Evolutionary Alternating Direction Method of Multipliers for Constrained Multi-Objective Optimization with Unknown Constraints

TL;DR

This work addresses constrained multi-objective optimization with unknown constraints by introducing Evolutionary ADMM (EADMM), an ADMM-inspired framework that decouples objective and constraint handling into two co-evolving populations that alternate directions to find feasible Pareto-optimal fronts. EADMM reformulates the problem additively and implements three modules: Module ❶ constrained-objective search using modified backbones (NSGA-II, IBEA, MOEA/D), Module ❷ unconstrained exploration, and Module ❸ a disruption-minimizing local search that aligns the two populations via $\mathbf{x}=\mathbf{y}$. The approach demonstrates faster convergence and robustness across various Pareto-front shapes on 120 synthetic benchmarks and two real-world problems, outperforming five peer algorithms while preserving diversity. Its plug-and-play compatibility with existing EMO backbones and its effectiveness under unknown constraints suggest strong practical impact for complex design and engineering optimization tasks.

Abstract

Constrained multi-objective optimization problems (CMOPs) pervade real-world applications in science, engineering, and design. Constraint violation has been a building block in designing evolutionary multi-objective optimization algorithms for solving constrained multi-objective optimization problems. However, in certain scenarios, constraint functions might be unknown or inadequately defined, making constraint violation unattainable and potentially misleading for conventional constrained evolutionary multi-objective optimization algorithms. To address this issue, we present the first of its kind evolutionary optimization framework, inspired by the principles of the alternating direction method of multipliers that decouples objective and constraint functions. This framework tackles CMOPs with unknown constraints by reformulating the original problem into an additive form of two subproblems, each of which is allotted a dedicated evolutionary population. Notably, these two populations operate towards complementary evolutionary directions during their optimization processes. In order to minimize discrepancy, their evolutionary directions alternate, aiding the discovery of feasible solutions. Comparative experiments conducted against five state-of-the-art constrained evolutionary multi-objective optimization algorithms, on 120 benchmark test problem instances with varying properties, as well as two real-world engineering optimization problems, demonstrate the effectiveness and superiority of our proposed framework. Its salient features include faster convergence and enhanced resilience to various Pareto front shapes.

Figures (17)

• Figure 1: Flowchart of our proposed EADMM flowchart.
• Figure 2: Illustrations of the working mechanism of EADMM/MOEAD, where $\dashrightarrow$ is the weight vector, [0.5ex]0.5cm0.5pt2pt is the contour line of the Tchebycheff function.
• Figure 3: Illustrations for the PFs and the feasible regions of the synthetic test problems built upon C1-DTLZ1. Subfigures (a) to (e) represents the cases ranging from Type-I to Type-V.
• Figure 4: Total Scott-Knott test ranks achieved by each of the three algorithm instances of our proposed framework (the smaller the rank is, the better performance achieved).
• Figure 5: Percentage of the large, medium, small, and equal $A_{12}$ effect size, respectively, when comparing EADMM/MOEA/D against EADMM/NSGA-II and EADMM/IBEA.

Theorems & Definitions (18)

• Definition 1
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• Definition 5
• Remark 1
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• Remark 5