Local Constrained Bayesian Optimization

Bayesian optimization (BO) for high-dimensional constrained problems remains a significant challenge due to the curse of dimensionality. We propose **L**ocal **C**onstrained **B**ayesian **O**ptimization (LCBO), a novel framework tailored for such settings. Unlike trust-region methods that are prone to premature shrinking when confronting tight or complex constraints, LCBO leverages the differentiable landscape of constraint-penalized surrogates to alternate between rapid local descent and uncertainty-driven exploration. Theoretically, we prove that LCBO achieves a convergence rate for the Karush-Kuhn-Tucker (KKT) residual that depends polynomially on the dimension $d$ for common kernels under mild assumptions, offering a rigorous alternative to global BO where regret bounds typically scale exponentially. Extensive evaluations on high-dimensional benchmarks (up to 100D) demonstrate that LCBO consistently outperforms state-of-the-art baselines.

Many important design problems are like tuning a complicated machine: each test is costly, there are many knobs to adjust, and the final design must obey safety or physical limits. As the number of variables grows, global Bayesian optimization methods often struggle because they must search too broadly across a vast design space. Existing local methods based on “trust regions” try to address this issue by focusing the search on smaller neighborhoods, but they can still be too rigid when the constraints are tight or complex. We propose Local Constrained Bayesian Optimization (LCBO), a method that focuses the search near the current promising design without relying on a rigid search box. LCBO alternates between two simple steps: collecting nearby measurements that best reduce uncertainty about how the objective and constraints change, and then taking a local improvement step using this learned information. In theory, its progress depends more gently on the number of variables than standard global approaches. In experiments on synthetic, engineering, and robotic-control tasks, LCBO finds better feasible solutions with fewer evaluations than strong existing methods.