Expensive multi-objective optimization problems (EMOPs) are common in real-world scenarios where evaluating objective functions is costly and involves extensive computations or physical experiments. Current Pareto set learning methods for such problems often rely on surrogate models like Gaussian processes to approximate the objective functions. These surrogate models can become fragmented, resulting in numerous small uncertain regions between explored solutions. When using acquisition functions such as the Lower Confidence Bound (LCB), these uncertain regions can turn into pseudo-local optima, complicating the search for globally optimal solutions. To address these challenges, we propose a novel approach called SVH-PSL, which integrates Stein Variational Gradient Descent (SVGD) with Hypernetworks for efficient Pareto set learning. Our method addresses the issues of fragmented surrogate models and pseudo-local optima by collectively moving particles in a manner that smooths out the solution space. The particles interact with each other through a kernel function, which helps maintain diversity and encourages the exploration of underexplored regions. This kernel-based interaction prevents particles from clustering around pseudo-local optima and promotes convergence towards globally optimal solutions. Our approach aims to establish robust relationships between trade-off reference vectors and their corresponding true Pareto solutions, overcoming the limitations of existing methods. Through extensive experiments across both synthetic and real-world MOO benchmarks, we demonstrate that SVH-PSL significantly improves the quality of the learned Pareto set, offering a promising solution for expensive multi-objective optimization problems.

Pareto Front Learning (PFL) was recently introduced as an effective approach to obtain a mapping function from a given trade-off vector to a solution on the Pareto front, which solves the multi-objective optimization (MOO) problem. Due to the inherent trade-off between conflicting objectives, PFL offers a flexible approach in many scenarios in which the decision makers can not specify the preference of one Pareto solution over another, and must switch between them depending on the situation. However, existing PFL methods ignore the relationship between the solutions during the optimization process, which hinders the quality of the obtained front. To overcome this issue, we propose a novel PFL framework namely PHN-HVI, which employs a hypernetwork to generate multiple solutions from a set of diverse trade-off preferences and enhance the quality of the Pareto front by maximizing the Hypervolume indicator defined by these solutions. The experimental results on several MOO machine learning tasks show that the proposed framework significantly outperforms the baselines in producing the trade-off Pareto front.