Distributionally Robust Optimization for Nonconvex QCQPs with Stochastic Constraints

The quadratically constrained quadratic program (QCQP) with stochastic constraints appears in a wide range of real-world problems, including but not limited to the control of power systems. The randomness in the constraints prohibits the application of classic stochastic optimization algorithms. In this work, we utilize the techniques from the distributionally robust optimization (DRO) and propose a novel optimization formulation to solve the QCQP problems under strong duality. The proposed formulation does not contain stochastic constraints. The solutions to the optimization formulation attain the optimal objective value among all solutions that satisfy the stochastic constraints with high probability under the data-generating distribution, even when only a few samples from the distribution are available. We design corresponding algorithms to solve the optimization problems under the new formulation. Numerical experiments are conducted to verify the theory and illustrate the empirical performance of the proposed algorithm. This work provides the first results on the application of DRO techniques to non-convex optimization problems with stochastic constraints and the approach can be extended to a broad class of optimization problems.