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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.)
Explanation of the Algorithm/Heuristic Methodology – How does it work?
Examples of Algorithm/Heuristic Usage – Discuss concrete examples of how it is used today in
the world.
Summary/Conclusions – Sum it all up! References – APA formatted sources…only correct
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) 716–728 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) 716–728 717 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 [11–13] for network location problems.
718
S. Pan, X. Li / Appl. Math. Comput. 167 (2005) 716–728
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) 716–728
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) 716–728
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) 716–728
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

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 Cauchy–Schwartz 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) 716–728
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) 716–728
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) 716–728
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) 716–728
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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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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