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This research advances multiobjective optimization under uncertainty by formulating the problem as a Bayesian decision task that maximizes the expected hypervolume rather than individual objective expectations. The authors demonstrate how stochastic gradient methods and Gaussian Process surrogate models can effectively handle non-differentiable constraints, while introducing active learning strategies to refine these surrogates efficiently. This work is significant for improving the robustness of optimization algorithms in complex, uncertain environments where traditional expectation-based approaches may fall short.
Expected Hypervolume Maximization for Multiobjective Optimization under Uncertainties
arXiv:2609.19858v1 Announce Type: new Abstract: The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.
Abstract:The problem of multiobjective optimization under uncertainties is often approached by taking the expectation of each objective. In this work, we propose instead to formulate this as a Bayesian decision problem and to rely on the expected value of the hypervolume, which is to be maximized with respect to a finite set of input points. We show that this can be performed using methods based on gradients in a stochastic optimization framework, provided that care is taken with respect to dominated points. Moreover, in the absence of readily available differentiable code, we propose to use Gaussian Processes as differentiable surrogate models, in order to perform the optimization. An additional contribution in this work are some active learning strategies, through acquisition functions which helps construct a surrogate model well-designed for the multiobjective optimization problem at stake. These strategies are compared on simple analytical problems to assess their performances.
Machine Learning (cs.LG)
arXiv:2609.19858 [cs.LG]
(or arXiv:2609.19858v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.19858 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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