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The Topic is Single Facility Euclidean Location Problem

Written Assignment Summary (in journal-style format):

Your assignment is to write a research paper using Microsoft Word with a minimum of 1500

words and a maximum of 2500 words (not including the References section) on the topic. It

must be word-processed, single-spaced and fully justified. Also, you must have a margin size

of 1 inch on all sides (i.e., where the top, bottom, left and right sides are even) of your submitted

assignment. Font size for text must be 12 pt and the font type should be Times New Roman

throughout this assignment unless otherwise specified below.

Your paper will address different topics/algorithms relevant to the field of Operations Research.

The title should be in 14 pt bold-type font and centered, and your name should appear beneath

the title (not in bold, but in 12 pt size font). Next, the Abstract section should summarize what

your research paper is about, followed by a line called Keywords where you list in horizontal

format (i.e., across the page as in many journal articles) 3-4 words/phrases that are significant to

your topic.

In addition, each research paper should have a minimum of the following headings:

Introduction Introduce the topic. Literature Review Research and review the literature

(When was it created? Who made it?, What are other researchers doing with it?, etc.)

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Examples of Algorithm/Heuristic Usage Discuss concrete examples of how it is used today in

the world.

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APA will be accepted

These headings should be in bold-type font with 12 pt size. You must also use a minimum

of 3 refereed journal articles (<=10 years old). Additional sources may be used as well and
all must be included in the References section.
Applied Mathematics and Computation 167 (2005) 716728
www.elsevier.com/locate/amc
An e?cient algorithm for the Euclidean
r-centrum location problemq
Shaohua Pan
a
a,*
, Xingsi Li
b
Department of Applied Mathematics, South China University of Technology,
Guangzhou 510641, China
b
State Key Laboratory of Structural Analysis of Industrial Equipment,
Dalian University of Technology, Dalian 116024, China
Abstract
In this paper we consider the single-facility Euclidean r-centrum location problem in
Rn , which generalizes and uni?es the classical 1-center and 1-median problem. Speci?cally, we reformulate this problem as a nonsmooth optimization problem only involving
the maximum function, and then develop a smoothing algorithm that is shown to be
globally convergent. The method transforms the original nonsmooth problem with certain combinatorial property into the solution of a deterministic smooth unconstrained
optimization problem. Numerical results are presented for some problems generated
randomly, indicating that the algorithm proposed here is extremely e?cient for large
problems.
Ó 2004 Elsevier Inc. All rights reserved.
Keywords: Euclidean r-centrum location problem; Sum of the r-largest functions; Nonsmooth;
Smoothing method
q
This work is supported by Natural Science Foundation of South China University of
Technology.
*
Corresponding author.
E-mail addresses: shhpan@scut.edu.cn (S. Pan), lixs@dlut.edu.cn (X. Li).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2004.06.122
S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728
717
1. Introduction
For a given r 2 {1, 2, . . . , m}, the single-facility Euclidean r-centrum location
problem in Rn concerns locating a new facility so as to minimize the sum of the
r largest weighted Euclidean distances to the existing m facilities. Let ti 2 Rn
and x 2 Rn respectively denote the known position of the ith facility and the
unknown position of the new facility, and de?ne
q?????????????????????????????????????????????????????????????????????????????????
2
2
2
gi ðxÞ ¼ xi ðx1 ti1 Þ þ ðx2 ti2 Þ þ þ ðxn tin Þ ; i ¼ 1; 2; . . . ; m
to be the weighted Euclidean distance between the new facility and the ith existing facility, where xi > 0 be the weight associated with ti. Then, this problem

can be formulated as the minimization of the sum of the r largest functions

m

of the collection fgi ðxÞgi¼1 :

minn Ur ðxÞ :¼

x2R

r

X

g½l ðxÞ;

ð1Þ

l¼1

where g[1](x), g[2](x), . . . , g[m](x) are the functions obtained through sorting the

m components of collection fgi ðxÞgmi¼1 in nonincreasing order.

At present, a host of models are available for continuous facility location

problems,

but most of them focus on the minimization of the total distance

Pm

i¼1 gi ðxÞ (the median concept) or the maximum distance g[1](x) (the center

concept); see [1,2]. The single-facility r-centrum location problem (1) to study

here does not only unify the classical Euclidean 1-center and 1-median problems, but also generalizes them from a view of solution concept. It is not

di?cult to see that the case r = 1 coincides with the weighted Euclidean 1-center problem and the case r = m de?nes the weighted Euclidean 1-median problem. To the best of our knowledge, the concept of r-centrum was ?rst

de?ned by Slater [3] and Andreatta and Mason [4,5] for the discrete singlefacility location problem on tree graph, and later was extended to general location problems covering discrete as well as continuous decisions in [6].

The solution of several special cases for r-centrum location problems has

been intensively studied. For example, for the Euclidean median problem in

Rn , many practical and e?cient algorithms have been designed since Weiszfeld

[7] presented a simple iterative algorithm in 1937; see the reference listed in

[8,9]. However, the solution methods for general continuous r-centrum problems are rarely seen in the literature. Recently, Ogryczak et al. [10] reformulated the single-facility rectilinear r-centrum problem as a linear program

through replacing each nonlinear constraint composed of a rectilinear distance

function by a set of 2m linear constraints, and then gave a polynomial time

algorithm with better complexity bound than those reported by [1113] for network location problems.

718

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

In this work, we intend to design an e?cient algorithm for the single-facility

Euclidean r-centrum problem (1). Note that this problem is a typical nonsmooth optimization problem due to the nondi?erentiability of Ur(x), so gradient-based algorithms cannot be directly used to solve it. To circumvent the

di?culty, we reduce this problem to an unconstrained nonsmooth problem

only involving the maximum function max{0, t}, and develop an e?cient

smoothing algorithm by means of the smooth approximation proposed by

[14,15]. This algorithm yields an optimal solution to the original problem via

a single smooth unconstrained minimization. Numerical results show that it

is extremely promising in solving large-scale problems.

The outline of this paper is as follows. Section 2 reformulates problem (1) as

an equivalent nonsmooth problem involving the maximum function, and then

develops a smoothing algorithm. The global convergence of algorithm is established in Section 3. Section 4 reports numerical results for some problems generated randomly. Finally, some conclusions are drawn.

2. A smoothing algorithm for nonsmooth problem (1)

To reduce problem (1) with certain combinatorial property to a deterministic smooth optimization problem, we rede?ne the sum Ur(x) of the r largest

functions by the maximum function. To this end, we ?rst show that

(

)

m

m

X

X

Ur ðxÞ ¼ max

zi gi ðxÞ :

zi ¼ r; 0 6 zi 6 1; i ¼ 1; 2; . . . ; m : ð2Þ

m

z2R

i¼1

i¼1

Pm

De?ne the set D :¼ fz 2 Rm j P

zi 6 1; i ¼ 1; 2; . . . ; mg. Then, for

i¼1 zi ¼ r; 0 6 P

any z 2 D, there always holds mi¼1 zi gi ðxÞ 6 rl¼1 g½l ðxÞ, which means

(

)

m

X

max

zi gi ðxÞ 6 Ur ðxÞ:

z2D

i¼1

On the other side, let ~zi ¼ 1 if gi(x) belongs to the r largest functions of

m

fg

set ~zi ¼ 0, then we have ~z ¼ ð~z1 ; . . . ; ~zm ÞT 2 D and

PimðxÞgi¼1 , andPotherwise

r

zi gi ðxÞ ¼ l¼1 g½l ðxÞ. This implies

i¼1~

(

)

m

m

X

X

max

zi gi ðxÞ P

~zi gi ðxÞ ¼ Ur ðxÞ:

z2D

i¼1

i¼1

Combining the two sides, we immediately prove the fact above. Particularly,

from (2), we conclude that Ur(x) is convex because it is the supremum of a family of convex functions.

Since the linear optimization problem in (2) is bounded and has a strictly

feasible point z0 = (r/m, r/m, . . . , r/m)T, it follows from the strong duality for lin-

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

719

ear programming that for any given x 2 Rn , its optimal value is same as that of

dual linear program. Thus, we have that

(

)

m

X

Ur ðxÞ ¼ min minm r- þ

ui : ui P gi ðxÞ þ -; ui P 0; i ¼ 1; . . . ; m ;

-2R u2R

i¼1

ð3Þ

m

where – 2 R and

Pmu 2 R are respectively Lagrange multipliers corresponding

to constraints i¼1 zi ¼ r and zi 6 1 (i = 1, 2, . . . , m). Observe the inner linear

program in (3)

m

X

ui

minm r- þ

u2R

s:t:

i¼1

ui P gi ðxÞ þ -; i ¼ 1; 2; . . . ; m

ui P 0; i ¼ 1; 2; . . . ; m

ð4Þ

and it is not di?cult to ?nd that each pair of constraints ui P gi(x) + – and

ui P 0 can be replaced by the constraint ui P max{0, gi(x) +P

-}. Considering

the monotonic nondecreasing of objective function r- þ mi¼1 ui again, we

infer that for any given x 2 Rn and – 2 R, the optimal condition for u* to be

a solution of problem (4) is

ui ¼ maxf0; gi ðxÞ þ -g;

i ¼ 1; 2; . . . ; m:

Substituting it into the cost of (4) and eliminating variable u directly yield

(

)

m

X

Ur ðxÞ ¼ min r- þ

maxð0; gi ðxÞ þ -Þ :

ð5Þ

-2R

i¼1

Accordingly, the original nonsmooth problem (1) is equivalent to

m

X

min minm r- þ

maxf0; gi ðxÞ þ -g:

-2R u2R

ð6Þ

i¼1

Note that the objective of nonsmooth problem (6) only involves the maximum function max{0, gi(x) + -}, so we exploit the smooth approximation

p 1 lnð1 þ expðptÞÞ given by [14,15] to approximate max{0, t}, and replace

gi(x) with its perturbed function

q????????????????????????????????????????????????????????????????????????????????????????????????????

2

2

2

2

gi ðx; pÞ ¼ xi ðx1 ti1 Þ þ ðx2 ti2 Þ þ þ ðxn tin Þ þ ð1=pÞ :

ð7Þ

Thus, combined the results of Lemma 1 below, problem (1) is converted into

the following smooth unconstrained optimization problem:

min m Ur ð-; x; pÞ :¼ r- þ p 1

-2R;u2R

m

X

i¼1

lnf1 þ exp½pðgi ðx; pÞ þ -Þ g:

ð8Þ

720

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

Lemma 1. The function Ur(-, x; p) defined in (8) has the following properties:

(1) For any – 2 R; x 2 Rn and p1, p2 satisfying 0 < p1 < p2, we have
Ur ð-; x; p1 Þ > Ur ð-; x; p2 Þ:

ð9Þ

n

(2) For any – 2 R; x 2 R and p > 0,

Ur ðxÞ 6 Ur ð-; x; pÞ 6 Ur ðxÞ þ ½mðln 2 þ 1Þ =p:

ð10Þ

(3) For any p > 0, Ur(-, x; p) is continuously differentiable and strictly convex.

Proof

(1) For any t 2 R, de?ne u(t, p) := p 1ln[1 + exp(pt)]. Then,

ouðt; pÞ expðptÞ ln½expðptÞ f1 þ expðptÞg lnf1 þ expðptÞg

¼

< 0;
op
p2 f1 þ expðptÞg
ouðt; pÞ=ot ¼ expðptÞ=½1 þ expðptÞ > 0:

Considering the strict decreasing of gi(x, p) with p again, so for any p1, p2

such that 0 < p1 < p2, we have the following inequality relationship
uðgi ðx; p1 Þ; p1 Þ > uðgi ðx; p2 Þ; p1 Þ > uðgi ðx; p2 Þ; p2 Þ;

i ¼ 1; 2; . . . ; m:

In terms of this inequality and the de?nition of Ur(-, x; p), we readily obtain the result in (9).

(2) For any – 2 R; x 2 Rn and p > 0, there obviously holds

0 6 maxf0; gi ðx; pÞ þ -g maxf0; gi ðxÞ þ -g 6 1=p

and

0 6 p 1 lnf1 þ exp½pðgi ðx; pÞ þ -Þ g maxf0; gi ðx; pÞ þ -g 6 ðln 2Þ=p:

Combining them with the de?nition of Ur(-, x; p), we have that

r- þ

m

X

maxf0; gi ðxÞ þ -g

i¼1

6 Ur ð-; x; pÞ 6 r- þ

m

X

i¼1

maxf0; gi ðxÞ þ -g þ

mðln 2 þ 1Þ

:

p

So, the inequality relation in (10) directly follows from (5).

(3) For any p > 0, clearly, Ur(-, x; p) is continuously differentiable. Now we

prove that it is strictly convex. From (7) and (8), we have

m

X

ki ð-; x; pÞ

ð11Þ

r- Ur ð-; x; pÞ ¼ r þ

i¼1

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

721

and

rx Ur ð-; x; pÞ ¼

m

X

ki ð-; x; pÞrgi ðx; pÞ;

ð12Þ

i¼1

where

ki ð-; x; pÞ ¼

exp½pgi ðx; pÞ þ p;

1 þ exp½pgi ðx; pÞ þ p-

i ¼ 1; 2; . . . ; m:

Write

Q ¼ r2xx Ur ð-; x; pÞ;

hi ðx; pÞ ¼

li ð-; x; pÞ ¼

p exp½pgi ðx; pÞ þ pð1 þ exp½pgi ðx; pÞ þ p- Þ

m

X

;

q????????????????????????????????????????????????????????????????????????????????????????????????????

ðx1 ti1 Þ2 þ ðx2 ti2 Þ2 þ þ ðxn tin Þ2 þ ð1=pÞ2 :

Then, from (11) and (12), we obtain

”

Pm

l ð-; x; pÞ

2

r Ur ð-; x; pÞ ¼ Pm i¼1 i

i¼1 li ð-; x; pÞrg i ðx; pÞ

Q¼

2

Pm

T

i¼1 li ð-; x; pÞðrgi ðx; pÞÞ

#

Q

;

li ðx; x; pÞrgi ðx; pÞrgi ðx; pÞT

i¼1

þ

h

i

T

m wi ki ðx; x; pÞ h2 ðx; pÞI ðx ti Þðx ti Þ

X

i

h3i ðx; pÞ

i¼1

where I is a unit matrix of n-by-n. Thus, for any s ¼ ðs0 ; sÞ 2 Rnþ1 with s 6¼ 0

and s 2 Rn ,

m

m

X

X

sT r2 Ur ð-;x;pÞ s ¼ s20

li ð-;x;pÞ þ 2s0

li ð-;x;pÞsT rgi ðx;pÞ þ sT Qs

i¼1

¼

m

X

i¼1

li ð-;x;pÞðs20 þ 2s0 sT rgi ðx;pÞ þ ðsT rgi ðx;pÞÞ2 Þ

i¼1

m

X

þ

h

i

xi ki ð-;x;pÞ½hi ðx;pÞ 3 sT h2i ðx;pÞI ðx ti Þðx ti ÞT s

i¼1

P

m

X

xi ki ð-;x;pÞ½hi ðx;pÞ 3 sT ½h2i ðx;pÞI ðx ti Þðx ti ÞT s

i¼1

P

m

X

xi ki ð-;x;pÞ½hi ðx;pÞ 3 h2i ðx;pÞksk2 kx ti k2 ksk2 > 0;

i¼1

where the second inequality follows from the CauchySchwartz inequality and

the third from the de?nition of hi(x, p). This shows that for any p > 0, Ur(-, x; p)

722

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

is a strict convex function of – and x. So far, we complete the proof of

lemma. h

Now, let us describe a speci?c algorithm for problem (1) based on the solution of smooth unconstrained problem (8), whose convergence analysis is left

to the next section.

2.1. A smoothing algorithm for the Euclidean r-centrum problem

Let r 2 (1, +1) and p0 > 0, select a starting point (-0, x0) and set k :¼ 0.

For k = 0, 1, 2, . . . , do

S1. Use an unconstrained minimization method to solve

min Ur ð-; x; pk Þ

ð13Þ

-2R;x2Rn

and let (-k, xk) denote its minimizer.

S2. Set pk+1 = rpk, let k

k + 1 and go to step S1.

End

Note that, as parameter p varies, the algorithm above consists of a sequence

of smooth unconstrained minimization. Nevertheless, taking account of the

uniform approximation of Ur(-, x; p) to Ur(x), we can set p to be a su?ciently

large but ?xed constant in the practical computation. As a result, a solution to

problem (1) arises from solving one single smooth unconstrained optimization

problem (13).

3. The global convergence analysis of algorithm

In what follows, we concentrate on establishing the global convergence of

algorithm. To this end, we ?rst show the following two lemmas.

n

Lemma 2. Let ffi ðxÞgm

i¼1 be a collection of convex functions from R to R and

f(x) be a vector-valued function with fi(x) as the ith component. Define an index

set I(x) :¼ {ijfi(x) P f[r](x)}, and then we have

! (

)

r

X

X

X

o

f½l ðxÞ ¼

qi ofi ðxÞ : 0 6 qi 6 1 for i 2 IðxÞ;

qi ¼ r :

l¼1

i2IðxÞ

i2IðxÞ

Pr

Proof. De?ne /ðyÞ :¼ l¼1 y ½l , where y[1], y[2], . . . , y[m] are the numbers

y1, y2, . . . , ym sorted in nonincreasing order. It is easily veri?ed that /(y) is convex and for a ?xed y 0 2 Rm , the following inequality

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

723

/ðyÞ P /ðy 0 Þ þ hn; y y 0 i

holds for any y if and only if n 2 rco{ei : i such that yi P y[r]}, where {ei} denotes the canonical basis of Rm and for a ?nite set {z1, z2, . . . , zm},

(

)

m

m

X

X

rcofz1 ; z2 ; . . . ; zm g ¼

ai zi :

ai ¼ r; 0 6 ai 6 1; i ¼ 1; 2; . . . ; m :

i¼1

i¼1

So, from the de?nition of subdi?erential, it follows that

o/ðyÞ ¼ rcofei : i such that y i P y ½r g:

By Theorem 4.3.1 [16], we thus have that

!

r

n[

o

X

o

f½l ðxÞ ¼ o/ðf ðxÞÞ ¼ rco

ofi ðxÞ : i 2 IðxÞ ;

l¼1

which is exactly the equality in Lemma 2.

h

Lemma 3. Let {-k, xk} be the sequence of points produced by the algorithm.

Then, any limit points of {xk} are optimal solutions to problem (1).

Proof. Let (-*, x*) be a limit point of {-k, xk}. Without loss of generality, we

suppose that {-k, xk} ! (-*, x*) as k ! +1. From the fact that {-k, xk} is a

solution of problem (13), we have the following equalities

m

X

ki ð-k ; xk ; pk Þ ¼ 0;

ð14Þ

r- Ur ð-k ; xk ; pk Þ ¼ r þ

i¼1

rx Ur ð-k ; xk ; pk Þ ¼

m

X

ki ð-k ; xk ; pk Þ

i¼1

xi ðxk ti Þ

¼0

hi ðxk ; pk Þ

and moreover, from Eqs. (5) and (10), we know

r

m

X

X

g½l ðxk Þ ¼ r-k þ

maxf0; gi ðxk Þ þ -k g;

Ur ðxk Þ ¼

l¼1

ð15Þ

ð16Þ

i¼1

where ki(-, x;p) and hi(x, p) is same as before. De?ne the index set

Iðx Þ ¼ fi : gi ðx Þ P g½r ðx Þ; i ¼ 1; 2; . . . ; mg;

then from the equality (16) and the continuity of g0i s, we can infer that

r

X

X

X

g½l ðx Þ ¼ r- þ

maxf0;gi ðx Þ þ – g þ

maxf0;gi ðx Þ þ – g

l¼1

which implies

i2Iðx Þ

i62Iðx Þ

724

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

gi ðx Þ þ – 6 0;

i 62 Iðx Þ:

ð17Þ

In the following, we prove that the strict inequality in (17) holds. From the continuity of giÕs, it follows that gi(xk) + -k > 0 for i 2 I (x*). Combining it with

(14) yields

lim

k!þ1

X

i62Iðx Þ

exp½pk ðgi ðxk ; pk Þ þ -k Þ

¼ 0:

1 þ exp½pk ðgi ðxk ; pk Þ þ -k Þ

ð18Þ

As a consequence of (17) and (18), we immediately obtain

gi ðx Þ þ – < 0;
i 62 Iðx Þ:
ð19Þ
Note that kki ¼ 1þexp½ pk ðg1i ðxk ;pk Þþ-k Þ ; i ¼ 1; 2; . . . ; m, so we have from (19) that
lim kki ¼ 0;
k!þ1
i 62 Iðx Þ:
ð20Þ
Pm
While from i¼1 kki ¼ r and kki > 0; i ¼ 1; . . . ; m, it follows that fkki g have a

convergent subsequence. Without loss of generality, we suppose that

lim kki ¼ ki ;

k!þ1

i 2 Iðx Þ:

Then, Eq. (20) implies that

X

ki ¼ r and 0 6 ki 6 1;

i 2 Iðx Þ:

i2Iðx Þ

For i 2 I(x*),

gi ¼ lim

k!þ1

xi ðxk ti Þ

2 ogi ðx Þ:

hi ðxk ; pk Þ

Thus, from the equality (15), it follows that

X

lim rx Ur ð-k ; xk ; pk Þ ¼

ki gi :

k!þ1

i2Iðx Þ

By the result of Lemma 2, this shows that 0 2 oUr(x*), and so x* is an optimal

solution of problem (1). h

Theorem 1. Let {-k, xk} be the sequence of points produced by the algorithm,

and x* be the unique optimal solution of problem (1). Then, we have

limk!+1xk = x*.

Proof. For any k P 1, by Lemma 1,

Ur ð-1 ; x1 ; p1 Þ > Ur ð-1 ; x1 ; pk Þ P Ur ð-k ; xk ; pk Þ P Ur ðxk Þ:

Since all giÕs are coercive, the following level set is bounded

L ¼ fx 2 Rn : Ur ðxÞ 6 Ur ð-1 ; x1 ; p1 Þg:

ð21Þ

S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716728

725

From (21), we have fxk g L. Hence, fxk g is bounded. As x* is the unique

optimal solution to (1), it follows from Lemma 3 that limk!+1xk = x*. Here,

we complete the convergence analysis of the algorithm. h

4. The implementation of algorithm and computational results

We implemented the algorithm described i …

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Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

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Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

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Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.