# Combining Progressive Hedging with a Frank-Wolfe Method to Compute Lagrangian Dual Bounds in Stochastic Mixed-Integer Programming

## TL;DR

This paper introduces a novel primal-dual algorithm combining progressive hedging with a Frank-Wolfe inspired linearization to efficiently compute Lagrangian dual bounds in stochastic mixed-integer programming, with proven convergence and superior empirical performance.

## Contribution

The paper presents a new algorithm that guarantees convergence to the optimal dual value and improves bound computation in SMIP by integrating Frank-Wolfe linearization steps into progressive hedging.

## Key findings

- Algorithm converges to the optimal Lagrangian dual value.
- Empirical results show improved bounds over standard progressive hedging.
- Demonstrates efficiency in solving stochastic mixed-integer programs.

## Abstract

We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical Frank-Wolfe method. Numerical results demonstrate that our new algorithm empirically outperforms the standard implementation of progressive hedging for obtaining bounds in SMIP.