An Operator Splitting Method for Large-Scale CVaR-Constrained Quadratic Programs

Abstract: We introduce a fast and scalable method for solving quadratic programs with conditional value-at-risk (CVaR) constraints. While these problems can be formulated as standard quadratic programs, the number of variables and constraints grows linearly with the number of scenarios, making general-purpose solvers impractical for large-scale problems. Our method combines operator splitting with a specialized $O(m\log m)$ algorithm for projecting onto CVaR constraints, where $m$ is the number of scenarios. The method alternates between solving a linear system and performing parallel projections: onto CVaR constraints using our specialized algorithm and onto box constraints with a closed-form solution. Numerical examples from several application domains demonstrate that our method outperforms general-purpose solvers by several orders of magnitude on problems with up to millions of scenarios. Our method is implemented in an open-source package called CVQP.