arXiv subjects

Marco Locatelli

Publications and source records attributed to Marco Locatelli.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Macroitem Sequences in the Precedence Constrained Knapsack Problem

The Precedence Constrained Knapsack Problem (PCKP) asks for a maximum-profit subset of items, subject to a knapsack capacity constraint and precedence constraints encoded by a directed acyclic graph. We study the structure of optimal solutions of the Linear Programming (LP) relaxation of the natural Integer Linear Programming formulation of the PCKP. We introduce the notion of macroitem and of feasible sequence of macroitems, which partitions the item set while respecting the precedence structure. We establish that an optimal LP solution is fully characterized by the optimal sequence of macroitems: items are packed in nonincreasing order of the profit-to-weight ratio of their macroitem, with at most one macroitem fractionally included. We further show that the breakpoints of the parametric Lagrangian function of the capacity constraint coincide with the profit-to-weight ratios of the macroitems in the optimal sequence, and provide a complete combinatorial characterization of optimal dual solutions in terms of a feasible flow within each macroitem. Finally, for the special case in which the precedence graph is a forest, we devise an O(n^2) algorithm to compute the optimal sequence, which improves to O(n log n) for in-trees or out-trees, where n denotes the number of items.

math.OC↗

arXiv2025

Expert-Guided POMDP Learning for Data-Efficient Modeling in Healthcare

Learning the parameters of Partially Observable Markov Decision Processes (POMDPs) from limited data is a significant challenge. We introduce the Fuzzy MAP EM algorithm, a novel approach that incorporates expert knowledge into the parameter estimation process by enriching the Expectation Maximization (EM) framework with fuzzy pseudo-counts derived from an expert-defined fuzzy model. This integration naturally reformulates the problem as a Maximum A Posteriori (MAP) estimation, effectively guiding learning in environments with limited data. In synthetic medical simulations, our method consistently outperforms the standard EM algorithm under both low-data and high-noise conditions. Furthermore, a case study on Myasthenia Gravis illustrates the ability of the Fuzzy MAP EM algorithm to recover a clinically coherent POMDP, demonstrating its potential as a practical tool for data-efficient modeling in healthcare.

cs.LG↗

arXiv2025

A convex reformulation for speed planning of a vehicle under the travel time and energy consumption objectives

In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted sum of two conflicting objectives, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the problem, we prove that the feasible region of this problem is a lattice. Moreover, we introduce a feasibility-based bound-tightening technique which allows us to derive the minimum and maximum element of the lattice, or establish that the feasible region is empty. We prove the exactness of a convex relaxation of the non-convex problem, obtained by replacing all constraints with the lower and upper bounds for the variables corresponding to the minimum and maximum elements of the lattice, respectively. After proving some properties of optimal solutions of the convex relaxation, we exploit them to develop a dynamic programming approach returning an approximate solution to the convex relaxation, and with time complexity $O(n^2)$, where $n$ is the number of points into which the continuous path is discretized.

math.OC↗

arXiv2025

Hidden convexity property of a speed planning problem

In this paper we address the speed planning problem for a vehicle along a predefined path. A weighted average of two (conflicting) terms, energy consumption and travel time, is minimized. After deriving a non-convex mathematical model of the problem, we introduce a convex relaxation of the model and show that, after the application of a suitable feasibility-based bound tightening procedure, the convex relaxation shares the same optimal value and solution of the non-convex problem. We also establish that the feasible region of the non-convex problem is a lattice and, through that, a necessary and sufficient condition for the non-emptiness of the feasible region.

math.OC↗

arXiv2025

Speed planning by minimizing travel time and energy consumption

In this paper we address the speed planning problem for a vehicle over an assigned path with the aim of minimizing a weighted sum of travel time and energy consumption under suitable constraints (maximum allowed speed, maximum traction or braking force, maximum power consumption). The resulting mathematical model is a non--convex optimization problem. We prove that, under some mild assumptions, a convex reformulation of the non--convex problem is exact. In particular, the convex reformulation is a Second Order Cone Programming (SOCP) problem, for which efficient solvers exist. Through the numerical experiments we confirm that the convex relaxation can be solved very efficiently and, moreover, we also provide the Pareto front of the trade-off between the two objectives (travel time and energy consumption).

math.OC↗

arXiv2024

Dynamic Programming based Local Search approaches for Multi-Agent Path Finding problems on Directed Graphs

Among sub-optimal Multi-Agent Path Finding (MAPF) solvers, rule-based algorithms are particularly appealing since they are complete. Even in crowded scenarios, they allow finding a feasible solution that brings each agent to its target, preventing deadlock situations. However, generally, rule-based algorithms provide much longer solutions than the shortest one. The main contribution of this paper is introducing a new local search procedure for improving a known feasible solution. We start from a feasible sub-optimal solution, and perform a local search in a neighborhood of this solution. If we are able to find a shorter solution, we repeat this procedure until the solution cannot be shortened anymore. At the end, we obtain a solution that is still sub-optimal, but generally of much better quality than the initial one. We propose two different local search policies. In the first, we explore all paths in which the agents positions remain in a neighborhood of the corresponding positions of the reference solution. In the second, we set an upper limit to the number of agents that can change their path with respect to the reference solution. These two different policies can also be alternated. We explore the neighborhoods by dynamic programming. The fact that our search is local is fundamental in terms of time complexity. Indeed, if the dynamic programming approach is applied to the full MAPF problem, the number of explored states grows exponentially with the number of agents. Instead, the introduction of a locality constraint allows exploring the neghborhoods in a time that grows polynomially with respect to the number of agents.

cs.MA↗

arXiv2024

A Dynamic Programming Approach for Road Traffic Estimation

We consider a road network represented by a directed graph. We assume to collect many measurements of traffic flows on all the network arcs, or on a subset of them. We assume that the users are divided into different groups. Each group follows a different path. The flows of all user groups are modeled as a set of independent Poisson processes. Our focus is estimating the paths followed by each user group, and the means of the associated Poisson processes. We present a possible solution based on a Dynamic Programming algorithm. The method relies on the knowledge of high order cumulants. We discuss the theoretical properties of the introduced method. Finally, we present some numerical tests on well-known benchmark networks, using synthetic data.

eess.SY↗

arXiv2024

Enabling pulse shape discrimination with commercial ASICs

Fast electronic readout for high-channel density scintillator-based systems is needed for radiation tracking and imaging in a wide range of applications, including nuclear physics, nuclear security and nonproliferation. Programmable electronics, like FPGAs and ASICs, provide a fast way of conditioning and processing the signal in real time. In this paper, we present a pulse shape discrimination (PSD) method based on the shaping circuit of a commercially available ASIC, the Citiroc1A by CAEN Technologies. We used two different shaping times per detector channel to calculate a shaping parameter that enables PSD. Using our new method, neutron and gamma-ray pulses detected by a d$_{12}$-stilbene scintillator can be effectively discriminated at light output values greater than 0.15 MeVee. While not achieving the PSD performance of traditional offline charge integration, our method does not require the transfer of data to a separate system for further processing and enables the direct deployment of high-channel density multi-particle detection systems. Moreover, the availability of a wider range of shaping times than those on the Citiroc1A can potentially further improve the PSD performance.

physics.ins-det↗

arXiv2024

A Branch and Bound method for the exact parameter identification of the PK/PD model for anesthetic drugs

We address the problem of parameter identification for the standard pharmacokinetic/pharmacodynamic (PK/PD) model for anesthetic drugs. Our main contribution is the development of a global optimization method that guarantees finding the parameters that minimize the one-step ahead prediction error. The method is based on a branch-and-bound algorithm, that can be applied to solve a more general class of nonlinear regression problems. We present some simulation results, based on a dataset of twelve patients. In these simulations, we are always able to identify the exact parameters, despite the non-convexity of the overall identification problem.

eess.SY↗

arXiv2024

Identification of Cyclists' Route Choice Criteria

The behavior of cyclists when choosing the path to follow along a road network is not uniform. Some of them are mostly interested in minimizing the travelled distance, but some others may also take into account other features such as safety of the roads or pollution. Individuating the different groups of users, estimating the numerical consistency of each of these groups, and reporting the weights assigned by each group to different characteristics of the road network, is quite relevant. Indeed, when decision makers need to assign some budget for infrastructural interventions, they need to know the impact of their decisions, and this is strictly related to the way users perceive different features of the road network. In this paper, we propose an optimization approach to detect the weights assigned to different road features by various user groups, leveraging knowledge of the true paths followed by them, accessible, for example, through data collected by bike-sharing services.

math.OC↗

arXiv2023

Is time-optimal speed planning under jerk constraints a convex problem?

We consider the speed planning problem for a vehicle moving along an assigned trajectory, under maximum speed, tangential and lateral acceleration, and jerk constraints. The problem is a nonconvex one, where nonconvexity is due to jerk constraints. We propose a convex relaxation, and we present various theoretical properties. In particular, we show that the relaxation is exact under some assumptions. Also, we rewrite the relaxation as a Second Order Cone Programming (SOCP) problem. This has a relevant practical impact, since solvers for SOCP problems are quite efficient and allows solving large instances within tenths of a second. We performed many numerical tests, and in all of them the relaxation turned out to be exact. For this reason, we conjecture that the convex relaxation is always exact, although we could not give a formal proof of this fact.

math.OC↗

arXiv2023

An algorithm with improved complexity for pebble motion/multi-agent path finding on trees

The pebble motion on trees (PMT) problem consists in finding a feasible sequence of moves that repositions a set of pebbles to assigned target vertices. This problem has been widely studied because, in many cases, the more general Multi-Agent path finding (MAPF) problem on graphs can be reduced to PMT. We propose a simple and easy to implement procedure, which finds solutions of length O(knc + n^2), where n is the number of nodes, $k$ is the number of pebbles, and c the maximum length of corridors in the tree. This complexity result is more detailed than the current best known result O(n^3), which is equal to our result in the worst case, but does not capture the dependency on c and k.

cs.MA↗

arXiv2022

Fix and Bound: An efficient approach for solving large-scale quadratic programming problems with box constraints

In this paper, we propose a branch-and-bound algorithm for solving nonconvex quadratic programming problems with box constraints (BoxQP). Our approach combines existing tools, such as semidefinite programming (SDP) bounds strengthened through valid inequalities, with a new class of optimality-based linear cuts which leads to variable fixing. The most important effect of fixing the value of some variables is the size reduction along the branch-and-bound tree, allowing to compute bounds by solving SDPs of smaller dimension. Extensive computational experiments over large dimensional (up to $n=200$) test instances show that our method is the state-of-the-art solver on large-scale BoxQPs. Furthermore, we test the proposed approach on the class of binary QP problems, where it exhibits competitive performance with state-of-the-art solvers.

math.OC↗

arXiv2022

Constrained Multi-Agent Path Finding on Directed Graphs

We discuss C-MP and C-MAPF, generalizations of the classical Motion Planning (MP) and Multi-Agent Path Finding (MAPF) problems on a directed graph G. Namely, we enforce an upper bound on the number of agents that occupy each member of a family of vertex subsets. For instance, this constraint allows maintaining a safety distance between agents. We prove that finding a feasible solution of C-MP and C-MAPF is NP-hard, and we propose a reduction method to convert them to standard MP and MAPF. This reduction method consists in finding a subset of nodes W and a reduced graph G/W, such that a solution of MAPF on G/W provides a solution of C-MAPF on G. Moreover, we study the problem of finding W of maximum cardinality, which is strongly NP-hard.

cs.MA↗

arXiv2022

Multi-Agent Path Finding on Strongly Connected Digraphs: feasibility and solution algorithms

On an assigned graph, the problem of Multi-Agent Pathfinding (MAPF) consists in finding paths for multiple agents, avoiding collisions. Finding the minimum-length solution is known to be NP-hard, and computation times grows exponentially with the number of agents. However, in industrial applications, it is important to find feasible, suboptimal solutions, in a time that grows polynomially with the number of agents. Such algorithms exist for undirected and biconnected directed graphs. Our main contribution is to generalize these algorithms to the more general case of strongly connected directed graphs. In particular, given a MAPF problem with at least two holes, we present an algorithm that checks the problem feasibility in linear time with respect to the number of nodes, and provides a feasible solution in polynomial time.

cs.MA↗

arXiv2022

Sharp and fast bounds for the Celis-Dennis-Tapia problem

In the Celis-Dennis-Tapia (CDT) problem a quadratic function is minimized over a region defined by two strictly convex quadratic constraints. In this paper we re-derive a necessary and optimality condition for the exactness of the dual Lagrangian bound (equivalent to the Shor relaxation bound in this case). Starting from such condition we propose to strengthen the dual Lagrangian bound by adding one or two linear cuts to the Lagrangian relaxation. Such cuts are obtained from supporting hyperplanes of one of the two constraints. Thus, they are redundant for the original problem but they are not for the Lagrangian relaxation. The computational experiments show that the new bounds are effective and require limited computing times. In particular, one of the proposed bounds is able to solve all but one of the 212 hard instances of the CDT problem presented in [Burer, Anstreicher, 2013].

math.OC↗

arXiv2021

The Bounded Acceleration Shortest Path problem: complexity and solution algorithms

The purpose of this work is to introduce and characterize the Bounded Acceleration Shortest Path (BASP) problem, a generalization of the Shortest Path (SP) problem. This problem is associated to a graph: the nodes represent positions of a mobile vehicle and the arcs are associated to pre-assigned geometric paths that connect these positions. BASP consists in finding the minimum-time path between two nodes. Differently from SP, we require that the vehicle satisfy bounds on maximum and minimum acceleration and speed, that depend on the vehicle position on the currently traveled arc. We prove that BASP is NP-hard and define solution algorithm that achieves polynomial time-complexity under some additional hypotheses on problem data.

cs.DS↗

arXiv2020

Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as CPLEX and GUROBI, are unable to solve to optimality even test instances with n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

math.OC↗