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An Enhanced Differential Grouping Method for Large-Scale Overlapping Problems

Maojiang Tian, Mingke Chen, Wei Du, Yang Tang, Yaochu Jin

TL;DR

The paper tackles large-scale overlapping problems (LSOP) where subcomponents share variables, complicating decomposition. It introduces OEDG, a two-stage differential grouping framework that first identifies all subcomponents and shared variables efficiently, then refines groupings with SUD and SD to correct unions of subcomponents. The authors also design diverse overlapping benchmarks (line, ring, complex topologies; MDO, NAO) to stress-test decomposition and optimization, and demonstrate that OEDG achieves high decomposition accuracy with reduced computational resources and, when paired with CBCCO, superior optimization on most benchmarks. While OEDG excels for line and ring topologies, complex topologies reveal challenges in shared-variable allocation, suggesting future work on more robust CC integration and handling of conflicting overlaps. Overall, the work provides a scalable, accurate structure-discovery approach for LSOP and expands evaluation infrastructure with new benchmarks.

Abstract

Large-scale overlapping problems are prevalent in practical engineering applications, and the optimization challenge is significantly amplified due to the existence of shared variables. Decomposition-based cooperative coevolution (CC) algorithms have demonstrated promising performance in addressing large-scale overlapping problems. However, current CC frameworks designed for overlapping problems rely on grouping methods for the identification of overlapping problem structures and the current grouping methods for large-scale overlapping problems fail to consider both accuracy and efficiency simultaneously. In this article, we propose a two-stage enhanced grouping method for large-scale overlapping problems, called OEDG, which achieves accurate grouping while significantly reducing computational resource consumption. In the first stage, OEDG employs a grouping method based on the finite differences principle to identify all subcomponents and shared variables. In the second stage, we propose two grouping refinement methods, called subcomponent union detection (SUD) and subcomponent detection (SD), to enhance and refine the grouping results. SUD examines the information of the subcomponents and shared variables obtained in the previous stage, and SD corrects inaccurate grouping results. To better verify the performance of the proposed OEDG, we propose a series of novel benchmarks that consider various properties of large-scale overlapping problems, including the topology structure, overlapping degree, and separability. Extensive experimental results demonstrate that OEDG is capable of accurately grouping different types of large-scale overlapping problems while consuming fewer computational resources. Finally, we empirically verify that the proposed OEDG can effectively improve the optimization performance of diverse large-scale overlapping problems.

An Enhanced Differential Grouping Method for Large-Scale Overlapping Problems

TL;DR

The paper tackles large-scale overlapping problems (LSOP) where subcomponents share variables, complicating decomposition. It introduces OEDG, a two-stage differential grouping framework that first identifies all subcomponents and shared variables efficiently, then refines groupings with SUD and SD to correct unions of subcomponents. The authors also design diverse overlapping benchmarks (line, ring, complex topologies; MDO, NAO) to stress-test decomposition and optimization, and demonstrate that OEDG achieves high decomposition accuracy with reduced computational resources and, when paired with CBCCO, superior optimization on most benchmarks. While OEDG excels for line and ring topologies, complex topologies reveal challenges in shared-variable allocation, suggesting future work on more robust CC integration and handling of conflicting overlaps. Overall, the work provides a scalable, accurate structure-discovery approach for LSOP and expands evaluation infrastructure with new benchmarks.

Abstract

Large-scale overlapping problems are prevalent in practical engineering applications, and the optimization challenge is significantly amplified due to the existence of shared variables. Decomposition-based cooperative coevolution (CC) algorithms have demonstrated promising performance in addressing large-scale overlapping problems. However, current CC frameworks designed for overlapping problems rely on grouping methods for the identification of overlapping problem structures and the current grouping methods for large-scale overlapping problems fail to consider both accuracy and efficiency simultaneously. In this article, we propose a two-stage enhanced grouping method for large-scale overlapping problems, called OEDG, which achieves accurate grouping while significantly reducing computational resource consumption. In the first stage, OEDG employs a grouping method based on the finite differences principle to identify all subcomponents and shared variables. In the second stage, we propose two grouping refinement methods, called subcomponent union detection (SUD) and subcomponent detection (SD), to enhance and refine the grouping results. SUD examines the information of the subcomponents and shared variables obtained in the previous stage, and SD corrects inaccurate grouping results. To better verify the performance of the proposed OEDG, we propose a series of novel benchmarks that consider various properties of large-scale overlapping problems, including the topology structure, overlapping degree, and separability. Extensive experimental results demonstrate that OEDG is capable of accurately grouping different types of large-scale overlapping problems while consuming fewer computational resources. Finally, we empirically verify that the proposed OEDG can effectively improve the optimization performance of diverse large-scale overlapping problems.
Paper Structure (37 sections, 2 theorems, 7 equations, 4 figures, 8 tables, 4 algorithms)

This paper contains 37 sections, 2 theorems, 7 equations, 4 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Types of Interaction
  4. Finite Differences Principle
  5. Overlapping Problems
  6. Topology Structure
  7. Shared Variable Character
  8. Overlapping Degree
  9. Grouping Methods for Overlapping Problems
  10. RDG3
  11. ORDG
  12. DG2
  13. Optimization Methods for Overlapping Problems
  14. The Proposed OEDG
  15. The Framework of OEDG
  16. ...and 22 more sections

Key Result

Proposition 1

Let $f(\boldsymbol{x})$ be an additively separable function. Two variables $x_p$ and $x_q$ in the function $f(\boldsymbol{x})$ interact with each other if the following condition holds: where

Figures (4)

  • Figure 1: An example to illustrate the grouping process of OEDG, with the legend depicting the initial two cycles of grouping.
  • Figure 2: An example illustrating how the selection of shared variables can impact the grouping results.
  • Figure 3: The UpSet diagrams of the line overlapping problems.
  • Figure 4: An example of a complex topology overlapping problem generated by CTOC with a parameter setting of $m=5$ and $p=0.2$.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Example 1
  • Proposition 1
  • Proposition 2
  • Definition 3