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

Tushar Rathi, Benjamin P. Riley, Angela Flores-Quiroz, Qi Zhang

TL;DR

The paper tackles the challenge of solving multistage stochastic MINLPs with discrete state variables and nonlinearities by developing a discretization-based Dantzig-Wolfe reformulation (M-MSSP) and a parallel column-generation algorithm. It introduces per-node pricing problems to generate columns, along with a column-sharing mechanism that enforces nonanticipativity and accelerates convergence. Through two case studies—multistage blending and mobile generator routing—the authors show that CG and especially CGCS achieve smaller optimality gaps and faster convergence than solving the full-space model, with CGCS often solving all large instances. The work demonstrates the practical potential of decomposed MINLP approaches for large-scale stochastic optimization in energy and process industries, facilitated by parallel computation.

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.

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

TL;DR

The paper tackles the challenge of solving multistage stochastic MINLPs with discrete state variables and nonlinearities by developing a discretization-based Dantzig-Wolfe reformulation (M-MSSP) and a parallel column-generation algorithm. It introduces per-node pricing problems to generate columns, along with a column-sharing mechanism that enforces nonanticipativity and accelerates convergence. Through two case studies—multistage blending and mobile generator routing—the authors show that CG and especially CGCS achieve smaller optimality gaps and faster convergence than solving the full-space model, with CGCS often solving all large instances. The work demonstrates the practical potential of decomposed MINLP approaches for large-scale stochastic optimization in energy and process industries, facilitated by parallel computation.

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.
Paper Structure (12 sections, 22 equations, 10 figures, 7 tables, 2 algorithms)

This paper contains 12 sections, 22 equations, 10 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Reformulation via discretization
  3. Column generation
  4. Column sharing
  5. Case studies
  6. Multistage blending problem
  7. Model formulation
  8. Results and discussion
  9. Mobile generator routing problem
  10. Model formulation
  11. Results and discussion
  12. Conclusions

Figures (10)

  • Figure 1: Schematic of a typical scenario tree for a multistage stochastic programming problem.
  • Figure 2: Illustrating column sharing amongst the first set of sibling nodes at the third stage of a scenario tree with branching structure $\mathcal{R}=\{R_1,R_2,R_3,R_4\}=\{1,2,4,2\}$, where $R_{t}$ is the number of children nodes of each node in stage $t-1$. If we price out one distinct column from each node in this scenario tree in iteration $k$, for a total of 27 distinct columns, we can obtain up to 42 additional columns by sharing them among sibling nodes.
  • Figure 3: A network of input (suppliers) and output (markets) tanks. The specification of the product delivered to each market is determined by the blending of input streams from different suppliers.
  • Figure 4: Illustrating the percentage of additional columns generated in CGCS versus CG, as well as the percentage of time spent in the CS step compared to the total pricing time (solving regular subproblems + CS).
  • Figure 5: (a) Convergence profile for an instance from 12 input/10 output with 7 time period case, and (b) average number of iterations for instances of different sizes.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3