MOEA/D-GM: Using probabilistic graphical models in MOEA/D for solving combinatorial optimization problems

Evolutionary algorithms based on modeling the statistical dependencies (interactions) between the variables have been proposed to solve a wide range of complex problems. These algorithms learn and sample probabilistic graphical models able to encode and exploit the regularities of the problem. This paper investigates the effect of using probabilistic modeling techniques as a way to enhance the behavior of MOEA/D framework. MOEA/D is a decomposition based evolutionary algorithm that decomposes a multi-objective optimization problem (MOP) in a number of scalar single-objective subproblems and optimizes them in a collaborative manner. MOEA/D framework has been widely used to solve several MOPs. The proposed algorithm, MOEA/D using probabilistic Graphical Models (MOEA/D-GM) is able to instantiate both univariate and multi-variate probabilistic models for each subproblem. To validate the introduced framework algorithm, an experimental study is conducted on a multi-objective version of the deceptive function Trap5. The results show that the variant of the framework (MOEA/D-Tree), where tree models are learned from the matrices of the mutual information between the variables, is able to capture the structure of the problem. MOEA/D-Tree is able to achieve significantly better results than both MOEA/D using genetic operators and MOEA/D using univariate probability models, in terms of the approximation to the true Pareto front.

💡 Research Summary

The paper introduces MOEA/D‑GM, a novel integration of probabilistic graphical models (PGMs) into the well‑known MOEA/D framework for solving discrete multi‑objective combinatorial problems. Traditional MOEA/D decomposes a multi‑objective optimization problem (MOP) into N scalar sub‑problems, each associated with a weight vector λ. Solutions are generated by selecting parents from a neighbourhood B(i) and applying crossover and mutation; the offspring replaces existing solutions in a replacement set R(i) if it improves the scalar aggregation function. This conventional variation mechanism ignores variable dependencies, which can be detrimental for deceptive problems where interactions between decision variables are crucial.

MOEA/D‑GM replaces the crossover‑mutation step with a model‑based sampling step. For each sub‑problem i, a subset S of the current population (typically the top 50 % by fitness) is selected. From S a probabilistic model M_i is learned and then sampled to produce a new candidate y. Two families of models are considered. The first is a univariate marginal distribution (UMDA/PBIL style), where each binary variable X_j is assumed independent and its probability p(X_j = 1) is estimated as the frequency of 1’s in S. New solutions are generated by independent Bernoulli draws. The second family is a tree‑structured model. Mutual information between every pair of variables is computed from S, and a maximum‑spanning‑tree algorithm yields a tree that maximizes the total mutual information. In the resulting tree, each variable has at most one parent; conditional probabilities p(X_j | parent) are estimated from the frequencies in S. Sampling proceeds from the root to the leaves, respecting the conditional dependencies.

The authors embed this learning‑and‑sampling pipeline into the standard MOEA/D loop, preserving the decomposition, neighbourhood, and external archive mechanisms. They also adopt the replacement strategy of Wang et al. (2015) that limits the number of solutions replaced by a new offspring, thereby preventing premature cloning.

Experimental evaluation focuses on a multi‑objective version of the deceptive Trap5 function, a binary problem where each block of five bits contributes a high fitness only when all bits are 1; otherwise the contribution is low. This structure creates strong epistatic interactions that are difficult for algorithms that assume independence. Three algorithmic variants are compared: (1) classic MOEA/D with crossover/mutation, (2) MOEA/D‑GM with univariate models (MOEA/D‑GM‑Uni), and (3) MOEA/D‑GM with tree models (MOEA/D‑GM‑Tree). Performance is measured using Inverted Generational Distance (IGD) and Hypervolume (HV) across 30 independent runs per configuration. Statistical significance is assessed with the Wilcoxon signed‑rank test.

Results show that MOEA/D‑GM‑Tree consistently achieves lower IGD and higher HV than both baselines. The tree model successfully captures the pairwise dependencies inherent in the Trap5 landscape, guiding the search toward the true Pareto front while maintaining diversity. The univariate model performs similarly to, or slightly worse than, the classic MOEA/D, confirming that ignoring dependencies hampers progress on deceptive problems. Computational overhead of learning the mutual‑information matrix (O(n²) where n is the number of variables) and constructing the tree is modest; overall runtime remains comparable to the baseline because model‑based sampling is cheap relative to crossover and mutation.

The paper positions MOEA/D‑GM as one of the first frameworks that couples decomposition‑based multi‑objective optimization with Estimation‑of‑Distribution Algorithms (EDAs) on a per‑sub‑problem basis. By allowing each sub‑problem to learn its own tailored graphical model, the approach can adapt to heterogeneous structures across the weight‑vector space. The authors discuss related work, noting that prior MOEA/D‑based EDAs have largely relied on univariate models and therefore could not exploit epistatic information.

Future research directions include extending the framework to richer PGMs such as Bayesian networks or Markov random fields, handling continuous or mixed‑type variables, dynamically adapting neighbourhood size and weight vectors, and scaling to many‑objective scenarios (more than three objectives). The authors also suggest investigating hybridization with local search and parallel implementations to further improve efficiency.

In summary, MOEA/D‑GM demonstrates that incorporating probabilistic graphical models—particularly tree‑structured models derived from mutual information—into the MOEA/D paradigm yields substantial gains on deceptive combinatorial multi‑objective problems. The work provides both theoretical insight into the benefits of modeling variable dependencies and practical evidence of improved convergence and diversity, opening a promising avenue for advanced multi‑objective evolutionary optimization.