Decision-Focused Surrogate Modeling for Mixed-Integer Linear Optimization

TL;DR

This work introduces decision-focused surrogate modeling for MILPs, constructing surrogate LPs (DFSOMs) by learning parametric linear inequalities that preserve original constraints while mimicking MILP solutions. Trained via a bilevel, penalty-based optimization grounded in KKT conditions, the method uses block coordinate descent to achieve data-efficient learning. In two case studies—hybrid vehicle control and production scheduling—the DFSOMs delivered high-accuracy predictions with minimal optimality loss and substantial speedups over solving MILPs or using neural-network proxies. The results demonstrate the practical potential of DFSM to enable real-time decision-making in online systems where MILP complexity would otherwise be prohibitive.

Abstract

Mixed-integer optimization is at the core of many online decision-making systems that demand frequent updates of decisions in real time. However, due to their combinatorial nature, mixed-integer linear programs (MILPs) can be difficult to solve, rendering them often unsuitable for time-critical online applications. To address this challenge, we develop a data-driven approach for constructing surrogate optimization models in the form of linear programs (LPs) that can be solved much more efficiently than the corresponding MILPs. We train these surrogate LPs in a decision-focused manner such that for different model inputs, they achieve the same or close to the same optimal solutions as the original MILPs. One key advantage of the proposed method is that it allows the incorporation of all the original MILP's linear constraints, which significantly increases the likelihood of obtaining feasible predicted solutions. Results from two computational case studies indicate that this decision-focused surrogate modeling approach is highly data-efficient and provides very accurate predictions of the optimal solutions. In these examples, it outperforms more commonly used neural-network-based optimization proxies.

Figures (13)

• Figure 1: Illustrative example in which for the given cost vector $c$, the MILP with integer variables $x_1$ and $x_2$ and constraints $Ax \leq b$ has the same optimal solution as the LPs with $\mathrm{conv}(\mathcal{S})$ and $\mathcal{S}'$ as their feasible regions.
• Figure 2: DFSOM performance compared to NN-based optimization proxies in the hybrid vehicle control problem.
• Figure 3: DFSOM performance for varying number of constraints learned ($|\mathcal{V}|$) in the hybrid vehicle control problem.
• Figure 4: Computational performance of DFSOMs evaluated across 1,000 random instances in the hybrid vehicle control case compared to directly solving the original MILPs and solving the MILPs to the same objective values achieved by the corresponding DFSOMs
• Figure 5: DFSOM performance compared to NN-based optimization proxies in the production scheduling problem.