Accelerating convergence in minisum location problem with ${\ell }_p$ norms

Abstract

This paper presents a procedure for accelerating convergence of the Weiszfeld algorithm in the classical single facility location median problem in which the distances are measured by ${\ell }_p\mathrm{-}\mathrm{norms}$. To this end, we combined Steffensen's method, a generic acceleration scheme applied to iterative processes for solving fixed point equations, with the acceleration methods based on the transformation of the Weiszfeld algorithm by a factor which is a function of the parameter p. The convergence of the proposed methodology and the conditions under which it is guaranteed are analyzed. The computational results show that the total number of iterations to meet a given stopping criterion will be reduced with respect to the results obtained in other algorithms proposed in the literature. The running times are either reduced or quite similar with respect to the existing algorithms for which no results of convergence are provided.

Introduction

In the classical single facility minisum location problem, we are given a finite set of points in a real normed space in order to minimize the weighted sum of the distances to these points (existing facilities or demand points). In this case, the use of ${\ell }_p\mathrm{-}\mathrm{norms}$ to measure distances has been proved to be particularly suitable (see [8], [11], [7]). Hence, given a set of demand points $A=\{a_1,\dots ,a_M\}\subseteq {\mathbb{R}}^n$, the formulation of the single facility minisum location problem in ${\mathbb{R}}^n$ with a ${\ell }_p\mathrm{-}\mathrm{norm}$ to measure distances is stated as follows:

$$\underset{x\in {\mathbb{R}}^n}{\min }F(x)≔\sum _{i=1}^M{w}_i\parallel x-a_i{\parallel }_p,$$

where $w_i\ge 0$ is the weight associated with the demand point a_i for $i=1,\dots ,M$ and

$$\parallel x-a_i{\parallel }_p={\left(\sum _{k=1}^n\right|x_k-a_{\mathit{ik}}{|}^p)}^{1/p}p\ge 1.$$

The restriction on the parameter p to a value greater than or equal to unity ensures that $\parallel ·{\parallel }_p$ has the properties of a norm. In a practical setting, the factor w_i would be proportional to the demand at customer i and the term $w_i\parallel x-a_i{\parallel }_p$ would be an estimate of the cost of serving the demand at customer i by the facility x. Thus, the objective function in (1) provides the total service (or distribution) cost. Regarding the existence and uniqueness of a solution for Problem (1), by the convexity of the objective function, we have that this problem has an optimal solution. Moreover, for $p>1$ the objective function is strictly convex when the demand points are not collinear (see [27]). Thus, in the following, we will assume that demand points are not collinear.

Problem (1) for $p\ge 1$ is a classical model in continuous location theory that has received much attention in the literature. The most popular method for solving this problem is given by a one-point iterative procedure that was first proposed for Euclidean distances (p=2) by Weiszfeld [53] and subsequently translated into English by Weiszfeld and Plastria [54]. This method was rediscovered by Miehle [34] and Cooper [19] approximately 20 years later. Its convergence was subsequently studied by Katz [30] and Kuhn [31], among others. The generalization to ${\ell }_p$ distances with $p\in [1,2]$ is given in Love et al. [33] and an analysis of its local convergence has been shown in Love et al. [33] and Brimberg and Love [9].

The global convergence result of this algorithm for ${\ell }_p$ norms was proved by Brimberg and Love [10] for $p\in [1,2]$. These results were extended to more general functions in Frenk et al. [27], in which the authors used a hyperbolic approximation of the ${\ell }_p$ norm in order to avoid the problem of singularities in the iteration function (for more details, see [4], [5], [15], [14], [31]).

Furthermore, these results have been extended to more general problems, including Banach spaces [24], [43], [44], the sphere [55], regional demand [17], [50], sets as demand facilities using closest Euclidean distances [12], radial distances [16], [20], [29], [35] and using different objective functions [27], [25], [46], [51], among others.

For the case $p>2$, Brimberg and Love [10] provided an example showing that, in general, the classical Weiszfeld algorithm does not converge. The literature contains several attempts to propose convergent algorithms by modifying the step size of the classical Weiszfeld algorithm, basically consisting of multiplying this step size by a factor, usually called the step size factor (see Section 2 for more details). Brimberg et al. [6] prove the local convergence of the algorithm on the plane for $p\ge 1$ when it is used as a step size factor $\lambda$ with $\lambda \le 1/(p-1)$, $p>1$. Uster and Love [49] show that the step-size factors $\lambda =2/p$ for $p\in [2,3]$ and $\lambda =2/(p-1)$ for $p>3$ yield good computational results although they do not prove the convergence of their proposal. Recently, Rodríguez-Chía and Valero-Franco [45] give a step-size factor for this algorithm that depends on each iterate and they provide a proof of the global convergence of this algorithm for $p>2$.

On the other hand, the solution of the single facility minisum location problem has been used as part of a solution procedure for many location problems. These procedures require the repeated application of the Weiszfeld algorithm, which is quite efficient in most cases. However, there are some cases in which convergence is very slow and may require thousands of iterations. Therefore, reducing the computational effort required for the solution of the single facility minisum location problem will improve the performance of these procedures. Actually, when a sequence or an iterative process is slowly converging, a convergence acceleration process must be used. This consists of transforming the slowly converging sequence into a new one that, under some assumptions, converges faster to the same limit. Since the Weiszfeld algorithm is an iterative procedure that is used to find the solution of a fixed point equation, the use of acceleration techniques for such fixed point algorithms may be a suitable option to improve its efficiency, see Brezinski [3], Burden et al. [13], Drezner [21] and Farnum [26].

The notion of accelerating the convergence of descent methods such as the Weiszfeld procedure requires an investigation of alternate step sizes [18]. A first attempt at accelerating the Weiszfeld procedure can be attributed to Katz [30], who suggested the use of Steffensen's iteration applied to each component for the case p=2. This method is not known to be globally convergent, but it may be used to accelerate the local convergence rate of the Weiszfeld procedure from linear to quadratic. However, the gain in the convergence rate may be offset by the longer computational time of Steffensen's iterations. Drezner [22] applies a factor, $\lambda$, to multiply the step size of the Weiszfeld procedure for the case of Euclidean distances $(p=2)$. In the Euclidean case, it is known that convergence will occur for a step size not exceeding double the original (see [42]). Drezner [22] uses an approximation function of F(x) to calculate a $\lambda$ at each iteration which may take on values considerably greater than 2. The computational results generally appear to indicate only a modest reduction in the number of iterations using this acceleration method as compared with a fixed $\lambda$ of 2. The net effect on running times is unclear, due to the increased computations associated with the variable step size factor. In addition, Drezner [22] claims that $\lambda =1.8$ is a good compromise for a fixed step size in handling different problems for p=2. Frenk et al. [27] recommend the use of a step size factor between 0 and 2 for a generalized problem, after showing that the descent property and convergence are still maintained for all $p\in [1,2]$. In Brimberg et al. [6], in addition to prove the local convergence of the iterative sequence with $\lambda =1/(p-1)$ for $p>1$, different step sizes factors that provided good computational results are suggested, but no convergence results are proved. An interesting outcome of this analysis is that the parameter p decreases to a value of 1 and $\lambda$ may be increased to $+\infty$ in the limit with local convergence of the sequence. Verkhovsky and Polyakov [52] proposed an accelerated algorithm to solve Problem (1) for p=2 using a feedback factor that can easily be implemented in engineering practices.

Similarly to Katz [30], Drezner [23] deals with a general acceleration method for the Weiszfeld procedure reducing its computational requirements for p=2. This acceleration method applied Aitken's ${\Delta }^2$ process to each component, also known as Steffensen's method when applied to fixed point problems, see Burden et al. [13] for further details.

Under some conditions, this method for the scalar case accelerates the convergence of any linearly convergent sequence. The approach suggested by Drezner [23] and Katz [30] for p=2 applies a different step size factor for the x-coordinate and the y-coordinate. This means that while the other methods extend the sequence of points in the same direction, this approach may be able to follow a converging sequence which forms a curved trajectory. As mentioned above, Drezner [23] and Katz [30] apply Aitken's ${\Delta }^2$ process to each component. However, Nievergelt [36] warned that Aitken's acceleration may fail if applied separately to each coordinate; therefore a more suitable generalization seems necessary.

The natural generalization to the vector case in the well-known Aitken's ${\Delta }^2$ process is called Henrici's transformation. This transformation is given by Sadok [47] that it is also called Steffensen's method for fixed point equations in several variables. Nievergelt [36] points out that Steffensen's acceleration of the iterations of a function of several variables lacked a theoretical foundation. Nievergelt [37] furnishes a partial result and Noda [38], [39], [40], [41] obtains several local convergence results for functions that are contracting maps in a neighborhood of their fixed point. Bellalij [1] proves a convergence acceleration result in the linear case under some natural conditions. Following the ideas presented by Jbilou and Sadok [28] concerning the vector extrapolation method, Le Ferrand [32] establishes the quadratic convergence of Steffensen's iteration. Nievergelt [36] extends this result to divergent sequences that must behave linearly by analyzing Henrici's transformation. Benchiboun [2] provides a convergence acceleration result that generalizes some results given by Nievergelt [36] and Sadok [47].

In this paper, we study the convergence of an acceleration method, based on the vectorial version of Steffensen's method, applied to different modifications of the Weiszfeld algorithms with a ${\ell }_p\mathrm{-}\mathrm{norm}$ with $p\ge 1$. Note that this paper is the first attempt in the literature to apply the vectorial version of Steffensen's method to the Weiszfeld algorithm, since in previous papers, an independent analysis was considered for each component. For the purposes of this analysis, we also assume that the optimal solution of Problem (1) occurs at non-singular points of the iteration functions. Alternatively, the singularities of the iteration functions may be avoided by introducing a smoothing function such as the hyperbolic approximation of the ${\ell }_p$ norm, see Love et al. [33] and Frenk et al. [27]. For a detailed discussion of the singularities see Brimberg and Love [9], [10].

The paper is organized in six sections. Section 2 provides a general formulation of a modification of the Weiszfeld algorithm by considering a step size factor. In Section 3, Steffensen's acceleration process is described. The conditions that guarantee the convergence of the proposed methodology are studied in Section 4. The computational analysis presented in Section 5 which illustrates the efficiency of the acceleration process applied to each of the modified Weiszfeld algorithms. The paper ends with some concluding remarks.