Maximum Mean Discrepancy Distributionally Robust Nonlinear Chance-Constrained Optimization with Finite-Sample Guarantee

TL;DR

This paper introduces MMD-DRCCP, a practical distributionally robust chance-constrained optimization method using kernel MMD ambiguity sets, capable of handling nonlinear constraints with finite-sample guarantees and without costly cross-validation.

Contribution

It proposes a novel MMD-based ambiguity set for DRCCP that handles general nonlinear constraints and provides finite-sample guarantees without ad-hoc reformulations.

Findings

Finite-sample guarantee of $rac{1}{\sqrt{N}}$ rate independent of dimension

Effective bootstrap scheme for constructing MMD ambiguity sets

Validated on portfolio optimization and control problems

Abstract

This paper is motivated by addressing open questions in distributionally robust chance-constrained programs (DRCCP) using the popular Wasserstein ambiguity sets. Specifically, the computational techniques for those programs typically place restrictive assumptions on the constraint functions and the size of the Wasserstein ambiguity sets is often set using costly cross-validation (CV) procedures or conservative measure concentration bounds. In contrast, we propose a practical DRCCP algorithm using kernel maximum mean discrepancy (MMD) ambiguity sets, which we term MMD-DRCCP, to treat general nonlinear constraints without using ad-hoc reformulation techniques. MMD-DRCCP can handle general nonlinear and non-convex constraints with a proven finite-sample constraint satisfaction guarantee of a dimension-independent $\mathcal{O}(\frac{1}{\sqrt{N}})$ rate, achievable by a practical algorithm. We further propose an efficient bootstrap scheme for constructing sharp MMD ambiguity sets in practice without resorting to CV. Our algorithm is validated numerically on a portfolio optimization problem and a tube-based distributionally robust model predictive control problem with non-convex constraints.