Robust Chance-Constrained Optimization using a Continuous Parameter Space Wasserstein-2 Ambiguity Set of Gaussian Mixtures

Traditional distributionally robust optimization often relies on finite-support ambiguity sets, which restrict uncertainty to a fixed set of empirical points. This paper addresses the limitation where such models fail to account for structural misspecification in the nominal Gaussian mixture model (GMM). The authors ask: can we develop a continuous-parameter Wasserstein-2 ambiguity set that allows for more flexible, endogenous selection of mixture components to improve out-of-sample reliability in chance-constrained optimization?

To solve this, the authors develop a novel Wasserstein-2-type metric in the parameter space, rather than the data space. This allows the ambiguity set to include distributions where the means and covariances of the Gaussian components are not fixed a priori. They prove strong duality for the inner worst-case chance-constraint problem and derive a semi-infinite reformulation. To solve this, they implement an adaptive cutting-surface algorithm that iteratively identifies the most violating mixture components, using a block-alternating local search to refine the component parameters.

The study demonstrates that the continuous-parameter approach (CDR) consistently outperforms finite-support (FDR) methods in out-of-sample chance constraint satisfaction. While FDR models often fail to meet prescribed probability targets, the CDR framework achieves these targets across various settings. Furthermore, the CDR model induces structural changes in decision-making—such as energy allocations in electric vehicle charging—whereas FDR solutions remain largely tethered to the nominal model. While the CDR model is more computationally intensive, it provides a more robust and reliable framework for service-level decisions.