A Decomposition Algorithm for Two-Stage Stochastic Programs with Nonconvex Recourse Functions

SIAM Journal on Optimization, Volume 34, Issue 1, Page 306-335, March 2024.
Abstract. In this paper, we have studied a decomposition method for solving a class of nonconvex two-stage stochastic programs, where both the objective and constraints of the second-stage problem are nonlinearly parameterized by the first-stage variables. Due to the failure of the Clarke regularity of the resulting nonconvex recourse function, classical decomposition approaches such as Benders decomposition and (augmented) Lagrangian-based algorithms cannot be directly generalized to solve such models. By exploring an implicitly convex-concave structure of the recourse function, we introduce a novel decomposition framework based on the so-called partial Moreau envelope. The algorithm successively generates strongly convex quadratic approximations of the recourse function based on the solutions of the second-stage convex subproblems and adds them to the first-stage master problem. Convergence has been established for both a fixed number of scenarios and a sequential internal sampling strategy. Numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm.