Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decoupling Constraint from Two Direction in Evolutionary Constrained Multi-objective Optimization

Ruiqing Sun, Dawei Feng, Xing Zhou, Lianghao Li, Sheng Qi, Bo Ding, Yijie Wang, Rui Wang, Huaimin Wang

TL;DR

This paper tackles CMOPs by exposing and exploiting constraint coupling to better locate the Constrained Pareto Front (CPF). It introduces SCPF, ICPF, and RCPF as geometric concepts and proposes DCF2D, a bidirectional, multi-population evolutionary framework that dynamically detects constraint couplings and spawns auxiliary populations to search both the feasible and infeasible boundaries. The method combines UPF-driven global exploration, boundary-focused constraint decoupling via auxiliary tasks, and a final convergence phase, supported by an infeasible archive (LI) that guides a resurrection mechanism. Empirical results across seven benchmark suites and 28 real-world CMOPs demonstrate that DCF2D consistently outperforms state-of-the-art CMOEAs, especially in highly constrained problems with complex coupling, validating the value of dual-direction decoupling for constraint-aware multi-objective optimization.

Abstract

Real-world Constrained Multi-objective Optimization Problems (CMOPs) often contain multiple constraints, and understanding and utilizing the coupling between these constraints is crucial for solving CMOPs. However, existing Constrained Multi-objective Evolutionary Algorithms (CMOEAs) typically ignore these couplings and treat all constraints as a single aggregate, which lacks interpretability regarding the specific geometric roles of constraints. To address this limitation, we first analyze how different constraints interact and show that the final Constrained Pareto Front (CPF) depends not only on the Pareto fronts of individual constraints but also on the boundaries of infeasible regions. This insight implies that CMOPs with different coupling types must be solved from different search directions. Accordingly, we propose a novel algorithm named Decoupling Constraint from Two Directions (DCF2D). This method periodically detects constraint couplings and spawns an auxiliary population for each relevant constraint with an appropriate search direction. Extensive experiments on seven challenging CMOP benchmark suites and on a collection of real-world CMOPs demonstrate that DCF2D outperforms five state-of-the-art CMOEAs, including existing decoupling-based methods.

Decoupling Constraint from Two Direction in Evolutionary Constrained Multi-objective Optimization

TL;DR

This paper tackles CMOPs by exposing and exploiting constraint coupling to better locate the Constrained Pareto Front (CPF). It introduces SCPF, ICPF, and RCPF as geometric concepts and proposes DCF2D, a bidirectional, multi-population evolutionary framework that dynamically detects constraint couplings and spawns auxiliary populations to search both the feasible and infeasible boundaries. The method combines UPF-driven global exploration, boundary-focused constraint decoupling via auxiliary tasks, and a final convergence phase, supported by an infeasible archive (LI) that guides a resurrection mechanism. Empirical results across seven benchmark suites and 28 real-world CMOPs demonstrate that DCF2D consistently outperforms state-of-the-art CMOEAs, especially in highly constrained problems with complex coupling, validating the value of dual-direction decoupling for constraint-aware multi-objective optimization.

Abstract

Real-world Constrained Multi-objective Optimization Problems (CMOPs) often contain multiple constraints, and understanding and utilizing the coupling between these constraints is crucial for solving CMOPs. However, existing Constrained Multi-objective Evolutionary Algorithms (CMOEAs) typically ignore these couplings and treat all constraints as a single aggregate, which lacks interpretability regarding the specific geometric roles of constraints. To address this limitation, we first analyze how different constraints interact and show that the final Constrained Pareto Front (CPF) depends not only on the Pareto fronts of individual constraints but also on the boundaries of infeasible regions. This insight implies that CMOPs with different coupling types must be solved from different search directions. Accordingly, we propose a novel algorithm named Decoupling Constraint from Two Directions (DCF2D). This method periodically detects constraint couplings and spawns an auxiliary population for each relevant constraint with an appropriate search direction. Extensive experiments on seven challenging CMOP benchmark suites and on a collection of real-world CMOPs demonstrate that DCF2D outperforms five state-of-the-art CMOEAs, including existing decoupling-based methods.
Paper Structure (24 sections, 17 equations, 5 figures, 4 tables, 3 algorithms)

This paper contains 24 sections, 17 equations, 5 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Traditional Constraint Handling Techniques
  4. Co-evolutionary and Multi-population Methods
  5. Multi-stage Methods
  6. Machine Learning-assisted Methods
  7. Constraint Decoupling Methods
  8. PROPOSED ALGORITHM
  9. Framework of DCF2D
  10. Key functions in DCF2D
  11. Update LI
  12. Update $AP_i$
  13. Analysis of DCF2D on a Running Instance
  14. Experimental Settings
  15. Comparative CMOEAs
  16. ...and 9 more sections

Figures (5)

  • Figure 1: Four types of coupling constraints. (a) The final CPF is composed of parts of two SCPFs. (b) The final CPF is composed of only a single SCPF, and the infeasible region of another constraint makes this SCPF partly infeasible. (c) The final CPF is partially composed of a SCPF, and the other part of the CPF (ICPF) is affected by the infeasible region of another constraint and is closely connected to the boundary of the infeasible region (we define it as RCPF). (d) Affected by the infeasible region of one of the constraints, the entire final CPF is ICPF that is only related to the RCPF.
  • Figure 2: The distribution of solutions in the Infeasible Archive ($LI$, marked as red circles) across four situations. $LI$ contains non-dominated infeasible solutions found during evolution, which are used to detect constraints that block the convergence of the CPF. Identifying the infeasible boundaries (RCPF) or SCPFs of these blocking constraints is crucial for guiding the search direction in DCF2D.
  • Figure 3: Evolutionary process of DCF2D on LIRCMOP2. (a) In Stage 1, $AT_0$ searches for the UPF. (b) At the beginning of Stage 2, $AT_1$ and $AT_2$ are activated in the Positive direction (triangles). (c) Since no feasible solutions are found, the direction switches to Negative, searching for the RCPF (inverted triangles). (d) $AT_1$ and $AT_2$ successfully approximate the infeasible boundaries closest to the CPF.
  • Figure 4: Impact of constraint count on the average IGD+ rank of DCF2D and comparison algorithms. The y-axis is inverted so that a higher position indicates a better ranking (lower numerical value).
  • Figure 5: Impact of the number of constraints ($n_{con}$) on the average runtime of DCF2D and comparative algorithms. The broken axis is used to accommodate the high runtime of MTOTC and FCDS.