arXiv subjects

Ambrogio Maria Bernardelli

Publications and source records attributed to Ambrogio Maria Bernardelli.

5 recordsLinked to original sources

arXiv2026

Comparative Analysis of Linear Battery Models for Carbon Emission Optimization in Solar Energy Systems

This work addresses the problem of minimizing equivalent carbon emissions in residential photovoltaic-battery energy storage systems (PV-BESS) under uncertainty. We develop and compare a hierarchy of linear optimization models that differ in their degree of anticipativity and feedback complexity, ranging from a rule-based self-consumption heuristic to fully stochastic formulations with linear feedback control. The proposed models explicitly incorporate the stochastic variability of household load, solar production, and grid carbon intensity through large scenario sets generated via principal component analysis of real operational data. Computational experiments on synthetic yet realistic scenarios show that direct stochastic optimization of expected emissions (Programmed Battery model) substantially outperforms heuristic control, achieving emission reductions close to the theoretical lower bound provided by the Omniscient Battery benchmark. Feedback-based models marginally improve training performance but do not generalize better on unseen data, while incurring higher computational costs. Overall, results demonstrate that linear stochastic programming provides an effective and tractable framework for emission-aware energy management in distributed PV-BESS systems.

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arXiv2026

The Cloven Traveling Salesman: Cycle Covers and the Integrality Gap of Small ATSP Instances

This work proposes a novel enumeration algorithm for computing the integrality gap of small instances of the subtour elimination formulation for the Asymmetric Traveling Salesman Problem (ATSP).The core idea is to enumerate pairs of cycle covers that can be filtered and mapped to half-integer vertices of the subtour elimination polytope. The two-cycle covers are encoded as lexicographically ordered partitions of $n$ numbers, with an encoding that prevents the generation of several isomorphic vertices. However, since not every cycle cover pair can be mapped to a vertex of the subtour elimination polytope, we have designed an efficient property-checking procedure to control whether a given point is a vertex of the asymmetric subtour elimination polytope. The proposed approach turns upside down the algorithms presented in the literature that first generate every possible vertex and later filter isomorphic vertices. With our approach, we can replicate state-of-the-art results for n<=9 in a tiny fraction of time, and we compute for the first time the exact integrality gap of half-integer vertices of the asymmetric subtour elimination polytope for n=10, 11, 12.

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arXiv2026

Theoretical Perspectives on Jabr-Type Convex Relaxations for AC Optimal Power Flow

The alternating current optimal power flow problem is a fundamental yet highly nonconvex optimization problem whose structure reflects both nonlinear power flow physics and the topology of the underlying network. Among convex relaxations, the second-order cone relaxation introduced by Jabr has proven particularly influential, serving as a computationally efficient alternative to semidefinite relaxations and a foundation for numerous strengthening techniques. In recent years, a variety of approaches have been proposed to tighten Jabr-type relaxations, including cycle-based constraints, convex envelopes of multilinear terms, and dual reformulations. However, these developments are often presented independently, concealing their common geometric and graph-theoretic foundations. This paper provides a structured review of strengthening techniques for the Jabr relaxation and develops a unifying perspective based on multilinear equalities. We reinterpret cycle constraints as multilinear consistency conditions, analyze their convexification through classical convex hull theory, and investigate the relationship between primal McCormick relaxations and dual extended formulations. In particular, we identify structural conditions under which these relaxations coincide and clarify the distinction between convexifying the interaction graph and convexifying the feasible set of the ACOPF. The resulting framework connects graph structure, multilinear convexification, and conic relaxations in a unified manner, offering both a conceptual synthesis of existing results and new insights for the design of stronger relaxations.

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arXiv2025

Lower bounds for the integrality gap of the bi-directed cut formulation of the Steiner Tree Problem

In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (BCR) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the BCR formulation on metric instances. A key contribution of our work is extending the Gap problem, previously explored in the context of the Traveling Salesman problems, to the metric Steiner Tree problem. To tackle the Gap problem for Steiner Tree instances, we first establish several structural properties of the CM formulation. We then classify the isomorphism classes of the vertices within the CM polytope, revealing a correspondence between the vertices of the BCR and CM polytopes. Computationally, we exploit these structural properties to design two complementary heuristics for finding nontrivial small metric Steiner instances with a large integrality gap. We present several vertices for graphs with a number of nodes <=10, which realize the best-known lower bounds on the integrality gap for the CM and the BCR formulations. We conclude the paper by presenting two new conjectures on the integrality gap of the BCR and CM formulations for small graphs.

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arXiv2024

Multi-Objective Linear Ensembles for Robust and Sparse Training of Few-Bit Neural Networks

Training neural networks (NNs) using combinatorial optimization solvers has gained attention in recent years. In low-data settings, state-of-the-art mixed integer linear programming solvers can train exactly a NN, avoiding intensive GPU-based training and hyper-parameter tuning and simultaneously training and sparsifying the network. We study the case of few-bit discrete-valued neural networks, both Binarized Neural Networks (BNNs), whose values are restricted to +-1, and Integer Neural Networks (INNs), whose values lie in a range {-P, ..., P}. Few-bit NNs receive increasing recognition due to their lightweight architecture and ability to run on low-power devices. This paper proposes new methods to improve the training of BNNs and INNs. Our contribution is a multi-objective ensemble approach based on training a single NN for each possible pair of classes and applying a majority voting scheme to predict the final output. Our approach results in training robust sparsified networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible. We compare this BeMi approach to the current state-of-the-art in solver-based NN training and gradient-based training, focusing on BNN learning in few-shot contexts. We compare the benefits and drawbacks of INNs versus BNNs, bringing new light to the distribution of weights over the {-P, ..., P} interval. Finally, we compare multi-objective versus single-objective training of INNs, showing that robustness and network simplicity can be acquired simultaneously, thus obtaining better test performances. While the previous state-of-the-art approaches achieve an average accuracy of 51.1% on the MNIST dataset, the BeMi ensemble approach achieves an average accuracy of 68.4% when trained with 10 images per class and 81.8% when trained with 40 images per class, having up to 75.3% NN links removed.

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