Boundary Unlocking Value for Safe Bayesian Optimization under Monotone Safety Constraints

Many safety-constrained optimization problems involve both a system configuration and an ordered intensity. When safety risk increases monotonically with intensity, each configuration is associated with an unknown maximum safe intensity. Under a limited evaluation budget, efficient optimization requires identifying which safe experiments are most valuable for subsequent optimization. Existing safe Bayesian optimization methods often guide query selection using criteria based on uncertainty, optimism, or information gain, but these criteria do not directly quantify the optimization value that may result from resolving different safety boundaries. The monotone configuration–intensity structure makes such value tractable to quantify: the feasibility uncertainty of each configuration can be represented by a latent maximum safe intensity, and each possible boundary location determines an accessible intensity range and the best objective value attainable within it. Building on this observation, we introduce Boundary Unlocking Value (BUV), which combines posterior boundary uncertainty with optimistic objective gains to quantify the exploration potential of each configuration. We further develop BUV-SafeBO, which evaluates candidate points within the certified safe set based on how informative they are about the safety frontiers of high-BUV configurations. It also incorporates expected improvement at each candidate to balance future safe expansion with immediate objective optimization. Experiments on multiple structured safe optimization benchmarks show that BUV-SafeBO achieves competitive optimization performance while maintaining high safe-query rates.