On the Minimum Norm Solution to Weber Problem

This paper investigates an unconstrained form of classical Weber problem. The main idea is to reformulate Weber problem as an unconstrained minimum norm problem. A result based on the representation of the objective function as a Lipschitzian function, which is necessarily a convex function, is proposed. The existence of global solution to such problem is proven using coercivity assumptions.


Introduction
The continuous single-facility location problem can be stated as follows: find the location for a new facility ( ) such that the sum of the weighted distances from X to n existing facility locations ( ) ( ) If we let ( ) i P X d , represent the distance between X and i P and let i w represent the positive weight associated with travel between i P and X, then the problem can be formulated as: Where distance is measured using the Euclidean-distance metric i.e., ( ) ( ) ( ) This problem is sometimes called the Weber problem.The Weber problem (WP) has two very important properties.First, ( ) X f is a convex function which ensures that any local optimum is also a global optimum.Second, the optimal location for the new facility must lie within the convex hull of the existing facility locations.
In recent times, research have been carried out on the minimum norm problems and resolved using several techniques, see ( [1,3,4,5,11]) to mention a few.Such problems have been found useful in approximation theory, statistical estimation problem [11], signal and image reconstruction as well as in other engineering applications [3].
In the [3], the author showed that minimum norm problem can be recast as fixed point problem and showed Tx x = .He further proved the existence and uniqueness of the minimum solution of operator equation exists and is unique, if T is non expansive.The research carried out by [4] was to find the minimum norm solution of a linear programs by a Newton-type method which was shown to be globally convergent.In [1], the equivalency of this type of problem was shown using duality principle.
A recent stride reported in [11] was directed at an estimation problem using simple random sampling technique.The idea was used in formulating the estimation problem as an equivalent minimum norm problem in the Hilbert space and resolved by an appropriate application of the classical projection theorem.
In this paper, we show that a facility location problem otherwise called Weber problem can be recast as a minimum norm problem and resolved as a global solution using coercivity assumptions.

Preliminary and Notation
Let X be a real Banach space with .and * X the dual of X.For a function ℜ → X f : and are the level set and strict level set of f at height λ .The indicator function of the subset A of X is . We conclude the section by introducing some definitions and a theorem (which is proved in [9]) that will be useful subsequently.

Definition 1
A set is said to be weakly closed (w-closed) if it is closed with respect to the topology ( )

Definition 2
A set X K ⊂ is said to be weakly compact (w-compact) if every sequence from K contains a weakly convergent subsequence.

Definition 3
A function f is lower semicontinuous (lsc) at 0 x if for every 0 ∈> there exists a neighbourhood U of 0 for all x in U. Equivalently, this can be expressed as ( ) ( ) A function f is w-lsc if its lower semicontinuous with respect to the topology ( )

Theorem 1 Weierstrass theorem
A lower semicontinuous functional on a compact subset K of a normed linear space X achieves a minimum on K.

Unconstrained Minimization Reformulation
In order to find an unconstrained minimization reformulation for the minimum norm solution for Weber problem (1), we exploit the result of Boyd and Vanderberghe [2] as follows: As an application, we can think of the points as locations of plants or warehouses of a company, and the links as the routes over which goods must be shipped.The goal is to find locations that minimize the total transportation cost.In another application, the points represent wires that connect pairs of cell.Here the goal might be to place the cells in such a way that the total length of materials needed to interconnect the cells is minimized.
In the simplest version of the problem, the cost ij f associated with arc (i,j) is the distance between nodes i and j: i.e., ( ) , such that we minimize ( ) We can use any norm, but the most commonly used is the Euclidean or the l 1 -norm.We can include nonnegative weights that reflect differences in the cost per unit distance along different arcs: By assigning a weight w ij =0 to pairs of nodes that are not connected and w ij =1, otherwise, we can express this problem more simply using the objective Thus, the Weber problem (1) can be reformulated as:

Let ( )
. , X D ⊂ be a non-empty set and consider the distance function

Existence and Uniqueness Theorems
Consider the constrained subject to C l To problem (13), we can associate an unconstrained problem: where we call value of problem ( 13) the extended real An optimal solution of problem ( 13) is an element C x ∈ with the property that ( ) ( ) . We denote by ( ) P S or ( ) C f S , the set of optimal solutions of problem (13).Therefore,

is denoted by f argmin
The most important result which assures the existence of minimum solution for (1) is the famous Weierstrass's theorem.But we may use for the same purposes, some coercivity conditions because the underlying spaces are not compact.
This completes the proof.
. Since f is lower semicontinuous (lsc) and convex, f is weakly lower semicontinuous.The conclusion follows using the Weierstrass theorem applied to the function [ ] Having X u ∈ , an important problem consists of determining the set minimum solution of x by elements of D. with Lipschitz constant 1.

.
weakly closed and X is reflexive, we have that [ ] The conclusion follows from (i).

.
Since f is Lipschitzian, then it is necessarily convex and lower semicontinuous.By theorem 4.1, there existsX u ∈ such that ( ) ( ) u f u f ≤ for every X u ∈ i.e.
It is obvious that f is coercive if and only if all level .e., f is coercive if and only if the level set is bounded. i