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

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

Problem Statement
The paper addresses Distributionally Robust Chance-Constrained Optimization (DRCCO) problems where the uncertain parameters follow a Gaussian Mixture Model (GMM). In standard DRCCO, decision-makers seek to minimize cost subject to a constraint that a linear inequality holds with at least a prescribed probability $\theta$, under the worst-case distribution within an ambiguity set.

Existing approaches often rely on finite-support ambiguity sets (e.g., based on empirical data points or a fixed set of mixture components). The authors argue that such finite-support constructions may fail to protect against realizations outside the observed support or underrepresent low-frequency (tail) regions, particularly when the underlying data is multimodal. Furthermore, fixing the means and covariances of mixture components to those estimated from data (as in Finite-Parameter DR or FDR) restricts the ambiguity set to a pre-selected finite set of parameters, potentially missing the true distribution if the parameter estimation is imperfect.

The core problem is to formulate and solve a DRCCO model where the ambiguity set allows the mixing weights and the parameters (means and covariances) of the Gaussian components to vary over a continuous parameter space, rather than being restricted to a finite set.

Methodology

1. Continuous Parameter Space Wasserstein-2 Metric:
   The authors develop a novel computable Wasserstein-2-type metric for GMMs defined over a continuous parameter space.

   • They establish an equivalence between the standard Wasserstein-2 metric in the data space (defined via optimal transport between probability measures) and a metric defined in the parameter space (defined via optimal transport between the mixing laws of the Gaussian parameters).
   • Specifically, they prove that the Wasserstein-2 distance between two GMMs can be computed by optimizing over couplings of their mixing laws (probability measures over the space of means and covariances), utilizing the Burer-Gelbrich distance (the closed-form Wasserstein-2 distance between two Gaussian distributions) as the cost function.
   • This allows the ambiguity set to be defined as a "ball" around a nominal GMM, where the radius $\rho$ controls the deviation allowed in the mixing law, permitting the worst-case distribution to place mass on Gaussian components with means and covariances that differ continuously from the nominal estimates.

2. Strong Duality and Semi-Infinite Reformulation:
   For the resulting distributionally robust chance-constrained problem (CDR), the authors prove strong duality for the inner worst-case problem under mild regularity conditions (specifically, that the covariance matrices are strictly positive definite and the parameter support is compact).

   • This duality result allows the reformulation of the robust chance constraint into a semi-infinite program (S-Inf). The reformulation introduces a finite number of dual variables but an infinite number of constraints, one for every possible parameter point in the continuous support set.

3. Adaptive Cutting-Surface Algorithm (CSA):
   To solve the semi-infinite program, the authors propose an adaptive cutting-surface algorithm.

   • Master Problem: Solves a finite relaxation of the S-Inf using a subset of constraints (cuts) generated from a finite set of parameter points.
   • Subproblem (Separation): Identifies the "worst-case" parameter point (mean and covariance) that violates the current solution's constraints the most. This involves maximizing a non-convex function involving the Gaussian cumulative distribution function (CDF) and the Burer-Gelbrich distance.
   • Approximation: Since the Gaussian CDF lacks a closed-form expression, the authors use piecewise-linear (PWL) inner and outer approximations to transform the subproblem and master problem into Mixed-Integer Linear/Nonlinear Programs (MILP/MINLP) solvable by commercial solvers.
   • Heuristic: A block-alternating local-search procedure is employed to solve the non-convex separation subproblem efficiently, identifying new Gaussian distributions to add to the pool.
   • Convergence: The authors prove that the algorithm attains any prescribed optimality gap in finitely many iterations, utilizing a finite number of linear segments to approximate the non-convexity of the Gaussian CDF.

Key Contributions

• Theoretical: The paper provides a rigorous theoretical foundation for defining and computing the Wasserstein-2 metric for GMMs over continuous parameter spaces, generalizing previous finite-set-based definitions. It proves the equivalence between the data-space and parameter-space formulations.
• Algorithmic: It develops a cutting-surface algorithm with proven finite convergence for solving the resulting semi-infinite robust chance-constrained problems. The method endogenously determines the locations of mixture components receiving mass and their parameters, rather than fixing them a priori.
• Computational: The study evaluates the Continuous-Support DR (CDR) model against Finite-Support DR (FDR) and nominal models using a real-world case study of electric vehicle (EV) charging station energy allocation.

Results

The computational study uses a dataset of EV charging demands to test the models under various target satisfaction probabilities ($\theta \in \{0.95, 0.97, 0.99\}$) and Wasserstein radii ($\rho \in \{0.001, 0.005, 0.01\}$).

• Out-of-Sample (OSS) Performance:
  • The FDR model (finite support) consistently fails to attain the prescribed target probability in out-of-sample testing, regardless of the radius or target level. Its performance remains close to the nominal model.
  • The CDR model exhibits a consistent improvement in OSS probability as the Wasserstein radius $\rho$ increases. For instance, at $\theta=0.95$, CDR satisfies the target even at the smallest radius ($\rho=0.001$) with OSS values between 95.04% and 95.87%. At $\theta=0.99$ and $\rho=0.01$, CDR achieves OSS values up to 99.17% (with mean-support uncertainty $\pm 10\%$), whereas FDR fails to reach 96%.
• Cost vs. Reliability Trade-off:
  • Achieving higher reliability with CDR comes at a cost. The objective value (cost) increases by approximately 1.5% to 5.4% relative to the nominal model and 1.5% to 5.0% relative to FDR.
  • The mean-support uncertainty parameter ($\varsigma$) is identified as the primary driver of robustness gains, while covariance scaling has a more limited effect.
• Structural Changes:
  • Unlike FDR, which produces allocations close to the nominal schedule, CDR induces structural changes in the energy allocation profile. The robustness is achieved not by uniformly increasing allocations, but by selectively redistributing energy to specific hours (e.g., shifting load to hours 11–12, 17, and 19–21) that are critical under distributional perturbations.

Significance and Claims

The paper claims that the proposed continuous-support robust framework is necessary when out-of-sample chance constraint satisfaction is critical, particularly in settings with multimodal uncertainty where parameter estimation errors are a concern.

• Superiority over Finite Support: The results demonstrate that restricting the ambiguity set to finite support points (FDR) is insufficient for achieving high service-level targets in the presence of multimodal data and parameter uncertainty. The continuous support allows the model to hedge against errors in the estimated means and covariances of the mixture components.
• Operational Value: The study shows that robustness is a "timing decision." The CDR model provides actionable insights by identifying specific time blocks where protection is most valuable, leading to structurally different and more reliable operational schedules compared to nominal or finite-support approaches.
• Practicality: Despite the increased computational complexity (solution times ranging from minutes to hours compared to seconds for nominal models), the framework is computationally approachable and yields solutions with significantly higher empirical reliability, justifying the additional computational overhead for high-stakes service-level decisions.

The authors conclude that while the current implementation relies on off-the-shelf solvers and incurs runtime costs, the framework successfully bridges the gap between theoretical distributional robustness and practical service-level management for multimodal uncertainties. Future work is suggested to focus on computational acceleration through warm-starting and tailored decomposition strategies.