Bayesian preference elicitation for decision support in multiobjective optimization

TL;DR

The paper tackles the challenge of identifying a decision-maker’s most preferred solution from the Pareto set in multiobjective optimization under uncertain preferences. It introduces a Bayesian, nonparametric approach using a Gaussian process prior over the DM’s utility with a logistic likelihood for noisy pairwise comparisons, and leverages the $qEUBO$ acquisition to select informative queries; it also provides a menu-based final decision aid by maximizing the expected utility of the best item. Key contributions include a scalable variational inference scheme for the posterior, a principled interactive elicitation strategy, and a principled menu-generation mechanism, demonstrated on problems with up to $m=9$ objectives and accompanied by an open-source implementation. The approach achieves high data efficiency and robustness to noise, enabling effective decision support in high-dimensional MO optimization and offering practical utility for interactive design and decision-making processes.

Abstract

We present a novel approach to help decision-makers efficiently identify preferred solutions from the Pareto set of a multi-objective optimization problem. Our method uses a Bayesian model to estimate the decision-maker's utility function based on pairwise comparisons. Aided by this model, a principled elicitation strategy selects queries interactively to balance exploration and exploitation, guiding the discovery of high-utility solutions. The approach is flexible: it can be used interactively or a posteriori after estimating the Pareto front through standard multi-objective optimization techniques. Additionally, at the end of the elicitation phase, it generates a reduced menu of high-quality solutions, simplifying the decision-making process. Through experiments on test problems with up to nine objectives, our method demonstrates superior performance in finding high-utility solutions with a small number of queries. We also provide an open-source implementation of our method to support its adoption by the broader community.

Figures (5)

• Figure 1: Mean regret and standard error for DTLZ7 with 5 decision variables and 3 objectives, DTLZ2 with 9 decision variables and 6 objectives, WFG3 with 14 decision variables and 9 objectives, and Car Cab Design with 7 decision variables and 9 objectives.
• Figure 2: Convergence of our algorithm in objective space for DTLZ7 with $5$ decision variables and $2$ objectives. The leftmost panel shows the Pareto front (orange) together with contour lines of the true utility function. The subsequent panels depict the $4$ initial queries (beige crosses), the algorithm’s selected queries (black crosses), and the model’s predicted expected utility after $1$, $6$, and $11$ user interactions.
• Figure 3: Effect of the monotonicity for DTLZ2 with 9 decision variables and 6 objectives, WFG3 with 14 decision variables and 9 objectives, and Car Cab Design with 7 decision variables and 9 objectives.
• Figure 4: Effect of noise for DTLZ7 with 5 decision variables and 3 objectives and DTLZ2 with 9 decision variables and 6 objectives, and Car Cab Design with 7 decision variables and 9 objectives. The noise levels corresponding to 0%, 15% and 30% mistakes of the DM at the top 1% of the utility values in the domain are indicated by the 'no noise', '15% noise', and '30% noise' labels.
• Figure 5: Regret for different menu sizes for DTLZ7 with 5 decision variables and 3 objectives (left), DTLZ2 with 9 decision variables and 6 objectives (center), and Car Cab Design with 7 decision variables and 9 objectives (right). The first row shows results for noise-free responses from the DM. The second row shows results for noisy responses with medium noise level (i.e., 15% mistakes of the DM at the top 1% of the utility values in the domain). Short, medium, and small dashes correspond to menus of size 1, 4, and 16, respectively.