# Stochastic Lipschitz Dynamic Programming

## TL;DR

This paper introduces a novel algorithm for multistage stochastic MILPs that uses Lipschitz cuts to improve lower bounds, demonstrated through case studies comparing with existing methods like SDDP and SDDiP.

## Contribution

It develops a new Lipschitz cut-based approach for stochastic MILPs that maintains problem class integrity and enhances solution quality.

## Key findings

- The proposed algorithm effectively approximates non-convex cost functions.
- Application to case studies shows competitive performance with existing methods.
- Lipschitz cuts derived from Augmented Lagrangian Duality are MILP representable.

## Abstract

We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to cutting plane methods, we construct nonlinear Lipschitz cuts to build lower approximations for the non-convex cost to go functions. An example of such a class of cuts are those derived using Augmented Lagrangian Duality for MILPs. The family of Lipschitz cuts we use is MILP representable, so that the introduction of these cuts does not change the class of the original stochastic optimization problem. We illustrate the application of this algorithm on two simple case studies, comparing our approach with the convex relaxation of the problems, for which we can apply SDDP, and for a discretized approximation, applying SDDiP.