# Regularized Stochastic Dual Dynamic Programming for convex nonlinear optimization problems

## TL;DR

This paper introduces regularized variants of dual dynamic programming algorithms, REDDP and SDDP-REG, for nonlinear stochastic optimization, demonstrating significant computational speedups in portfolio management problems with market impact costs.

## Contribution

It extends dual dynamic programming methods with regularization to nonlinear and stochastic problems, providing convergence proofs and demonstrating substantial efficiency improvements.

## Key findings

- REDDP is up to 184 times faster than traditional DDP.
- SDDP-REG outperforms standard SDDP in portfolio problems with market impact costs.
- Numerical results confirm the efficiency and convergence of the proposed algorithms.

## Abstract

We define a regularized variant of the Dual Dynamic Programming algorithm called REDDP (REgularized Dual Dynamic Programming) to solve nonlinear dynamic programming equations. We extend the algorithm to solve nonlinear stochastic dynamic programming equations. The corresponding algorithm, called SDDP-REG, can be seen as an extension of a regularization of the Stochastic Dual Dynamic Programming (SDDP) algorithm recently introduced which was studied for linear problems only and with less general prox-centers. We show the convergence of REDDP and SDDP-REG. We assess the performance of REDDP and SDDP-REG on portfolio models with direct transaction and market impact costs. In particular, we propose a risk-neutral portfolio selection model which can be cast as a multistage stochastic second-order cone program. The formulation is motivated by the impact of market impact costs on large portfolio rebalancing operations. Numerical simulations show that REDDP is much quicker than DDP on all problem instances considered (up to 184 times quicker than DDP) and that SDDP-REG is quicker on the instances of portfolio selection problems with market impact costs tested and much faster on the instance of risk-neutral multistage stochastic linear program implemented (8.2 times faster).