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Efficient state transition algorithm with guaranteed optimality

Xiaojun Zhou, Chunhua Yang, Weihua Gui

TL;DR

This work tackles slow late-stage convergence and the lack of automatic termination in the state transition algorithm (STA) by introducing ESTA and EXSTA, which integrate predictive translation, expanded and axesion refinements, adaptive parameter control, and a termination condition that guarantees optimality (via the rotation bound and no-progress criterion). The authors develop ARIMA-inspired first- and second-order translation models, supplement them with new expansion/axesion operators, and implement two parameter-control strategies including an inexact line-search for step size selection. Experimental results across diverse unconstrained benchmarks show that ESTA/EXSTA deliver faster convergence, greater robustness to flat landscapes, and reliable stopping at optimal points, outperforming STA variants and many contemporary metaheuristics under fixed and budgeted evaluation limits. While effective, the method acknowledges limitations on certain low-dimensional Griewank problems and suggests extending to population-based STA in future work to enhance global search capabilities.

Abstract

As a constructivism-based intelligent optimization method, state transition algorithm (STA) has exhibited powerful search ability in optimization. However, the standard STA still shows slow convergence at a later stage for flat landscape and a user has to preset its maximum number of iterations (or function evaluations) by experience. To resolve these two issues, efficient state transition algorithm is proposed with guaranteed optimality. Firstly, novel translation transformations based on predictive modeling are proposed to generate more potential candidates by utilizing historical information. Secondly, parameter control strategies are proposed to accelerate the convergence. Thirdly, a specific termination condition is designed to guarantee that the STA can stop automatically at an optimal point, which is equivalent to the zero gradient in mathematical programming. Experimental results have demonstrated the effectiveness and superiority of the proposed method.

Efficient state transition algorithm with guaranteed optimality

TL;DR

This work tackles slow late-stage convergence and the lack of automatic termination in the state transition algorithm (STA) by introducing ESTA and EXSTA, which integrate predictive translation, expanded and axesion refinements, adaptive parameter control, and a termination condition that guarantees optimality (via the rotation bound and no-progress criterion). The authors develop ARIMA-inspired first- and second-order translation models, supplement them with new expansion/axesion operators, and implement two parameter-control strategies including an inexact line-search for step size selection. Experimental results across diverse unconstrained benchmarks show that ESTA/EXSTA deliver faster convergence, greater robustness to flat landscapes, and reliable stopping at optimal points, outperforming STA variants and many contemporary metaheuristics under fixed and budgeted evaluation limits. While effective, the method acknowledges limitations on certain low-dimensional Griewank problems and suggests extending to population-based STA in future work to enhance global search capabilities.

Abstract

As a constructivism-based intelligent optimization method, state transition algorithm (STA) has exhibited powerful search ability in optimization. However, the standard STA still shows slow convergence at a later stage for flat landscape and a user has to preset its maximum number of iterations (or function evaluations) by experience. To resolve these two issues, efficient state transition algorithm is proposed with guaranteed optimality. Firstly, novel translation transformations based on predictive modeling are proposed to generate more potential candidates by utilizing historical information. Secondly, parameter control strategies are proposed to accelerate the convergence. Thirdly, a specific termination condition is designed to guarantee that the STA can stop automatically at an optimal point, which is equivalent to the zero gradient in mathematical programming. Experimental results have demonstrated the effectiveness and superiority of the proposed method.

Paper Structure

This paper contains 20 sections, 1 theorem, 13 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Basics of STA
  4. The principle behind the STA
  5. Analysis of the termination conditions used in intelligent optimization
  6. Proposed efficient STA
  7. Novel translation transformation based on predictive modeling
  8. New expansion and axesion transformation for supplementation
  9. Proposed parameter control strategy
  10. An intuitive parameter adaptation strategy
  11. Parameter optimal selection strategy
  12. Proposed efficient STA with guaranteed optimality
  13. Experimental results and analysis
  14. Experimental setup
  15. Comparison among different predictive models
  16. ...and 5 more sections

Key Result

Theorem 1

If the sample size SE is sufficiently large and the rotation factor $\alpha$ is sufficiently small at later stage, the sequence $\{f({\bm{Best}}_{k})\}$ generated by the STA can at least converge to a local minimum, i.e., where $\bm x^{*}$ is a local minimum.

Figures (6)

  • Figure 1: Illustration of the 2D plot and 3D histogram of state transformations
  • Figure 2: Illustration and comparison of the two new translation transformations
  • Figure 3: The illustration of an extreme case
  • Figure 4: An intuitive parameter adaptation strategy
  • Figure 5: Iterative curves of STA variants for 50-dimensional problems
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2