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

TL;DR

This paper introduces a novel decomposition algorithm for nonconvex two-stage stochastic programs with complex recourse functions, leveraging an implicit convex-concave structure and partial Moreau envelopes to enable convergence and practical solution finding.

Contribution

It develops a new decomposition framework based on partial Moreau envelopes for nonconvex stochastic programs, addressing limitations of classical methods.

Findings

Algorithm converges under fixed scenarios and interior sampling.

Numerical experiments demonstrate the method's effectiveness.

Successfully handles nonconvex recourse functions in stochastic programming.

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 variable. 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 under both fixed scenarios and interior samplings is established. Numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm.