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An Evolutionary Algorithm with Probabilistic Annealing for Large-scale Sparse Multi-objective Optimization

Shuai Shao, Yuhao Sun, Xing Chen, Ye Tian, Guan Wang, Jin Li

Abstract

Large-scale sparse multi-objective optimization problems (LSMOPs) are prevalent in real-world applications, where optimal solutions typically contain only a few nonzero variables, such as in adversarial attacks, critical node detection, and sparse signal reconstruction. Since the function evaluation of LSMOPs often relies on large-scale datasets involving a large number of decision variables, the search space becomes extremely high-dimensional. The coexistence of sparsity and high dimensionality greatly intensifies the conflict between exploration and exploitation, making it difficult for existing multi-objective evolutionary algorithms (MOEAs) to identify the critical nonzero decision variables within limited function evaluations. To address this challenge, this paper proposes an evolutionary algorithm with probabilistic annealing for large-scale sparse multi-objective optimization. The algorithm is driven by two probability vectors with distinct entropy characteristics: a convergence-oriented probability vector with relatively low entropy ensures stable exploitation, whereas an annealed probability vector with gradually decreasing entropy enables an adaptive transition from global exploration to local refinement. By integrating these complementary search dynamics, the proposed algorithm achieves a dynamic equilibrium between exploration and exploitation. Experimental results on benchmark problems and real-world applications demonstrate that the proposed algorithm outperforms state-of-the-art evolutionary algorithms in terms of both convergence and diversity.

An Evolutionary Algorithm with Probabilistic Annealing for Large-scale Sparse Multi-objective Optimization

Abstract

Large-scale sparse multi-objective optimization problems (LSMOPs) are prevalent in real-world applications, where optimal solutions typically contain only a few nonzero variables, such as in adversarial attacks, critical node detection, and sparse signal reconstruction. Since the function evaluation of LSMOPs often relies on large-scale datasets involving a large number of decision variables, the search space becomes extremely high-dimensional. The coexistence of sparsity and high dimensionality greatly intensifies the conflict between exploration and exploitation, making it difficult for existing multi-objective evolutionary algorithms (MOEAs) to identify the critical nonzero decision variables within limited function evaluations. To address this challenge, this paper proposes an evolutionary algorithm with probabilistic annealing for large-scale sparse multi-objective optimization. The algorithm is driven by two probability vectors with distinct entropy characteristics: a convergence-oriented probability vector with relatively low entropy ensures stable exploitation, whereas an annealed probability vector with gradually decreasing entropy enables an adaptive transition from global exploration to local refinement. By integrating these complementary search dynamics, the proposed algorithm achieves a dynamic equilibrium between exploration and exploitation. Experimental results on benchmark problems and real-world applications demonstrate that the proposed algorithm outperforms state-of-the-art evolutionary algorithms in terms of both convergence and diversity.
Paper Structure (17 sections, 3 equations, 4 figures, 3 tables, 6 algorithms)

This paper contains 17 sections, 3 equations, 4 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work and Motivation
  3. Large-scale Sparse Multi-objective Optimization
  4. Representative LSMOEAs
  5. Motivation of This Work
  6. The Proposed Algorithm
  7. Procedure of PAMEA
  8. Exploitation Search Strategy
  9. Annealing-Based Variable Grouping
  10. Annealing Search Strategy
  11. Empirical Studies
  12. Comparative Algorithms
  13. Test Problems
  14. Results on Benchmark Problems
  15. Results on Real-world Applications
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Illustration of the proposed PAMEA built on the dual-entropy scheme, where an exploitation search strategy and an annealing search strategy are integrated to balance exploitation and exploration for searching sparse Pareto optimal solutions.
  • Figure 2: Objectives values obtained by TS-SparseEA, SCEA, S-NSGA-II, MOEA/CKF, DKCA, and the Proposed PAMEA on SMOP2, SMOP3, and SMOP7 with 5 000 decision variables.
  • Figure 3: IGD variations obtained by TS-SparseEA, SCEA, S-NSGA-II, MOEA/CKF, DKCA, and the Proposed PAMEA on SMOP2, SMOP3, and SMOP7 with 5 000 decision variables
  • Figure 4: HV variations obtained by TS-SparseEA, SCEA, S-NSGA-II, MOEA/CKF, DKCA, and the Proposed PAMEA on PM2.