Fusing Backdoors, Machine Learning, and Optimization for Large-Scale Parametric Mixed-Integer Programs

Decision-focused learning has shown great promise for addressing predict-then-optimize problems, particularly in the presence of under-specified models. However, its practical deployment is often hindered by high computational costs and limited scalability, as it requires solving a constrained optimization problem for each training instance at every iteration. To address these challenges, we propose a novel framework that incorporates Lagrangian decomposition into the decision-focused learning paradigm. Specifically, we introduce a new surrogate objective along with two loss functions for evaluating and training the underlying prediction model. We further propose two variants of our approach, which offer different trade-offs between computational efficiency and solution quality. Our framework can be seamlessly integrated with standard decision-focused learning methods, including Smart Predict-then-Optimize (SPO+) and Implicit Maximum Likelihood Estimation (IMLE). Through experiments on two standard benchmarks, the multi-dimensional knapsack problem and quadratic portfolio optimization, we demonstrate that our approach achieves competitive performance while remaining amenable to parallelization. In particular, it consistently outperforms traditional decision-focused learning methods on large-scale instances, involving up to eight times more variables than those typically considered in related work. The implementation is available at https://github.com/corail-research/DFL-LD.

Optimization models developed by operations research (OR) experts are often deployed as decision-support systems in industrial settings. However, real-world environments are dynamic, with evolving business rules and unforeseen perturbations. In such contexts, end users should ideally re-optimize models to recover feasible and implementable solutions, often without access to the original model developers. This paper introduces an agentic re-optimization framework in which a large language model (LLM) acts as an OR expert, dynamically supporting end users through natural-language interaction. The LLM translates user prompts into structured updates of the underlying optimization model, selects suitable re-optimization techniques from an optimization toolbox, and solves the resulting instance to return implementable solutions. The toolbox leverages primal information, including historical solutions, valid inequalities, solver configurations, and metaheuristics, to accelerate re-optimization while preserving solution quality. The proposed framework enables interactive and continuous adaptation of deployed optimization models, reducing dependence on OR experts, and improving the sustainability of decision-support systems. Extensive experiments on two complementary large-scale real-world case studies demonstrate the effectiveness and scalability of the proposed framework. The first considers online supply chain re-optimization, where solutions must be generated rapidly while remaining close to the deployed plan, whereas the second focuses on offline university exam scheduling, where solution quality is prioritized over runtime. Results show that the toolbox-driven architecture significantly improves computational efficiency through primal-based and solver-aware re-optimization techniques, while the structured patch-based updates improve interpretability and traceability of model modifications.

Benders decomposition is a fundamental framework for solving large-scale mixed-integer optimization problems with complicating variables that, when fixed, yield significantly easier subproblems. However, classical Benders decomposition repeatedly solves highly similar subproblems and often exhibits zigzagging behavior across iterations, leading to slow convergence in large-scale settings. Motivated by the repetitive structure and parametric nature of Benders subproblems, this paper introduces the proxy Benders decomposition (Proxy-BD), a new decomposition framework in which subproblem optimization is replaced by certified optimization proxies rather than repeated exact solves. The proposed proxy follows a self-supervised predict-project-and-complete mechanism that produces dual-feasible solutions for generating provably valid Benders cuts. The framework preserves the theoretical validity of the decomposition independently of prediction quality through a projection-and-completion certification layer. A formal characterization of proxy-induced cuts is established, and the framework naturally extends to modern decomposition schemes, including branch-and-Benders-cut algorithms. Computational experiments on large-scale facility location and network design problems demonstrate that Proxy-BD substantially reduces the computational effort of subproblems while maintaining near-optimal solution quality. On large-scale uncapacitated facility location instances up to 2000x2000, Proxy-BD achieves median optimality gaps below 0.5%, yields up to 161x median speedups, and reduces the number of generated cuts by more than 240x on the largest instances. The computational gains consistently increase with recourse complexity, indicating that proxy-based inference scales substantially more favorably than repeated exact subproblem optimization in large-scale decomposition settings.