Convex Risk Control with Exact Probabilities: The CVaR-Chance-Constraint Approach

Chance-constrained optimization (CCO) offers exact control of failure probabilities but becomes numerically prohibitive for large scenario sets. The buffered failure probability, also known as the Conditional Value-at-Risk (CVaR), is convex and therefore tractable, but it typically leads to overly conservative designs. We introduce a new formulation, the CVaR-Chance-Constraint (CVaR-CC), which preserves the probabilistic guarantee of CCO while leveraging the convex-analytic structure of the superquantile. We develop three scalable algorithms: (i) a secant root-finding scheme that iteratively adjusts the CVaR right-hand side until the chance constraint is met; (ii) a unit-slope quasi-Newton iteration whose local convergence holds under mild assumptions; and (iii) an active-set procedure that retains only tail scenarios, shrinking the master problem and accelerating convergence for very large scenario sets. For each algorithm we establish convergence and provide explicit sufficient conditions. Numerical experiments on illustrative examples and energy-portfolio benchmarks show that CVaR-CC attains the required reliability with objective values close to the CCO solution while solving up to an order of magnitude faster than mixed-integer state-of-the-art methods. The framework reconciles risk fidelity with computational efficiency, enabling chance-constrained design in large, data-driven applications.