Parametric Pareto Set Learning for Expensive Multi-Objective Optimization

TL;DR

The paper tackles expensive parametric multi-objective optimization by learning a parametric Pareto-set model that maps both preferences and exogenous parameters to Pareto-optimal solutions. It introduces PPSL-MOBO, combining a hypernetwork with Low-Rank Adaptation (LoRA) to generate task-conditioned Pareto models, Gaussian-process surrogates over augmented inputs, and a hypervolume-based acquisition strategy to minimize evaluations. The approach demonstrates strong performance in two challenging domains: multi-objective optimization with shared components and dynamic multi-objective optimization, achieving near-instant Pareto-set inference after a single training phase and reducing evaluations by orders of magnitude compared to re-optimization per parameter. This parametric learning framework enables real-time adaptation and efficient exploration across parameter space, with implications for modular design, personalized optimization, and dynamic decision-making in engineering and beyond.

Abstract

Parametric multi-objective optimization (PMO) addresses the challenge of solving an infinite family of multi-objective optimization problems, where optimal solutions must adapt to varying parameters. Traditional methods require re-execution for each parameter configuration, leading to prohibitive costs when objective evaluations are computationally expensive. To address this issue, we propose Parametric Pareto Set Learning with multi-objective Bayesian Optimization (PPSL-MOBO), a novel framework that learns a unified mapping from both preferences and parameters to Pareto-optimal solutions. PPSL-MOBO leverages a hypernetwork with Low-Rank Adaptation (LoRA) to efficiently capture parametric variations, while integrating Gaussian process surrogates and hypervolume-based acquisition to minimize expensive function evaluations. We demonstrate PPSL-MOBO's effectiveness on two challenging applications: multi-objective optimization with shared components, where certain design variables must be identical across solution families due to modular constraints, and dynamic multi-objective optimization, where objectives evolve over time. Unlike existing methods that cannot directly solve PMO problems in a unified manner, PPSL-MOBO learns a single model that generalizes across the entire parameter space. By enabling instant inference of Pareto sets for new parameter values without retraining, PPSL-MOBO provides an efficient solution for expensive PMO problems.

Figures (8)

• Figure 1: The PPSL-MOBO Framework. A hypernetwork adapts a PS model to a task parameter $\bm{t}$. The PS model generates approximate Pareto optimal solution $\bm{x}$ from preference $\bm{\lambda}$. The entire model is trained using predictions from surrogate models. A Bayesian optimization loop acquires new data by selecting PS-generated candidates that maximize HVI, which in turn refines the surrogate model.
• Figure 2: Learned PFs for the Rocket Injector Design Problem (RE37) with Shared Components. We demonstrate PFs for several specific parameter values, while PPSL-MOBO can obtain PS for any parameter in milliseconds. The transparent points denote the PFs for the problem without shared component. The shared components correspond to the decision variable $x_1$ (hydrogen flow angle), $x_2$ (hydrogen area), $x_3$ (oxygen area), and $x_4$ (oxidizer post tip thickness).
• Figure 3: Comparison of generated solutions by DNSGA-II (green) and PPSL-MOBO (red) at each time step on DMOPs. 200 evaluations are allowed for each time step.
• Figure 4: Learned Pareto fronts for the Four-Bar Truss Design Problem (RE21) with shared components. The dashed black line represents the unconstrained Pareto front. The color coding indicates different parameter values, illustrating how the Pareto front geometry evolves as the shared component values change.
• Figure 5: Learned Pareto fronts for the Disc Brake Design Problem (RE33) with shared components.

Theorems & Definitions (5)

• Definition 1: Pareto Dominance
• Definition 2: Pareto Optimality
• Definition 3: Pareto Set and Pareto Front
• Theorem 1: choo1983proper
• Theorem 2: lin2024smooth