Analyzing and Overcoming Local Optima in Complex Multi-Objective Optimization by Decomposition-Based Evolutionary Algorithms

TL;DR

This work tackles the persistent problem of local optima in decomposition-based MOEAs when facing complex, non-convex Pareto fronts. It provides a rigorous geometric analysis showing that the traditional min-based reference point is a primary cause of diversity loss and stagnation, and it introduces the normW method, combining a novel reference point $Z_w$ aligned with weight directions and a Gaussian-based timing strategy to balance diversity and convergence. The authors validate the theory with an ablation study across 14 MOEA/D variants and demonstrate that normW and $Z_w$ outperform both traditional RP and the latest DRP approach in population diversity and convergence across 16 benchmark problems, including IMOP, WFG, and DTLZ suites. The findings underscore the critical role of RP design in global search capabilities and offer a practical, theory-backed method that can enhance MOEA/D-based optimizers in real-world, complex MOPs. Overall, the work provides a principled pathway to mitigating local optima and improving Pareto-front coverage in high-dimensional, non-convex multi-objective optimization.

Abstract

When addressing the challenge of complex multi-objective optimization problems, particularly those with non-convex and non-uniform Pareto fronts, Decomposition-based Multi-Objective Evolutionary Algorithms (MOEADs) often converge to local optima, thereby limiting solution diversity. Despite its significance, this issue has received limited theoretical exploration. Through a comprehensive geometric analysis, we identify that the traditional method of Reference Point (RP) selection fundamentally contributes to this challenge. In response, we introduce an innovative RP selection strategy, the Weight Vector-Guided and Gaussian-Hybrid method, designed to overcome the local optima issue. This approach employs a novel RP type that aligns with weight vector directions and integrates a Gaussian distribution to combine three distinct RP categories. Our research comprises two main experimental components: an ablation study involving 14 algorithms within the MOEADs framework, spanning from 2014 to 2022, to validate our theoretical framework, and a series of empirical tests to evaluate the effectiveness of our proposed method against both traditional and cutting-edge alternatives. Results demonstrate that our method achieves remarkable improvements in both population diversity and convergence.

Figures (16)

• Figure 1: Illustration of the population results obtained by MOEA/D on the IMOP2 problem with $FE=20000$: (a) linear coordinate axis; (b) logarithmic coordinate axis.
• Figure 2: The Pareto Front of IMOP2 and Positions of $F, Z, G$ under the objective function coordinate plane.
• Figure 3: Objective function coordinate plane under the condition $k_{OW} > k_{ZF}$
• Figure 4: The objective function coordinate plane under the condition $k_{OW} \leq k_{ZF}$
• Figure 5: The objective function coordinate plane under the condition $k_{OW} > k_{ZF}$

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6