Column generation for multistage stochastic mixed-integer nonlinear programs with discrete state variables

TL;DR

This paper develops a column generation method for solving large-scale multistage stochastic mixed-integer nonlinear programs with discrete state variables, improving solution efficiency and quality.

Contribution

It introduces a tailored column generation approach that decomposes the problem into smaller MINLP subproblems and generates additional columns to satisfy nonanticipativity constraints.

Findings

Significantly improved convergence and solution quality.

Effective handling of large-scale multistage stochastic MINLPs.

Successful application to power network routing and blending problems.

Abstract

Stochastic programming provides a natural framework for modeling sequential optimization problems under uncertainty; however, the efficient solution of large-scale multistage stochastic programs remains a challenge, especially in the presence of discrete decisions and nonlinearities. In this work, we consider multistage stochastic mixed-integer nonlinear programs (MINLPs) with discrete state variables, which exhibit a decomposable structure that allows its solution using a column generation approach. Following a Dantzig-Wolfe reformulation, we apply column generation such that each pricing subproblem is an MINLP of much smaller size, making it more amenable to global MINLP solvers. We further propose a method for generating additional columns that satisfy the nonanticipativity constraints, leading to significantly improved convergence and optimal or near-optimal solutions for many large-scale instances in a reasonable computation time. The effectiveness of the tailored column generation algorithm is demonstrated via computational case studies on a multistage blending problem and a problem involving the routing of mobile generators in a power distribution network.