Fermat Weber Location Problem Research Articles

AoI-Minimal Task Assignment and Trajectory Optimization in Multi-UAV-Assisted IoT Networks

This article investigates an unmanned-aerial-vehicle (UAV)-assisted Internet of Things system, where multiple UAVs take off from a data center (DC), collect data from multiple ground sensor nodes (SNs), distribute data for multiple users, and finally return to DC. A centralized information-sharing mechanism is designed among UAVs before data distribution, such that UAVs’ repeatedly visiting the same user can be avoided to save energy and time. A nonconvex Age-of-Information (AoI) minimization problem is formulated, where the task assignment, selection of interaction point (IPT), and UAVs’ trajectories are optimized jointly. It is decomposed into two subproblems: one is a joint task assignment and UAV trajectory optimization problem, which is solved by a multipopulation-based genetic algorithm (MPGA), as well as a hybrid algorithm combining dynamic programming (DP) with an improved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -means clustering; and the other is an IPT selection problem, which is modeled as a Fermat–Weber location problem and is solved by the convex optimization technique. Simulation results show that the MPGA performs better than the hybrid algorithm as well as several benchmark algorithms.

New Convergence Results for Inertial Krasnoselskii–Mann Iterations in Hilbert Spaces with Applications

We consider inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. To do these, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the inertial Krasnoselskii–Mann iteration for fixed point of nonexpansive operators in infinite dimensional real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. One of our contributions is that the convergence analysis and rate of convergence results are obtained using conditions which appear not complicated and restrictive as assumed in other previous related results in the literature. We then show that Fermat–Weber location problem and primal–dual three-operator splitting are special cases of fixed point problem of nonexpansive mapping and consequently obtain the convergence analysis of inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. Some numerical implementations are drawn from primal–dual three-operator splitting to support the theoretical analysis.

Pointing Consensus and Bearing-Based Solutions to the Fermat–Weber Location Problem

The pointing consensus problem asks each agent in a multiagent system to agree on their headings toward a common target. This paper proposes a decentralized approach to the pointing consensus problem by simultaneously solving three smaller problems: bearing-only measurement based network localization, target decision, and heading coordination. The proposed solution guarantees that all agents’ headings almost globally asymptotically target any weighted centroid of the agents’ positions. Furthermore, based on this approach, two decentralized solutions for the Fermat–Weber location problem are proposed and analyzed. Simulation results are also provided to support the analysis.

A coevolution model of population distribution and road networks

This paper proposes a coevolution model of population distribution and road networks. In the model, new roads are built with the formation of new communities, and these are connected to existing zones with higher population density with higher probability. The relative neighborhood graph and a Fermat–Weber location problem are introduced as the connection mechanism to capture the characteristics of population distribution and road networks. Moreover, two kinds of economic factors, accessibility and housing price, are considered to reflect the impacts on the coevolution between population distribution and road networks. Simulations and empirical experiments are conducted to demonstrate the interaction between population distribution and road networks. The topological attributes of the road network are characterized in the experiments using degree distribution, betweenness centrality, coverage, circuitness, and treeness. The experimental results show that the proposed model has reflect on the simulation.

Population-driven Urban Road Evolution Dynamic Model

In this paper, we propose a road evolution model by considering the interaction between population distribution and urban road network. In the model, new roads need to be constructed when new zones are built, and existing zones with higher population density have higher probability to connect with new roads. The relative neighborhood graph and a Fermat-Weber location problem are introduced as the connection mechanism to capture the characteristics of road evolution. The simulation experiment is conducted to demonstrate the effects of population on road evolution. Moreover, the topological attributes for the urban road network are evaluated using degree distribution, betweenness centrality, coverage, circuitness and treeness in the experiment. Simulation results show that the distribution of population in the city has a significant influence on the shape of road network, leading to a growing heterogeneous topology.

The Fermat–Weber location problem in single integrator dynamics using only local bearing angles

This paper proposes a modified gradient control law to solve the Fermat–Weber location problem. The control law may guide an autonomous agent to the point that minimizes the weighted sum distance with regard to some stationary beacons using only bearing measurements. Based on previous results from Fermat–Weber location problem and Lyapunov stability theory, we prove that the agent will globally asymptotically reach the destination only using local bearing measurements. We also consider a scenario when the beacons are not stationary. Simulation results are provided to verify the mathematical analysis.

Revisiting several problems and algorithms in continuous location with $$\ell _\tau $$ ℓ τ norms

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different $$\ell _\tau $$ l ? norms, $$\tau \ge 1$$ ? ? 1 , in the demand points. We analyze the difficulty of this family of problems and revisit convergence properties of some well-known algorithms. The ultimate goal is to provide a common approach to solve the family of continuous $$\ell _\tau $$ l ? ordered median location problems Nickel and Puerto (Facility location: a unified approach, 2005) in dimension $$d$$ d (including of course the $$\ell _\tau $$ l ? minisum or Fermat-Weber location problem for any $$\tau \ge 1$$ ? ? 1 ). We prove that this approach has a polynomial worst case complexity for monotone lambda weights and can be also applied to constrained and even non-convex problems.

Accelerating the convergence in the single-source and multi-source Weber problems

The modified Weiszfeld method [Y. Vardi, C.H. Zhang, A modified Weiszfeld algorithm for the Fermat–Weber location problem, Mathematical Programming 90 (2001) 559–566] is perhaps the most widely-used algorithm for the single-source Weber problem (SWP). In this paper, in order to accelerate the efficiency for solving SWP, a new numerical method, called Weiszfeld–Newton method, is developed by combining the modified Weiszfeld method with the well-known Newton method. Global convergence of the new Weiszfeld–Newton method is proved under mild assumptions. For the multi-source Weber problem (MWP), a new location-allocation heuristic, Cooper–Weiszfeld–Newton method, is presented in the spirit of Cooper algorithm [L. Cooper, Heuristic methods for location-allocation problems, SIAM Review 6 (1964) 37–53], using the new Weiszfeld–Newton method in the location phase to locate facilities and adopting the nearest center reclassification algorithm (NCRA) in the allocation phase to allocate the customers. Preliminary numerical results are reported to verify the evident effectiveness of Weiszfeld–Newton method for SWP and Cooper–Weiszfeld–Newton method for MWP.

Contour approximation of data: A duality theory

Given a dataset D partitioned in clusters, the joint distance function (JDF) J ( x ) at any point x is the harmonic mean of the distances between x and the cluster centers. The JDF is a continuous function, capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. In particular, contour approximation allows a compact representation of the data: for a dataset in R n with N points organized in K clusters, the JDF requires K centers and covariances (if Mahalanobis distances are used), for a total of Kn ( n + 3 ) / 2 parameters, and a considerable reduction of storage if N ≫ K , n . The JDF of the whole dataset, J ( D ) ≔ ∑ { J ( x ) : x ∈ D } , is a measure of the classifiability of the data, and can be used to determine the “right” number of clusters for D . A duality theory for the JDF J ( D ) is given, in analogy with Kuhn’s geometric duality theory for the Fermat–Weber location problem. The JDF J ( D ) is the optimal value of a primal problem (P), for which a dual problem (D) is given, with a sharp lower bound on J ( D ) .

On the convergence of the Weiszfeld algorithm

In this work we analyze the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited. Mathematical Programming 71, 71–76” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm. Some counterexamples are shown to the proofs showed in Brimberg’s paper.

A modified Weiszfeld algorithm for the Fermat-Weber location problem

This paper gives a new, simple, monotonically convergent, algorithm for the Fermat-Weber location problem, with extensions covering more general cost functions.

Accelerating convergence in the Fermat–Weber location problem

This paper presents a simple procedure for accelerating convergence in a generalized Fermat–Weber problem with l p distances. The main idea is to multiply the predetermined step size of the Weiszfeld algorithm by a factor which is a function of the parameter p. The form of this function is derived from the local convergence properties of the iterative sequence. Computational results are obtained which demonstrate that the total number of iterations to meet a given stopping criterion will be reduced substantially by the new step size, with the most dramatic results being observed for values of p close to 1.

The Fermat—Weber location problem revisited

The Fermat—Weber location problem requires finding a point in ℝN that minimizes the sum of weighted Euclidean distances tom given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld's algorithm was proven in a seminal paper by Kuhn in 1973. However, since them given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever them given points are not collinear, Weiszfeld's algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of ℝN. We resolve this open question by proving that Weiszfeld's algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimensionN.

ON A NEW PROOF OF DURIER'S THEOREM

Abstract A new proof of Durier's Theorem [3] concerning a characterization of Hilbert spaces in terms of the Fermat-Weber location problem will be presented.

A polynomial time algorithm for solving the fermat-weber location problem with mixed norms

This paper discusses the Fermat-Weber location problem, manages to apply the ellipsoid method to this problem and proves the ellipsoid method can be terminated at an approximately optimal location in polynomial time, verifies the ellipsoid method is robust for the lower dimensional location problem

Algebraic optimization: The Fermat-Weber location problem

The Fermat-Weber location problem is to find a point in ℝ n that minimizes the sum of the (weighted) Euclidean distances fromm given points in ℝ n . In this work we discuss some relevant complexity and algorithmic issues. First, using Tarski's theory on solvability over real closed fields we argue that there is an infinite scheme to solve the problem, where the rate of convergence is equal to the rate of the best method to locate a real algebraic root of a one-dimensional polynomial. Secondly, we exhibit an explicit solution to the strong separation problem associated with the Fermat-Weber model. This separation result shows that ane-approximation solution can be constructed in polynomial time using the standard Ellipsoid Method.

Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem

The Fermat—Weber location problem is to find a point in ℝ n that minimizes the sum of the weighted Euclidean distances fromm given points in ℝ n . A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.

A primal-dual algorithm for the fermat-weber problem involving mixed gauges

We give a new algorithm for solving the Fermat-Weber location problem involving mixed gauges. This algorithm, which is derived from the partial inverse method developed by J.E. Spingarn, simultaneously generates two sequences globally converging to a primal and a dual solution respectively. In addition, the updating formulae are very simple; a stopping rule can be defined though the method is not dual feasible and the entire set of optimal locations can be obtained from the dual solution by making use of optimality conditions. When polyhedral gauges are used, we show that the algorithm terminates in a finite number of steps, provided that the set of optimal locations has nonepty interior and a counterexample to finite termination is given in a case where this property is violated. Finally, numerical results are reported and we discuss possible extensions of these results.