Strengthened and Faster Linear Approximation to Joint Chance Constraints with Wasserstein Ambiguity

Abstract

Many real-world decision-making problems have uncertain parameters in constraints. Wasserstein distributionally robust joint chance constraints (WDRJCC) offer a promising solution by explicitly guaranteeing the probability of the simultaneous constraint satisfaction. However, WDRJCC are computationally demanding, and practical applications often require more tractable approaches, especially for large-scale problems such as power system unit commitment problems and multilevel problems with chance constraints in lower levels. To address this, this paper proposes a convex inner-approximation for WDRJCC with right-hand-side uncertainties (RHS-WDRJCC). We propose a Strengthened and Faster Linear Approximation (SFLA) by strengthening an existing convex inner-approximation. This strengthening process reduces the number of constraints and tightens the feasible region for ancillary variables, leading to significant computational speedup. We prove that the proposed SFLA does not introduce extra conservativeness and can be less conservative compared to common approximations such as W-CVaR. We then extend the proposed SFLA to a more interpretable decision-making paradigm: robustness maximization, where the risk level and the Wasserstein radius are determined by maximizing solution robustness subject to a utility degradation limit. We discuss the connection between risk minimization and radius maximization as two formulations of robustness maximization, and show the advantage of radius maximization. In power system unit commitment, the proposed SFLA achieves up to 10x computational speedup compared to the strengthened and exact reformulation. In a bilevel strategic bidding problem where the exact reformulation is not applicable due to non-convexity, the proposed SFLA leads to 90x speedup than W-CVaR. In robustness maximization, the proposed SFLA demonstrated over 100x speedup.