Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Ilias Mitrai, Prodromos Daoutidis

TL;DR

This work tackles the online solution challenge of mixed integer MPC by marrying Generalized Benders Decomposition with machine learning surrogates. The core idea is to learn surrogate models for the subproblem value function and its Lagrange multipliers, enabling single-pass master problem solves and rapid cut generation during branch-and-check. Applied to integrated scheduling and dynamic optimization of a chemical process, the approach achieves up to about 97% reduction in solution time with modest objective-function error (around 1%), across cases with multiple products and transitions. The method promises significant practical impact for real-time control of chemically driven processes by enabling faster, feasible solutions without sacrificing much accuracy, and it outlines pathways for data-efficient surrogate training and feasibility restoration in more challenging problem classes.

Abstract

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient and robust online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50x) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

TL;DR

This work tackles the online solution challenge of mixed integer MPC by marrying Generalized Benders Decomposition with machine learning surrogates. The core idea is to learn surrogate models for the subproblem value function and its Lagrange multipliers, enabling single-pass master problem solves and rapid cut generation during branch-and-check. Applied to integrated scheduling and dynamic optimization of a chemical process, the approach achieves up to about 97% reduction in solution time with modest objective-function error (around 1%), across cases with multiple products and transitions. The method promises significant practical impact for real-time control of chemically driven processes by enabling faster, feasible solutions without sacrificing much accuracy, and it outlines pathways for data-efficient surrogate training and feasibility restoration in more challenging problem classes.

Abstract

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient and robust online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50x) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.
Paper Structure (20 sections, 1 theorem, 27 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 20 sections, 1 theorem, 27 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Machine learning based branch and Benders cut algorithm
  3. Generalized Benders Decomposition
  4. Branch and check solution approach
  5. Machine learning based branch and check Generalized Benders Decomposition
  6. Mixed Integer Model Predictive Control for Integrated Scheduling and Optimization of a Chemical Process
  7. Scheduling model
  8. Dynamic model
  9. Integrated problem
  10. Decomposition-based solution approach
  11. Feasibility of the mixed integer MPC problem
  12. Numerical case studies
  13. Case study 1: 5 products
  14. Learning the surrogate models
  15. Transitions between products
  16. ...and 5 more sections

Key Result

Theorem 3.1

The mixed integer optimization problem in Eq. intgr prob is feasible if the following condition holds: where $\mathcal{I}_{trans}$ contains the transitions that occur between products and $i^{*}$ is the product that is manufactured in the first slot such that the total transition time is minimized.

Figures (3)

  • Figure 1: Schematic of mixed integer MPC problem
  • Figure 2: Transition cost from the intermediate state to the steady state of product 1 for different transition times. The x-axis is the transition time, the y-axis the initial concentration in the reactor $(x_0)$, and the z-axis is the transition cost divided by 1000.
  • Figure 3: Solution time for 20 randomly generated instances using the multicut GBD (M-GBD), hybrid multicut GBD (hM-GBG) from mitrai2022multicut and the proposed method using different surrogate models, Neural Network (NN-GBD), Decision Tree (DT-GBD), and Random Forest (RF-GBD) for the different number of products.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 3.1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4