arXiv subjects

Andrea Lodi

Publications and source records attributed to Andrea Lodi.

At least 19 recordsLinked to original sources

arXiv2026

Fair Online Resource Allocation

We study the problem of fair online resource allocation, motivated by applications such as refugee resettlement and airline scheduling, where agents arrive sequentially and must be assigned to facilities with limited capacities. We introduce a model that maximizes the overall welfare subject to resource constraints and a Lipschitz fairness requirement, which ensures that similar agents arriving in the same batch receive similar expected outcomes. We first analyze the offline problem, proving that the value of the optimal fair allocation is at least an $\Omega(1/\gamma)$ fraction of the optimal unfair allocation, where $\gamma$ is the fairness coefficient, thereby bounding the price of fairness. For the online setting, we propose an algorithm based on dual mirror descent that enforces fairness constraints within batches while estimating optimal dual variables. We prove that this algorithm achieves sublinear regret relative to the optimal offline fluid benchmark. Finally, we validate our theoretical results using real-world data from the Refugee Economies Programme, demonstrating the algorithm's performance and examining the trade-offs between welfare maximization and fairness enforcement.

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arXiv2026

Learning to reoptimize: a GNN-aided fix-and-optimize approach and an application to the Lot Sizing problem

In many operational contexts, solutions to NP-hard combinatorial optimization problems, modeled by means of Mixed-Integer Linear Programming (MILP), may become infeasible due to unpredictable disruptions. Typically, reoptimizing by solving the MILP formulation on the perturbed instance is not possible as new solutions must be obtained in a very short computing time, while simple repairing heuristics may result in low-quality solutions. To bridge this gap, we propose a learning-to-reoptimize framework, and apply it to the Lot Sizing Problem (LSP) under machine breakdown disruptions. We design a fix-and-optimize strategy aided by a Graph Neural Network (GNN) that efficiently computes a new solution within the neighborhood of a repaired solution. By representing the instance, the original solution and the disruption as a feature graph, we train a GNN to predict the likelihood that specific binary variables require to be modified. These predictions guide the selection of a small subset of variables to be reoptimized by an MILP solver, while the other variables are hard-fixed. Numerical experiments on a large dataset demonstrate that our approach handles effectively different problem sizes, and that it significantly outperforms a baseline alternative approach, yielding larger cost reductions within the same limited time budget.

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arXiv2026

From Sequential Nodes to GPU Batches: Parallel Branch and Bound for Optimal $k$-Sparse GLMs

GPUs have significantly accelerated first-order methods for large-scale optimization, especially in continuous optimization. However, this success has not transferred cleanly to problems with discrete variables, combinatorial structure, and nonlinear objectives, such as certifying optimal solutions for cardinality-constrained generalized linear models. Major challenges include the sequential processing of heterogeneous nodes in branch and bound (BnB) and frequent data movement between the CPU and GPU. We propose a simple, generic, and modular CPU--GPU framework that processes multiple BnB nodes in batches on GPUs. The framework is built around a small set of GPU-efficient routines and uses padding together with lightweight custom kernels to handle irregular node data structures. Experiments show one to two orders of magnitude speedups and zero optimality gap on challenging instances. The framework can also be extended to collect the entire Rashomon set, enabling downstream statistical analysis such as variable-importance analysis and model selection under secondary user-specific measures (e.g., AUC in classification).

cs.LG↗

arXiv2026

Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

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arXiv2026

Enforcing TSP-Optimality in Fair Vehicle Routing by Cutting Planes

We study the fair capacitated vehicle routing problem, in which a fleet of vehicles must serve a set of customers such that the difference between the longest and shortest route, the range, is minimized. A key challenge is that the range objective is non-monotonic: it can be reduced by artificially lengthening routes, leading to solutions that violate TSP-optimality of individual routes. Existing exact methods struggle to handle this efficiently. We propose a branch-price-and-cut framework that enforces TSP-optimality through TSP-optimality cuts, which forbid TSP-dominated arc sequences. We strengthen the cuts through a dedicated lifting procedure. Computational experiments on benchmark instances with up to 25 customers show the method solves nearly all instances to optimality, achieving an average gap of 0.27% on the hardest configurations.

math.OC↗

arXiv2026

Chv\'atal-Gomory Rounding of Eigenvector Inequalities for QCQPs

We introduce and analyze a class of valid inequalities for nonconvex quadratically constrained optimization problems (QCQPs) which we call Eigen-CG inequalities. These inequalities are obtained by applying a Chv\'atal-Gomory (CG) rounding to the well-known eigenvector inequalities for QCQPs, and transferring binary-valid inequalities to the continuous setting via a result of Burer and Letchford (2009). We define three nested subfamilies and prove that they are strictly contained in one another. However, we show that the convex conic closure of two of these subfamilies is equal and, in fact, coincides with the Boros-Hammer inequalities -- a powerful family of inequalities that include, in particular, the triangle and McCormick inequalities. Using this CG perspective, we also prove that dense Eigen-CG inequalities are ineffective when used with the standard SDP+McCormick relaxation. This provides a complementary perspective on what is observed in practice: that sparse inequalities are impactful. Finally, based on these insights, we develop a computational strategy to find sparse Eigen-CG cuts and verify their effectiveness in nonconvex QCQP instances. Our results confirm that density quickly degrades effectiveness, but that including sparse inequalities beyond triangle inequalities can provide significant improvements in dual bounds.

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arXiv2026

Hardness of some optimization problems over correlation polyhedra

We prove the \textbf{NP}-hardness, using Karp reductions, of some problems related to the correlation polytope and its corresponding cone, spanned by all of the $n\times n$ rank-one matrices over $\{0,1\}$. The problems are: membership, rank of the decomposition, and a ``relaxed rank'' obtained from relaxing the zero-norm expression for the rank to an $\ell_1$ norm. While membership and rank are natural problems for any matrix cone, the relaxed rank problem occurs in some signal processing and statistical applications.

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arXiv2026

Sparse Cuts for the Positive Semidefinite Cone

We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite cone and are sparse in the sense that they are supported only on the variables corresponding to products of variables present in quadratic functions. We show that these sparse linear inequalities yield an LP relaxation that gives the same bound as the SDP relaxation. We demonstrate how to identify these inequalities via a separation procedure that involves solving a structured ``projection'' SDP. In a computational study, we find that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.

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arXiv2026

GPU-friendly and Linearly Convergent First-order Methods for Certifying Optimal $k$-sparse GLMs

We investigate the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through a cardinality constraint. While Branch-and-Bound (BnB) frameworks can certify optimality using perspective relaxations, existing methods for solving these relaxations are computationally intensive, limiting their scalability. To address this challenge, we reformulate the relaxations as composite optimization problems and develop a unified proximal framework that is both linearly convergent and computationally efficient. Under specific geometric regularity conditions, our analysis links primal quadratic growth to dual quadratic decay, yielding error bounds that make the Fenchel duality gap a sharp proxy for progress towards the solution set. This leads to a duality gap-based restart scheme that upgrades a broad class of sublinear proximal methods to provably linearly convergent methods, and applies beyond the sparse GLM setting. For the implicit perspective regularizer, we further derive specialized routines to evaluate the regularizer and its proximal operator exactly in log-linear time, avoiding costly generic conic solvers. The resulting iterations are dominated by matrix--vector multiplications, which enables GPU acceleration. Experiments on synthetic and real-world datasets show orders-of-magnitude faster dual-bound computations and substantially improved BnB scalability on large instances.

math.OC↗

arXiv2026

Batched First-Order Methods for Parallel LP Solving in MIP

We present a batched first-order method for solving multiple linear programs in parallel on GPUs. Our approach extends the primal-dual hybrid gradient algorithm to efficiently solve batches of related linear programming problems that arise in mixed-integer programming techniques such as strong branching and bound tightening. By leveraging matrix-matrix operations instead of repeated matrix-vector operations, we obtain significant computational advantages on GPU architectures. We demonstrate the effectiveness of our approach on various case studies and identify the problem sizes where first-order methods outperform traditional simplex-based solvers depending on the computational environment one can use. This is a significant step for the design and development of integer programming algorithms tightly exploiting GPU capabilities where we argue that some specific operations should be allocated to GPUs and performed in full instead of using light-weight heuristic approaches on CPUs.

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arXiv2025

Cutting Planes for Binarized Network Flow Problems

We consider integer programming problems with bounded general-integer variables belonging to the general class of network flow problems. For those, we computationally investigate the effect on mixed-integer linear programming (MIP) solvers of the different ways of producing extended formulations that replace a bounded general integer variable by a linear combination of a set of auxiliary binary variables linked by additional linear constraints. We show that MILP solvers perform very differently depending on which extended formulations is used and we interpret that different performance through the lens of cutting planes generation. Finally, we discuss a simple family of mixed-integer rounding inequalities that especially benefit from the reformulation, and we show its benefit within different MIP solvers. This provides methodological and practical guidelines for the use of those extended formulations in MIP and, to the best of our knowledge, this is the first extensive computational analysis of the topic. All our data and tables are publicly available at https://github.com/anton-derkach1/binarizations.

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arXiv2025

SMiLE: Provably Enforcing Global Relational Properties in Neural Networks

Artificial Intelligence systems are increasingly deployed in settings where ensuring robustness, fairness, or domain-specific properties is essential for regulation compliance and alignment with human values. However, especially on Neural Networks, property enforcement is very challenging, and existing methods are limited to specific constraints or local properties (defined around datapoints), or fail to provide full guarantees. We tackle these limitations by extending SMiLE, a recently proposed enforcement framework for NNs, to support global relational properties (defined over the entire input space). The proposed approach scales well with model complexity, accommodates general properties and backbones, and provides full satisfaction guarantees. We evaluate SMiLE on monotonicity, global robustness, and individual fairness, on synthetic and real data, for regression and classification tasks. Our approach is competitive with property-specific baselines in terms of accuracy and runtime, and strictly superior in terms of generality and level of guarantees. Overall, our results emphasize the potential of the SMiLE framework as a platform for future research and applications.

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arXiv2025

Electric Vehicle Scheduling and Vehicle-to-Grid Integration in Microgrids

The logistical challenges and high costs associated with procuring and transporting fuel to remote military bases underscore the need for sustainable and resilient energy solutions. Integrating renewable energy sources and electric vehicles into military microgrids offers a promising approach to enhance energy security and operational readiness. This paper explores the optimization of travel, charging, and discharging schedules for a fleet of military electric trucks, aiming to minimize reliance on fuel-generated electricity while ensuring that mission-critical transportation needs are met. We extend the classical Vehicle Scheduling Problem by incorporating Electric Vehicle Scheduling Problem dynamics and Vehicle-to-Grid capabilities, developing a comprehensive optimization model that addresses both logistical and energy demands within a military microgrid context. Utilizing a column generation approach, we efficiently solve large-scale instances and demonstrate significant improvements in fuel efficiency, renewable energy utilization, and overall operational cost. Computational experiments using realistic demand and solar generation data illustrate that integration of vehicle-to-grid enabled electric vehicles substantially reduces fuel consumption and can generate surplus energy returned to the grid. The results indicate that while battery constraints may require an increased fleet size, strategic scheduling of charging and discharging yields considerable economic and operational benefits. Our findings provide valuable insights for planners aiming to optimize energy use, reduce dependence on traditional fuel sources, and enhance operational resilience in remote environments.

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arXiv2025

How hard is learning to cut? Trade-offs and sample complexity

In the recent years, branch-and-cut algorithms have been the target of data-driven approaches designed to enhance the decision making in different phases of the algorithm such as branching, or the choice of cutting planes (cuts). In particular, for cutting plane selection two score functions have been proposed in the literature to evaluate the quality of a cut: branch-and-cut tree size and gap closed. In this paper, we present new sample complexity lower bounds, valid for both scores. We show that for a wide family of classes $\mathcal{F}$ that maps an instance to a cut, learning over an unknown distribution of the instances to minimize those scores requires at least (up to multiplicative constants) as many samples as learning from the same class function $\mathcal{F}$ any generic target function (using square loss). Our results also extend to the case of learning from a restricted set of cuts, namely those from the Simplex tableau. To the best of our knowledge, these constitute the first lower bounds for the learning-to-cut framework. We compare our bounds to known upper bounds in the case of neural networks and show they are nearly tight. We illustrate our results with a graph neural network selection evaluated on set covering and facility location integer programming models and we empirically show that the gap closed score is an effective proxy to minimize the branch-and-cut tree size. Although the gap closed score has been extensively used in the integer programming literature, this is the first principled analysis discussing both scores at the same time both theoretically and computationally.

math.OC↗

arXiv2025

Chance-constrained battery management strategies for the electric bus scheduling problem

The global transition to battery electric buses (EBs) presents an opportunity to reduce air and noise pollution in urban areas. However, the adoption of EBs introduces challenges related to limited driving range, extended charging times, and battery degradation. This study addresses these challenges by proposing a novel chance-constrained model for the electric vehicle scheduling problem (E-VSP) that accounts for stochastic energy consumption and battery degradation. The model ensures compliance with recommended state-of-charge (SoC) ranges while optimizing operational costs. A tailored branch-and-price heuristic with stochastic pricing problems is developed. Computational experiments on realistic instances demonstrate that the stochastic approach can provide win-win solutions compared to deterministic baselines in terms of operational costs and battery wear. By limiting the probability of operating EBs outside the recommended SoC range, the proposed framework supports fleet management practices that align with battery leasing company and manufacturer guidelines for battery health and longevity.

math.OC↗

arXiv2025

Accelerated windowing for the crew rostering problem with machine learning

The crew rostering problem (CRP) for pilots is a complex crew scheduling task assigning pairings, or sequences of flights starting and ending at the same airport, to pilots to create a monthly schedule. In this paper, we propose an innovative solution method for the CRP that uses a windowing approach. First, using a combination of machine learning (ML) and combinatorial optimisation (CO), we quickly generate an initial solution. The solution is obtained with a sequential assignment procedure (\textit{seqAsg}) based on a neural network trained by an evolutionary algorithm. Then, this initial solution is reoptimized using a branch-and-price algorithm that relies on a windowing scheme to quickly obtain a CRP solution. This windowing method consists of decomposing the optimization horizon into several overlapping windows, and then optimizing each one sequentially. Although windowing has been successfully used in other airline applications, it had never been implemented for the CRP, due to its large number of horizontal constraints involving the whole planning horizon. We test our approach on two large real-world instances, and show that our method is over ten times faster than the state-of-the-art branch-and-price CRP solver GENCOL while providing solutions on average less than 1% away from optimality. We show that our windowing approach greatly benefits from being initialized with good-quality ML-based solutions. This is because the initial solution provides reliable information on the following windows, allowing the solver to better optimize the current one. For this reason, this approach outperforms other naive heuristics, including stand-alone ML or windowing.

math.OC↗

arXiv2025

Learning Large Neighborhood Search for Maritime Inventory Routing Optimization

Maritime inventory routing optimization is an important yet challenging combinatorial optimization problem. We propose a machine learning-based local search approach for finding feasible solutions of large-scale maritime inventory routing optimization problems. Given the combinatorial complexity of the problems, we integrate a graph neural network-based neighborhood selection method to enhance local search efficiency. Our approach enables a structured exploration of different neighborhoods by imitating an optimization-based expert neighborhood selection policy, improving solution quality while maintaining computational efficiency. Through extensive computational experiments on realistic instances, we demonstrate that our method outperforms direct mixed-integer programming as well as benchmark local search approaches in solution time and solution quality.

math.OC↗

arXiv2025

Scalable First-order Method for Certifying Optimal k-Sparse GLMs

This paper investigates the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through an $\ell_0$ cardinality constraint. While branch-and-bound (BnB) frameworks can certify optimality by pruning nodes using dual bounds, existing methods for computing these bounds are either computationally intensive or exhibit slow convergence, limiting their scalability to large-scale problems. To address this challenge, we propose a first-order proximal gradient algorithm designed to solve the perspective relaxation of the problem within a BnB framework. Specifically, we formulate the relaxed problem as a composite optimization problem and demonstrate that the proximal operator of the non-smooth component can be computed exactly in log-linear time complexity, eliminating the need to solve a computationally expensive second-order cone program. Furthermore, we introduce a simple restart strategy that enhances convergence speed while maintaining low per-iteration complexity. Extensive experiments on synthetic and real-world datasets show that our approach significantly accelerates dual bound computations and is highly effective in providing optimality certificates for large-scale problems.

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