arXiv subjects

Luca Consolini

Publications and source records attributed to Luca Consolini.

At least 19 recordsLinked to original sources

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.

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arXiv2025

System Realizations by Mammillary Models with an Application to Propofol Pharmacokinetics

This work addresses the problem of linear system realizations by mammillary models, offering necessary and sufficient conditions under which a given transfer function can be represented in this form. While standard identifi cation techniques may yield transfer functions without an explicit connection to underlying physiological processes, com partmental models, particularly mammillary ones, reflect the physiological dynamics of distribution and elimination. This feature is especially relevant in clinical pharmacology, where model parameters must correspond to meaningful biological processes to support interpretability, personalization, and safe drug delivery, such as in total intravenous anesthesia. To conclude, an application to a propofol infusion model illustrates how mammillary realizations can support physiologically inter pretable system representations.

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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.

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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).

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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.

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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.

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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.

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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.

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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

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.

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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.

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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.

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arXiv2021

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].

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arXiv2020

Structured identification for network reconstruction of RC-models

Resistive-capacitive (RC) networks are used to model various processes in engineering, physics or biology. We consider the problem of recovering the network connection structure from measured input-output data. We address this problem as a structured identification one, that is, we assume to have a state-space model of the system (identified with standard techniques, such as subspace methods) and find a coordinate transformation that puts the identified system in a form that reveals the nodes connection structure. We characterize the solution set, that is, the set of all possible RC-networks that can be associated to the input-output data. We present a possible solution algorithm and show some computational experiments.

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arXiv2019

A Second-Order Lower Bound for Globally Optimal 2D Registration

The problem of planar registration consists in finding the transformation that better aligns two point sets. In our setting, the search domain is the set of planar rigid transformations and the objective function is the sum of the distances between each point of the transformed source set and the destination set. We consider a Branch and Bound (BnB) method for finding the globally optimal solution. The algorithm recursively splits the search domain into boxes and computes an upper and a lower bound for the minimum value of the restricted problem. The main contribution of this work is the introduction of a novel lower bound, the relaxation bound, which corresponds to the solution of a concave relaxation of the objective function based on the linearization of the distance. In the BnB we also employ the so called cheap bound, equal to to the sum of the minimum distances between each point of source point set, transformed according to current box, and all the candidate points in the destination point set. We prove, both theoretically and practically, that the novel relaxation bound dominates the cheap bound over small boxes. More precisely, from the theoretical point of view, we prove that the relaxation bound is a second-order approximation of the minimum value, i.e., its distance from the minimum value decreases quadratically with respect to the diameter of the box (see Theorem 1), while the cheap bound is a first-order one (see Proposition 3). From the practical point of view, we show through different computational experiments that the addition of the relaxation bound considerably enhances the performance of the BnB algorithm, compensating the higher cost of its computation with respect to the cheap bound with a strong reduction of the number of BnB nodes to be explored.

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arXiv2018

Graph-based algorithms for the efficient solution of a class of optimization problems

In this paper, we address a class of specially structured problems that include speed planning, for mobile robots and robotic manipulators, and dynamic programming. We develop two new numerical procedures, that apply to the general case and to the linear subcase. With numerical experiments, we show that the proposed algorithms outperform generic commercial solvers.

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arXiv2018

A solution of the minimum-time velocity planning problem based on lattice theory

For a vehicle on an assigned path, we find the minimum-time speed law that satisfies kinematic and dynamic constraints, related to maximum speed and maximum tangential and transversal acceleration. We present a necessary and sufficient condition for the feasibility of the problem and a simple operator, based on the solution of two ordinary differential equations, which computes the optimal solution. Theoretically, we show that the problem feasible set, if not empty, is a lattice, whose supremum element corresponds to the optimal solution.

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