Optimization under rare chance constraints

TL;DR

This paper introduces a novel sampling-free approach for solving rare chance constrained optimization problems with Gaussian mixture uncertainties, improving computational efficiency and applicability across various fields.

Contribution

It develops a new method integrating large deviation theory with convex analysis, avoiding sampling and handling non-convex constraints for rare event optimization.

Findings

Method is independent of event rarity and complexity.

Applicable to linear, nonlinear, and PDE-constrained problems.

Outperforms classical sampling methods in accuracy and efficiency.

Abstract

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.