Stochastic Quadratic Dynamic Programming

TL;DR

The paper introduces SQDP, a quadratic extension of SDDP for multistage stochastic optimization with strongly convex recourse, demonstrating faster performance in large-scale and deterministic problems.

Contribution

It develops SQDP, replacing affine cuts with quadratic cuts, and proves its convergence and complexity, improving solution speed over existing methods.

Findings

SQDP outperforms SDDP for large strong convexity constants.

QCSC is faster than competing optimizers on deterministic problems.

Numerical experiments confirm the correctness and efficiency of the proposed methods.

Abstract

We introduce an algorithm called SQDP (Stochastic Quadratic Dynamic Programming) to solve some multistage stochastic optimization problems having strongly convex recourse functions. The algorithm extends the classical Stochastic Dual Dynamic Programming (SDDP) method replacing affine cuts by quadratic cuts. We provide conditions ensuring strong convexity of the recourse functions and prove the convergence of SQDP. In the special case of a single stage deterministic problem, we call QCSC (Quadratic Cuts for Strongly Convex optimization) the method and prove its complexity. Numerical experiments illustrate the performance and correctness of SQDP, with SQDP being much quicker than SDDP for large values of the constants of strong convexity both for a multistage problem and a two-stage assembly recourse model. We also present the results of numerical experiments on deterministic problems where QCSC is much quicker than several popular competing optimizers for solving 6 strongly convex optimization problems from the literature.