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