# Dual Dynamic Programming with cut selection: convergence proof and numerical experiments

## TL;DR

This paper proves convergence of a dual dynamic programming method with cut selection strategies for convex optimization, and demonstrates its efficiency through numerical experiments on large-scale portfolio and inventory problems.

## Contribution

It introduces a limited memory variant of Level 1 cut selection and proves convergence for nonlinear problems, with finite convergence for linear cases.

## Key findings

- The proposed method converges for nonlinear convex problems.
- Finite convergence is achieved for linear programs.
- Numerical results show significant speed-ups over simplex algorithms.

## Abstract

We consider convex optimization problems formulated using dynamic programming equations. Such problems can be solved using the Dual Dynamic Programming algorithm combined with the Level 1 cut selection strategy or the Territory algorithm to select the most relevant Benders cuts. We propose a limited memory variant of Level 1 and show the convergence of DDP combined with the Territory algorithm, Level 1 or its variant for nonlinear optimization problems. In the special case of linear programs, we show convergence in a finite number of iterations. Numerical simulations illustrate the interest of our variant and show that it can be much quicker than a simplex algorithm on some large instances of portfolio selection and inventory problems.