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A Performance Analysis of Basin Hopping Compared to Established Metaheuristics for Global Optimization

Marco Baioletti, Valentino Santucci, Marco Tomassini

TL;DR

Comparisons with readily available implementations of the well known metaheuristics Differential Evolution, Particle Swarm Optimization, and Covariance Matrix Adaptation Evolution Strategy show that Basin Hopping can be considered a good candidate for global numerical optimization problems along with the more established metaheuristics.

Abstract

During the last decades many metaheuristics for global numerical optimization have been proposed. Among them, Basin Hopping is very simple and straightforward to implement, although rarely used outside its original Physical Chemistry community. In this work, our aim is to compare Basin Hopping, and two population variants of it, with readily available implementations of the well known metaheuristics Differential Evolution, Particle Swarm Optimization, and Covariance Matrix Adaptation Evolution Strategy. We perform numerical experiments using the IOH profiler environment with the BBOB test function set and two difficult real-world problems. The experiments were carried out in two different but complementary ways: by measuring the performance under a fixed budget of function evaluations and by considering a fixed target value. The general conclusion is that Basin Hopping and its newly introduced population variant are almost as good as Covariance Matrix Adaptation on the synthetic benchmark functions and better than it on the two hard cluster energy minimization problems. Thus, the proposed analyses show that Basin Hopping can be considered a good candidate for global numerical optimization problems along with the more established metaheuristics, especially if one wants to obtain quick and reliable results on an unknown problem.

A Performance Analysis of Basin Hopping Compared to Established Metaheuristics for Global Optimization

TL;DR

Comparisons with readily available implementations of the well known metaheuristics Differential Evolution, Particle Swarm Optimization, and Covariance Matrix Adaptation Evolution Strategy show that Basin Hopping can be considered a good candidate for global numerical optimization problems along with the more established metaheuristics.

Abstract

During the last decades many metaheuristics for global numerical optimization have been proposed. Among them, Basin Hopping is very simple and straightforward to implement, although rarely used outside its original Physical Chemistry community. In this work, our aim is to compare Basin Hopping, and two population variants of it, with readily available implementations of the well known metaheuristics Differential Evolution, Particle Swarm Optimization, and Covariance Matrix Adaptation Evolution Strategy. We perform numerical experiments using the IOH profiler environment with the BBOB test function set and two difficult real-world problems. The experiments were carried out in two different but complementary ways: by measuring the performance under a fixed budget of function evaluations and by considering a fixed target value. The general conclusion is that Basin Hopping and its newly introduced population variant are almost as good as Covariance Matrix Adaptation on the synthetic benchmark functions and better than it on the two hard cluster energy minimization problems. Thus, the proposed analyses show that Basin Hopping can be considered a good candidate for global numerical optimization problems along with the more established metaheuristics, especially if one wants to obtain quick and reliable results on an unknown problem.
Paper Structure (19 sections, 11 equations, 7 figures, 4 tables, 4 algorithms)

This paper contains 19 sections, 11 equations, 7 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Metaheuristics
  4. Basin Hopping
  5. Population-based Basin Hopping
  6. Differential Evolution
  7. Covariance Matrix Adaptation Evolution Strategy
  8. Particle Swarm Optimization
  9. Experimental Setup
  10. Implementations of the algorithms
  11. Benchmark functions
  12. Design of the experiments
  13. Experimental Results
  14. Preliminary comparison between BHPOP and PBH
  15. Fixed Budget Analysis
  16. ...and 4 more sections

Figures (7)

  • Figure 1: Boxplots of the logscores obtained by the five algorithms, grouped by dimension $D \in \{5,10,20,40\}$ and benchmark function group, considering a budget of $200\,000$ evaluations.
  • Figure 2: Heatmap showing all the pairwise comparisons among the five algorithms in all the $4 \times 24 = 96$ problems, considering a budget of $200\,000$ evaluations. The entries are greenish or reddish when the row-algorithm is, respectively, better or worse than the column-algorithm. The grade of green or red is set on the basis of the p-value computed according to the Conover post-hoc test with Benjamini-Hochberg adjustment. These p-values are also provided in the heatmap. The omnibus Friedman test rejected the equivalence of effectiveness among the five algorithms with a p-value smaller than $10^{-11}$.
  • Figure 3: Boxplots of the logscores obtained by the five algorithms in all the executions over all the problem instances, considering different budgets of evaluations.
  • Figure 4: Average success rates achieved by the five algorithms, grouped by dimension and target precision.
  • Figure 5: The curves represent the empirical cumulative density function (ECDF) for each algorithm inset as a function of the number of function evaluations on the x-axis (log scale). ECDF curves show the increase in percentage of successful (target,execution) pairs as the number of function evaluations increases.
  • ...and 2 more figures