Predicting Accurate Lagrangian Multipliers for Mixed Integer Linear Programs

Lagrangian relaxation stands among the most efficient approaches for solving a Mixed Integer Linear Programs (MILP) with difficult constraints. Given any duals for these constraints, called Lagrangian Multipliers (LMs), it returns a bound on the optimal value of the MILP, and Lagrangian methods seek the LMs giving the best such bound. But these methods generally rely on iterative algorithms resembling gradient descent to maximize the concave piecewise linear dual function: the computational burden grows quickly with the number of relaxed constraints. We introduce a deep learning approach that bypasses the descent, effectively amortizing the local, per instance, optimization. A probabilistic encoder based on a graph convolutional network computes high-dimensional representations of relaxed constraints in MILP instances. A decoder then turns these representations into LMs. We train the encoder and decoder jointly by directly optimizing the bound obtained from the predicted multipliers. Numerical experiments show that our approach closes up to 85~\% of the gap between the continuous relaxation and the best Lagrangian bound, and provides a high quality warm-start for descent based Lagrangian methods.

Combinatorial Optimization enriched Machine Learning to solve the Dynamic Vehicle Routing Problem with Time Windows

With the rise of e-commerce and increasing customer requirements, logistics service providers face a new complexity in their daily planning, mainly due to efficiently handling same day deliveries. Existing multi-stage stochastic optimization approaches that allow to solve the underlying dynamic vehicle routing problem are either computationally too expensive for an application in online settings, or -- in the case of reinforcement learning -- struggle to perform well on high-dimensional combinatorial problems. To mitigate these drawbacks, we propose a novel machine learning pipeline that incorporates a combinatorial optimization layer. We apply this general pipeline to a dynamic vehicle routing problem with dispatching waves, which was recently promoted in the EURO Meets NeurIPS Vehicle Routing Competition at NeurIPS 2022. Our methodology ranked first in this competition, outperforming all other approaches in solving the proposed dynamic vehicle routing problem. With this work, we provide a comprehensive numerical study that further highlights the efficacy and benefits of the proposed pipeline beyond the results achieved in the competition, e.g., by showcasing the robustness of the encoded policy against unseen instances and scenarios.

Learning-based Online Optimization for Autonomous Mobility-on-Demand Fleet Control

Autonomous mobility-on-demand systems are a viable alternative to mitigate many transportation-related externalities in cities, such as rising vehicle volumes in urban areas and transportation-related pollution. However, the success of these systems heavily depends on efficient and effective fleet control strategies. In this context, we study online control algorithms for autonomous mobility-on-demand systems and develop a novel hybrid combinatorial optimization enriched machine learning pipeline which learns online dispatching and rebalancing policies from optimal full-information solutions. We test our hybrid pipeline on large-scale real-world scenarios with different vehicle fleet sizes and various request densities. We show that our approach outperforms state-of-the-art greedy, and model-predictive control approaches with respect to various KPIs, e.g., by up to 17.1% and on average by 6.3% in terms of realized profit.

Explainable Data-Driven Optimization: From Context to Decision and Back Again

Data-driven optimization uses contextual information and machine learning algorithms to find solutions to decision problems with uncertain parameters. While a vast body of work is dedicated to interpreting machine learning models in the classification setting, explaining decision pipelines involving learning algorithms remains unaddressed. This lack of interpretability can block the adoption of data-driven solutions as practitioners may not understand or trust the recommended decisions. We bridge this gap by introducing a counterfactual explanation methodology tailored to explain solutions to data-driven problems. We introduce two classes of explanations and develop methods to find nearest explanations of random forest and nearest-neighbor predictors. We demonstrate our approach by explaining key problems in operations management such as inventory management and routing.

Learning with Combinatorial Optimization Layers: a Probabilistic Approach

Combinatorial optimization (CO) layers in machine learning (ML) pipelines are a powerful tool to tackle data-driven decision tasks, but they come with two main challenges. First, the solution of a CO problem often behaves as a piecewise constant function of its objective parameters. Given that ML pipelines are typically trained using stochastic gradient descent, the absence of slope information is very detrimental. Second, standard ML losses do not work well in combinatorial settings. A growing body of research addresses these challenges through diverse methods. Unfortunately, the lack of well-maintained implementations slows down the adoption of CO layers. In this paper, building upon previous works, we introduce a probabilistic perspective on CO layers, which lends itself naturally to approximate differentiation and the construction of structured losses. We recover many approaches from the literature as special cases, and we also derive new ones. Based on this unifying perspective, we present InferOpt.jl, an open-source Julia package that 1) allows turning any CO oracle with a linear objective into a differentiable layer, and 2) defines adequate losses to train pipelines containing such layers. Our library works with arbitrary optimization algorithms, and it is fully compatible with Julia's ML ecosystem. We demonstrate its abilities using a pathfinding problem on video game maps.

Robust Counterfactual Explanations for Random Forests

Counterfactual explanations describe how to modify a feature vector in order to flip the outcome of a trained classifier. Several heuristic and optimal methods have been proposed to generate these explanations. However, the robustness of counterfactual explanations when the classifier is re-trained has yet to be studied. Our goal is to obtain counterfactual explanations for random forests that are robust to algorithmic uncertainty. We study the link between the robustness of ensemble models and the robustness of base learners and frame the generation of robust counterfactual explanations as a chance-constrained optimization problem. We develop a practical method with good empirical performance and provide finite-sample and asymptotic guarantees for simple random forests of stumps. We show that existing methods give surprisingly low robustness: the validity of naive counterfactuals is below $50\%$ on most data sets and can fall to $20\%$ on large problem instances with many features. Even with high plausibility, counterfactual explanations often exhibit low robustness to algorithmic uncertainty. In contrast, our method achieves high robustness with only a small increase in the distance from counterfactual explanations to their initial observations. Furthermore, we highlight the connection between the robustness of counterfactual explanations and the predictive importance of features.

Learning structured approximations of operations research problems

The design of algorithms that leverage machine learning alongside combinatorial optimization techniques is a young but thriving area of operations research. If trends emerge, the literature has still not converged on the proper way of combining these two techniques or on the predictor architectures that should be used. We focus on operations research problems for which no efficient algorithms are known, but that are variants of classic problems for which ones efficient algorithm exist. Elaborating on recent contributions that suggest using a machine learning predictor to approximate the variant by the classic problem, we introduce the notion of structured approximation of an operations research problem by another. We provide a generic learning algorithm to fit these approximations. This algorithm requires only instances of the variant in the training set, unlike previous learning algorithms that also require the solution of these instances. Using tools from statistical learning theory, we prove a result showing the convergence speed of the estimator, and deduce an approximation ratio guarantee on the performance of the algorithm obtained for the variant. Numerical experiments on a single machine scheduling and a stochastic vehicle scheduling problem from the literature show that our learning algorithm is competitive with algorithms that have access to optimal solutions, leading to state-of-the-art algorithms for the variant considered.

Optimal Counterfactual Explanations in Tree Ensembles

Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.

Learning to solve the single machine scheduling problem with release times and sum of completion times

In this paper, we focus on the solution of a hard single machine scheduling problem by new heuristic algorithms embedding techniques from machine learning field and scheduling theory. These heuristics transform an instance of the hard problem into an instance of a simpler one solved to optimality. The obtained schedule is then transposed to the original problem. Computational experiments show that they are competitive with state-of-the-art heuristics, notably on large instances.