Tightening CVaR Approximations via Scenario-Wise Scaling for Chance-Constrained Programming

TL;DR

This paper introduces a scenario-wise scaling approach to tighten CVaR-based approximations in chance-constrained programming, improving solution quality while managing computational complexity.

Contribution

It proposes a novel scenario-wise scaling method that enhances CVaR approximations, providing conditions for optimality and developing algorithms to improve computational efficiency.

Findings

The scaled CVaR approximation can match the original CCP's optimal value.

Proposed algorithms improve solution quality over standard CVaR methods.

Numerical results show reduced conservativeness and maintained tractability.

Abstract

Chance-constrained programs (CCPs) provide a powerful modeling framework for decision-making under uncertainty, but their nonconvex feasible regions make them computationally challenging. A widely used convex inner approximation replaces chance constraints with Conditional Value-at-Risk (CVaR) constraints; however, the resulting solutions can be overly conservative and suboptimal. We propose a scenario-wise scaling approach that strengthens CVaR approximations for CCPs with finitely supported uncertainty. The method introduces scaling factors that reweight individual scenarios within the CVaR constraint, yielding a family of potentially tighter inner approximations. We establish sufficient conditions under which, for a suitable choice of scaling factors, the scaled CVaR approximation attains the same optimal value as the original CCP and admits a (near-)optimal solution of the CCP. We show that these conditions are tight and further relax them in the convex setting. We also show that optimizing over scenario-wise scaling factors is NP-hard. To address this computational challenge, we develop efficient heuristic and sequential convex approximation algorithms that iteratively update the scaling factors and generate improved feasible solutions. Numerical experiments demonstrate that the proposed methods consistently improve upon standard CVaR and state-of-the-art convex approximations, often reducing conservativeness while maintaining tractability.