A Hybrid Learning-to-Optimize Framework for Mixed-Integer Quadratic Programming

TL;DR

This work addresses real-time solution of parametric mixed-integer quadratic programs (MIQPs) common in mixed-integer model predictive control (MI-MPC). It proposes a hybrid learning-to-optimize (L2O) framework that combines a neural network predicting integer decisions with a differentiable QP layer to obtain continuous decision variables, enabling backpropagation through the optimization problem. A hybrid loss blends supervised optimality with self-supervised feasibility, supported by a slack-variable QP relaxation to ensure differentiability even for infeasible intermediate problems, with theoretical guarantees on exactness when feasible and minimal violation when not. Empirical evaluation on two MI-MPC benchmarks shows substantial runtime gains (approximately 9x–12x faster than GUROBI) while balancing constraint satisfaction and near-global optimality, indicating strong potential for real-time control applications and for extending to broader problem classes in the future.

Abstract

In this paper, we propose a learning-to-optimize (L2O) framework to accelerate solving parametric mixed-integer quadratic programming (MIQP) problems, with a particular focus on mixed-integer model predictive control (MI-MPC) applications. The framework learns to predict integer solutions with enhanced optimality and feasibility by integrating supervised learning (for optimality), self-supervised learning (for feasibility), and a differentiable quadratic programming (QP) layer, resulting in a hybrid L2O framework. Specifically, a neural network (NN) is used to learn the mapping from problem parameters to optimal integer solutions, while a differentiable QP layer is integrated to compute the corresponding continuous variables given the predicted integers and problem parameters. Moreover, a hybrid loss function is proposed, which combines a supervised loss with respect to the global optimal solution, and a self-supervised loss derived from the problem's objective and constraints. The effectiveness of the proposed framework is demonstrated on two benchmark MI-MPC problems, with comparative results against purely supervised and self-supervised learning models.

Figures (3)

• Figure 1: Architecture of the proposed hybrid framework (c) compared with supervised learning (a) and self-supervised learning (b). In our framework, the NN takes the problem parameters $\boldsymbol{\theta}$ to predict the integer solution $\boldsymbol{\delta}$, while the QP layer computes the continuous solution $\boldsymbol{x}$ based on $\boldsymbol{\theta}$ and $\boldsymbol{\delta}$. In conventional SL and SSL, the NN is trained to predict the integer solution without considering the continuous solution or to predict both the integer and continuous solutions, respectively.
• Figure 2: Statistical comparison of the three models: hybrid L2O (H-L2O), supervised learning (SL), and self-supervised learning (SSL), for the robot navigation example.
• Figure 3: Statistical comparison of the three models: hybrid L2O (H-L2O), supervised learning (SL), and self-supervised learning (SSL), for thermal energy tank example.

Theorems & Definitions (4)

• remark 1
• theorem 1
• theorem 2
• proof