Optimization under Rare Chance Constraints

Chance constraints provide a principled framework to mitigate the risk of high-impact extreme events by modifying controllable properties system. The low probability and rare occurrence such events, however, impose severe sampling computational requirements on classical solution methods that render them impractical. This work proposes novel sampling-free method for solving chance constrained optimization problems affected uncertainties follow general Gaussian mixture distributions. By integrating modern developments in large deviation theory with tools from convex analysis bilevel optimization, we propose tractable formulations can be solved off-the-shelf solvers. Our enjoy several advantages compared methods: their size complexity is independent event rarity, they do not require linearity or convexity assumptions system constraints, under easily verifiable conditions, serve as safe conservative approximations asymptotically exact reformulations true problem. Computational experiments linear, nonlinear, PDE-constrained applications portfolio management, structural engineering, fluid dynamics illustrate broad applicability our its over sampling-based approaches terms both accuracy efficiency.