A Cutting-plane and Benders' Decomposition Algorithm for Two-Stage Distributionally Robust Convex programs

TL;DR

This paper introduces a convergent cutting-plane and Benders' decomposition algorithm for two-stage distributionally robust convex programs, demonstrating practical efficiency with small optimality gaps in computational tests.

Contribution

It develops a novel algorithm combining cutting planes and Benders' decomposition for robust convex programs, extending to practical mixed-integer cases.

Findings

Achieves less than 5% optimality gap in 12 hours on test problems.

Outperforms commercial solvers with over 17% gap.

Demonstrates practical applicability of the proposed method.

Abstract

We present a finitely convergent cutting-plane algorithm for solving a general mixed-integer convex program given an oracle for solving a general convex program. This method is extended to solve a family of two-stage mixed-integer convex programs using cutting planes, with applications to solving distributionally-robust two-stage stochastic mixed-integer convex programs. Analysis is also given for the case where convex programming oracle provides an $epsilon$-optimal solution. We combine the cut generation with a branch-and-union scheme to develop a more practical algorithm. Computational results on generated test problems show the practicality of our algorithm. Specifically, results show that in the tested problems our algorithm achieves < 5% optimality gap in 12 hours. This gap is >17% with a commercial solver.