MOBO-CAPS: Multi-objective Bayesian Optimization with Cardinality-Aware Pareto Selection

Multi-objective Bayesian optimization (MOBO) traditionally aims to approximate the entire Pareto front, but real-world deployment is often limited to a small, representative subset of solutions due to resource constraints. Current two-stage methods first approximate the full front before selecting solutions, wasting evaluation budget. We formalize this problem as Multi-objective Bayesian Optimization with Cardinality-aware Pareto Selection (MOBO-CAPS), where the goal is to identify a set of exactly $M$ solutions with high hypervolume. We propose two practical cardinality-aware acquisition strategies: REHVI, a replacement-aware approximation to expected hypervolume improvement, and CHO-UCB, which constructs an optimistic $M$-point set by optimizing a collective UCB-based hypervolume objective. We prove a sublinear simple regret bound for the batched version of CHO-UCB, and empirically show that cardinality-aware acquisition design improves over standard MOBO followed by post-hoc subset selection across nine benchmarks.