Dealing With Structure Constraints in Evolutionary Pareto Set Learning

In the past few decades, many multiobjective evolutionary optimization algorithms (MOEAs) have been proposed to find a finite set of approximate Pareto solutions for a given problem in a single run. However, in many real-world applications, it could be desirable to have structure constraints on the entire optimal solution set, which define the patterns shared among all solutions. The current population-based MOEAs cannot properly handle such requirements. In this work, we make a first attempt to add structure constraints on the solution set. Specifically, we propose to model such a multiobjective optimization problem as a set optimization problem with structure constraints. The structure constraints define some patterns that all the solutions are required to share. Such patterns can be fixed components shared by all solutions, specific relations among decision variables, and the required shape of the Pareto set. In addition, we develop a simple yet efficient evolutionary stochastic optimization method to learn the set model, which only requires a low computational budget similar to classic MOEAs. With our proposed method, the decision-makers can easily trade off the Pareto optimality with preferred structures, which is not supported by other MOEAs. A set of experiments on benchmark test suites and real-world application problems demonstrates that our proposed method is effective. © 2025 IEEE.