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A Composite Decomposition Method for Large-Scale Global Optimization

Maojiang Tian, Minyang Chen, Wei Du, Yang Tang, Yaochu Jin, Gary G. Yen

TL;DR

The paper tackles the challenge of decomposing large-scale global optimization problems by introducing Composite Separability Grouping (CSG), a framework that unifies DG and GSG strengths. It adds two novel components, MSVD for detecting multiplicatively separable variables and NVG for efficient non-separable grouping, and it introduces the BMS benchmark to stress multiple separability types. Empirical results show CSG achieves higher decomposition accuracy at lower computational cost than state-of-the-art methods and yields favorable optimization performance within a cooperative co-evolution framework. This approach offers a scalable, accurate pathway for solving LSGO problems and sets a benchmark for evaluating separability-detection methods. The work also suggests promising avenues for extending separability concepts and leveraging advanced modeling for problem structure discovery.

Abstract

Cooperative co-evolution (CC) algorithms, based on the divide-and-conquer strategy, have emerged as the predominant approach to solving large-scale global optimization (LSGO) problems. The efficiency and accuracy of the grouping stage significantly impact the performance of the optimization process. While the general separability grouping (GSG) method has overcome the limitation of previous differential grouping (DG) methods by enabling the decomposition of non-additively separable functions, it suffers from high computational complexity. To address this challenge, this article proposes a composite separability grouping (CSG) method, seamlessly integrating DG and GSG into a problem decomposition framework to utilize the strengths of both approaches. CSG introduces a step-by-step decomposition framework that accurately decomposes various problem types using fewer computational resources. By sequentially identifying additively, multiplicatively and generally separable variables, CSG progressively groups non-separable variables by recursively considering the interactions between each non-separable variable and the formed non-separable groups. Furthermore, to enhance the efficiency and accuracy of CSG, we introduce two innovative methods: a multiplicatively separable variable detection method and a non-separable variable grouping method. These two methods are designed to effectively detect multiplicatively separable variables and efficiently group non-separable variables, respectively. Extensive experimental results demonstrate that CSG achieves more accurate variable grouping with lower computational complexity compared to GSG and state-of-the-art DG series designs.

A Composite Decomposition Method for Large-Scale Global Optimization

TL;DR

The paper tackles the challenge of decomposing large-scale global optimization problems by introducing Composite Separability Grouping (CSG), a framework that unifies DG and GSG strengths. It adds two novel components, MSVD for detecting multiplicatively separable variables and NVG for efficient non-separable grouping, and it introduces the BMS benchmark to stress multiple separability types. Empirical results show CSG achieves higher decomposition accuracy at lower computational cost than state-of-the-art methods and yields favorable optimization performance within a cooperative co-evolution framework. This approach offers a scalable, accurate pathway for solving LSGO problems and sets a benchmark for evaluating separability-detection methods. The work also suggests promising avenues for extending separability concepts and leveraging advanced modeling for problem structure discovery.

Abstract

Cooperative co-evolution (CC) algorithms, based on the divide-and-conquer strategy, have emerged as the predominant approach to solving large-scale global optimization (LSGO) problems. The efficiency and accuracy of the grouping stage significantly impact the performance of the optimization process. While the general separability grouping (GSG) method has overcome the limitation of previous differential grouping (DG) methods by enabling the decomposition of non-additively separable functions, it suffers from high computational complexity. To address this challenge, this article proposes a composite separability grouping (CSG) method, seamlessly integrating DG and GSG into a problem decomposition framework to utilize the strengths of both approaches. CSG introduces a step-by-step decomposition framework that accurately decomposes various problem types using fewer computational resources. By sequentially identifying additively, multiplicatively and generally separable variables, CSG progressively groups non-separable variables by recursively considering the interactions between each non-separable variable and the formed non-separable groups. Furthermore, to enhance the efficiency and accuracy of CSG, we introduce two innovative methods: a multiplicatively separable variable detection method and a non-separable variable grouping method. These two methods are designed to effectively detect multiplicatively separable variables and efficiently group non-separable variables, respectively. Extensive experimental results demonstrate that CSG achieves more accurate variable grouping with lower computational complexity compared to GSG and state-of-the-art DG series designs.
Paper Structure (24 sections, 1 theorem, 16 equations, 2 figures, 7 tables, 5 algorithms)

This paper contains 24 sections, 1 theorem, 16 equations, 2 figures, 7 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Basic Definitions
  4. Problem Decomposition Methods
  5. Differential Grouping
  6. Minimum Points Shift Principle
  7. The Proposed CSG
  8. The Framework of CSG
  9. Multiplicatively Separable Variable Detection
  10. Non-separable Group Detection
  11. Computational Resource Consumption Analysis
  12. Experimental Study
  13. Benchmark Functions
  14. Comparison on Decomposition
  15. Accuracy Metric
  16. ...and 9 more sections

Key Result

Proposition 1

If $f(\boldsymbol{x})$ is an additively separable function consisting of a multiplicatively separable subfunction $f_{i}(\boldsymbol{X}_i)$ with a multiplicatively separable variable $x_i$. Let $\boldsymbol{x}_1=({x}_1,...,{x}_i,...,{x}_n)$ and $\boldsymbol{x}_2=({x}_1,...,{x}_i/2,...,{x}_n)$, then

Figures (2)

  • Figure 1: An example to illustrate the grouping process of CSG with a problem $f(\boldsymbol{x})=x_{1}+x_{2} \cdot x_{3}+\sqrt{x_{4}+x_{5}}+\left(x_{6}-x_{7}-1\right)^{2}$
  • Figure 2: Convergence curves for the CSG, GSG, RDG2, and MDG methods when embedded into the DECC framework to solve the 1000-D $f_9$ and $f_{11}$ in BMS benchmark with the different termination conditions. The horizontal axis represents the number of FEs used in the decomposition and optimization process. The vertical axis represents the average fitness value.

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Example 1
  • Definition 4
  • Example 2
  • Proposition 1
  • proof