Multiobjective Optimization under Uncertainties using Conditional Pareto Fronts

TL;DR

The paper tackles multiobjective optimization under parametric uncertainties by introducing Conditional Pareto Fronts (CPF) and Conditional Pareto Sets (CPS) to capture environment-dependent optima and a robustness metric called the probability of coverage $P_U[x \\in \\mathcal{P}^*_\\mathcal{X}(U)]$. It proposes Gaussian Process surrogates over the joint space $\\mathcal{X}\\times\\mathcal{U}$ and three acquisition functions—Profile EHVI (PEHVI), Integrated EHVI (IEHVI), and Weight-augmented PEHVI (WPEHVI)—to estimate and optimize the CPF/CPS under uncertainty, with CPF estimation via GP mean using a bounding parameter $\\beta$. The method is validated on analytical 4D and 10D toy problems and on EnergyPlus-based cabin design, showing that IEHVI generally provides the most accurate and robust guidance, while PEHVI offers strong exploration in some settings; results highlight the trade-off between computational cost and robustness in design decisions. Overall, the approach delivers a principled framework to quantify and optimize design robustness under uncertainty, enabling more informed, risk-conscious decisions with limited expensive evaluations.

Abstract

In this work, we propose a novel method to tackle the problem of multiobjective optimization under parameteric uncertainties, by considering the Conditional Pareto Sets and Conditional Pareto Fronts. Based on those quantities we can define the probability of coverage of the Conditional Pareto Set which can be interpreted as the probability for a design to be optimal in the Pareto sense. Due to the computational cost of such an approach, we introduce an Active Learning method based on Gaussian Process Regression in order to improve the estimation of this probability, which relies on a reformulation of the EHVI. We illustrate those methods on a few toy problems of moderate dimension, and on the problem of designing a cabin to highlight the differences in solutions brought by different formulations of the problem.

Figures (14)

• Figure 1: Illustration of the different domination regions in Multiobjective optimization
• Figure 1: Sketch of the different behaviors: Both $x_1$ and $x_2$ provide the same mean objective, the marginals of the objective vector are the same, but the behavior in a multiobjective optimization problem is not the same
• Figure 1: Illustration of discretized Conditional Pareto Sets and Fronts, depending on the environmental variable
• Figure 1: Illustration of PEHVI. The shaded region in cyan corresponds to the HV improvement corresponding to the choice $x_1$, the fuchsia to $x_2$
• Figure 1: Mean objective multiobjective optimization: the mean of the first and second objective are shown on the top figures. The Pareto front is shown on the bottom figure.