the expected runtime of GOMEA with a truthful linkage model, showing that it can solve the problem in $O(m^{3}2^k)$ with high probability, where $m$ is the number of subfunctions and $k$ is the subfunction length. This is a significant speedup compared to the (1+1) EA, which requires $O(ln{(m)}(mk)^{k})$ expected evaluations.

Analyzing the Runtime of the Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) on the Concatenated Trap Function

TL;DR

This work provides the first runtime analysis of GOMEA on the concatenated trap function, a modular, deceptive optimization benchmark. By proving that a truthful FOS enables efficient recombination of optimal subsolutions, the authors derive an upper bound of on the expected runtime, achieving a substantial speedup over the EA’s . They validate the theory with experiments showing that, for a suitably large population , GOMEA reliably finds the global optimum with high probability, and that the Optimal Mixing operator is key to preserving building blocks. The continuation work extends these results to generalized trap functions, highlighting the role of the optimal-region probability and offering bounds that scale as , with empirical evidence supporting robustness across parameter choices. Overall, the paper advances theoretical understanding of linkage learning and informed variation in modular, deceptive optimization problems, with implications for complex real-world domains that exhibit subfunction structure.

Abstract

The Gene-pool Optimal Mixing Evolutionary Algorithm (GOMEA) is a state of the art evolutionary algorithm that leverages linkage learning to efficiently exploit problem structure. By identifying and preserving important building blocks during variation, GOMEA has shown promising performance on various optimization problems. In this paper, we provide the first runtime analysis of GOMEA on the concatenated trap function, a challenging benchmark problem that consists of multiple deceptive subfunctions. We derived an upper bound on the expected runtime of GOMEA with a truthful linkage model, showing that it can solve the problem in with high probability, where is the number of subfunctions and is the subfunction length. This is a significant speedup compared to the (1+1) EA, which requires expected evaluations.
Paper Structure (22 sections, 5 theorems, 20 equations, 6 figures, 2 algorithms)

This paper contains 22 sections, 5 theorems, 20 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions
  3. Concatenated Trap Function
  4. (1+1) Evolutionary Algorithm
  5. Family Of Subsets
  6. Gene-pool Optimal Mixing EA
  7. Analysis
  8. (1+1) EA runtime on the Concatenated Trap Function
  9. GOMEA runtime on the Concatenated Trap Function
  10. Experiment
  11. Verification of Theorem \ref{['thm:gomea_runtime']}
  12. Success Rate with Constant $c$
  13. Future Work
  14. Conclusion
  15. Continuation
  16. ...and 7 more sections

Key Result

theorem 1

The expected optimization time of (1+1)EA on concatenated trap function with $m$ substrings of length $k$ is $O(\ln{m}(mk)^k)$ with a mutation rate of $\frac{1}{mk}$

Figures (6)

  • Figure 1: A GOM process with a truthful MP FOS on the concatenated trap function with k=3, m=3
  • Figure 2: A GOM process with an untruthful MP FOS on the concatenated trap function with k=3, m=3
  • Figure 3: (a) to (d) Upper bound vs worst case performance (e) Comparison with (1+1) EA with k=4 in logarithmic scale
  • Figure 4: (a) The success rate of GOMEA and ($\mu$+1)EA with deterministic crowding (b) Corresponding average runtime of succeeded runs.$k=4$ ,$m = 6$, over 1000 runs (c) Average runtime including ($\mu$ + 1) GA in logarithmic scale
  • Figure 5: (a) to (e) Upper bound vs worst case performance (e) Comparison between different $p_{*}$
  • ...and 1 more figures

Theorems & Definitions (10)

  • theorem 1
  • proof
  • lemma 1
  • proof
  • theorem 2
  • proof
  • lemma 2
  • proof
  • theorem 3
  • proof