Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 245458, 11 pages
doi:10.1155/2012/245458
Research Article
Global Error Bound Estimation for
the Generalized Nonlinear Complementarity
Problem over a Closed Convex Cone
Hongchun Sun1 and Yiju Wang2
1
2
School of Sciences, Linyi University, Shandong, 276005 Linyi, China
School of Management Science, Qufu Normal University, Shandong, 276800 Rizhao, China
Correspondence should be addressed to Yiju Wang, wyiju@hotmail.com
Received 10 February 2012; Accepted 28 April 2012
Academic Editor: Zhenyu Huang
Copyright q 2012 H. Sun and Y. Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The global error bound estimation for the generalized nonlinear complementarity problem over a
closed convex cone GNCP is considered. To obtain a global error bound for the GNCP, we first
develop an equivalent reformulation of the problem. Based on this, a global error bound for the
GNCP is established. The results obtained in this paper can be taken as an extension of previously
known results.
1. Introduction
Let mappings F, G : Rn → Rm , H : Rn → Rl , and the generalized nonlinear complementarity
problem, abbreviated as GNCP, is to find vector x∗ ∈ Rn such that
Fx∗ ∈ K,
Gx∗ ∈ K0 ,
Fx∗ T Gx∗ 0,
Hx∗ 0,
1.1
where K is a nonempty closed convex cone in Rm and K◦ is its dual cone, that is, K◦ {u ∈
Rm | uT v ≥ 0, for all v ∈ K}. We denote the solution set of the GNCP by X ∗ , and assume that
it is nonempty throughout this paper.
The GNCP is a direct generalization of the classical nonlinear complementarity
problem which finds applications in engineering, economics, finance, and robust optimization operations research 1–3. For example, the balance of supply and demand is
central to all economic systems; mathematically, this fundamental equation in economics
is often described by a complementarity relation between two sets of decision variables.
2
Journal of Applied Mathematics
Furthermore, the classical Walrasian law of competitive equilibria of exchange economies
can be formulated as a generalized nonlinear complementarity problem in the price and
excess demand variables 2. Up to now, the issues of numerical methods and existence of
the solution for the problem were discussed in the literature 4.
Among all the useful tools for theoretical and numerical treatment to variational
inequalities, nonlinear complementarity problems, and other related optimization problems,
the global error bound, that is, an upper bound estimation of the distance from a given point
in Rn to the solution set of the problem in terms of some residual functions, is an important
one 5, 6. The error bound estimation for the generalized linear complementarity problems
over a polyhedral cone was analyzed by Sun et al. 7. Using the natural residual function,
Pang 8 obtained a global error bound for the strongly monotone and Lipschitz continuous
classical nonlinear complementarity problem with a linear constraint set. Xiu and Zhang 9
also presented a global error bound for general variational inequalities with the mapping
being strongly monotone and Lipschitz continuous in terms of the natural residual function.
If Fx x, Gx is γ-strongly monotone and H ölder continuous, the local error bound for
classical variational inequality problems was given by Solodov 6.
To our knowledge, the global error bound for the problem 1.1 with the mapping
being γ-strongly monotone and H ölder-continuous hasn’t been investigated. Motivated by
this fact, The main contribution of this paper is to establish a global error bound for the GNCP
via the natural residual function under milder conditions than those needed in 6, 8, 9. The
results obtained in this paper can be taken as an extension of the previously known results in
6, 8, 9.
We give some notations used in this paper. Vectors considered in this paper are all
taken in Euclidean space equipped with the standard inner product. The Euclidean norm of
vector in the space is denoted by · . The inner product of vector in the space is denoted by
·, ·.
2. The Global Error Bound for GNCP
In this section, we would give error bound for GNCP, which can be viewed as extensions
of previously known results. To this end, we will in the following establish an equivalent
reformulation of the GNCP and state some well-known properties of the projection operator
which is crucial to our results.
In the following, we first give the equivalent reformulation of the GNCP.
Theorem 2.1. A point x∗ is a solution of 1.1 if and only if x∗ is a solution of the following problem:
Gx∗ T Fx − Fx∗ ≥ 0,
Hx∗ 0.
∀Fx ∈ K,
2.1
Proof. Suppose that x∗ is a solution of 2.1. Since vector 0 ∈ K, by substituting Fx 0
into 2.1, we have Gx∗ T Fx∗ ≤ 0. On the other hand, since Fx∗ ∈ K, then 2Fx∗ ∈
K. By substituting Fx 2Fx∗ into 2.1, we obtain Gx∗ T Fx∗ ≥ 0. Consequently,
Gx∗ T Fx∗ 0. For any Fx ∈ K, we have Gx∗ T Fx Gx∗ T Fx − Fx∗ ≥ 0,
that is, Gx∗ ∈ K◦ . Combining Hx∗ 0, thus, x∗ is a solution of 1.1.
Journal of Applied Mathematics
3
On the contrary, suppose that x∗ is a solution of 1.1, since Gx∗ ∈ K◦ , for any Fx ∈
K, we have Gx∗ T Fx ≥ 0, and from Gx∗ T Fx∗ 0, we have Gx∗ T Fx − Fx∗ ≥ 0,
combining Hx∗ 0. Therefor, x∗ is a solution of 2.1.
Now, we give the definition of projection operator and some related properties 10.
For nonempty closed convex set K ⊂ Rm and any vector x ∈ Rm , the orthogonal projection of
x onto K, that is, argmin{y − x | y ∈ K}, is denoted by PK x.
Lemma 2.2. For any u ∈ Rm , v ∈ K, then
i PK u − u, v − PK u ≥ 0,
ii PK u − PK v ≤ u − v.
For 2.1, β > 0 is a constant, ex : Fx − PK Fx − βGx is called projection-type
residual function, and let rx : ex. The following conclusion provides the relationship
between the solution set of 2.1 and that of projection-type residual function 11, which is
due to Noor 11.
Lemma 2.3. x is a solution of 2.1 if and only if ex 0, Hx 0.
To establish the global error bound of GNCP, we also need the following definition.
Definition 2.4. The mapping F : Rn → Rm is said to be
1 γ-strongly monotone with respect to G : Rn → Rm if there are constants μ > 0, γ > 1
such that
1γ
Fx − F y , Gx − G y ≥ μx − y ,
∀x, y ∈ Rn ;
2.2
2 H ölder-continuous if there are constants L > 0, v ∈ 0, 1 such that
Fx − F y ≤ Lx − yv ,
∀x, y ∈ Rn .
2.3
In this following, based on Lemmas 2.2 and 2.3, we establish error bound for GNCP in
the set Ω : {x ∈ Rn | Hx 0}.
Theorem 2.5. Suppose that F is γ-strongly monotone with respect to G and with positive constants
μ, γ, both F and G are H ölder continuous with positive constants L1 > 0, L2 > 0, v1 , v2 ∈ 0, 1,
4
Journal of Applied Mathematics
respectively, and βμ ≤ L1 L2 β2L1 L2 β holds. Then for any x ∈ Ω : {x ∈ Rn | Hx 0},
there exists a solution x∗ of 1.1 such that
rx
2L1 L2 β
1/ min{v1 ,v2 }
≤ x − x∗ ≤
L1 L2 β
rx
βμ
1/1γ−min{v1 ,v2 }
,
βμ
.
if rx ≤
L1 L2 β
rx
2L1 L2 β
1/ max{v1 ,v2 }
≤ x − x∗ ≤
L1 L2 β
rx
βμ
2.4
1/1γ−max{v1 ,v2 }
,
2.5
if rx ≥ 2L1 L2 β.
rx
2L1 L2 β
1/ min{v1 ,v2 }
≤ x − x∗ ≤
L1 L2 β
rx
βμ
1/1γ−max{v1 ,v2 }
,
βμ
< rx < 2L1 L2 β.
if
L1 L2 β
2.6
Proof. Since
Fx − ex PK Fx − βGx ∈ K,
2.7
by the first inequality of 2.1,
Fx − ex − Fx∗ T βGx∗ ≥ 0.
2.8
Combining Fx∗ ∈ K with Lemma 2.2i, we have
Fx∗ − PK Fx − βGx , PK Fx − βGx − Fx − βGx ≥ 0.
2.9
Substituting PK Fx − βGx in 2.9 by Fx − ex leads to that
Fx − Fx∗ − exT ex − βGx ≥ 0.
2.10
Using 2.8 and 2.10, we obtain
Fx − Fx∗ − exT ex βGx∗ − Gx ≥ 0,
2.11
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5
that is,
βFx − Fx∗ T Gx∗ − Gx exT Fx − Fx∗ − βGx∗ − Gx − exT ex ≥ 0.
2.12
Base on Definition 2.4, a direct computation yields that
x − x∗ 1γ ≤
≤
1
Fx − Fx∗ T Gx − Gx∗ μ
1 exT Fx − Fx∗ βGx − Gx∗ − exT ex
βμ
1 ≤
ex Fx − Fx∗ βGx − Gx∗ βμ
≤
2.13
1 rx L1 x − x∗ v1 βL2 x − x∗ v2 .
βμ
Combining this, we have
⎧
1/1γ−min{v1 ,v2 }
⎪
L1 L2 β
⎪
⎪
rx
,
⎪
⎨
βμ
x − x∗ ≤ 1/1γ−max{v1 ,v2 }
⎪
⎪ L1 L 2 β
⎪
⎪
rx
,
⎩
βμ
if x − x∗ ≤ 1,
2.14
if x − x∗ ≥ 1.
On the other hand, for the first inequality of 2.1, by Lemmas 2.3 and 2.2ii, we have
rx ex − ex∗ Fx − PK Fx − βGx − Fx∗ PK Fx∗ − βGx∗ Fx − Fx∗ PK Fx − βGx − PK Fx∗ − βGx∗ ≤ Fx − Fx∗ Fx − βGx − Fx∗ − βGx∗ 2.15
2Fx − Fx∗ βGx − Gx∗ ≤ 2L1 x − x∗ v1 L2 βx − x∗ v2 .
Thus,
⎧
1/ min{v1 ,v2 }
⎪
rx
⎪
⎪
,
⎨
2L L2 β
x − x∗ ≥ 1
1/ max{v1 ,v2 }
⎪
rx
⎪
⎪
,
⎩
2L1 L2 β
if x − x∗ ≤ 1,
∗
if x − x ≥ 1.
2.16
6
Journal of Applied Mathematics
Combining 2.14 with 2.16 for any x ∈ Rn , if x − x∗ ≤ 1, then
rx
2L1 L2 β
1/1γ−min{v1 ,v2 } L1 L 2 β
rx
,1 .
βμ
2.17
1/1γ−max{v1 ,v2 }
1/ max{v1 ,v2 } L1 L2 β
∗
rx
, 1 ≤ x − x ≤
.
βμ
2.18
1/ min{v1 ,v2 }
∗
≤ x − x ≤ min
For any x ∈ Rn , if x − x∗ ≥ 1, then
max
rx
2L1 L2 β
If rx ≤ βμ/L1 L2 β, then L1 L2 β/βμrx ≤ 1, by 2.17, we have x − x∗ ≤ 1,
and using 2.17 again, we obtain that 2.4 holds.
If rx ≥ L2 β 2L1 , then rx/2L1 L2 β ≥ 1, combining this with 2.18, we have
x − x∗ ≥ 1, and using 2.18 again, we conclude that 2.5 holds.
If βμ/L1 L2 β < rx < 2L1 L2 β, then
L1 L 2 β
rx > 1,
βμ
rx
< 1.
L2 β 2L1
2.19
Combining 2.17 with 2.18, we conclude that 2.6 holds.
Definition 2.6. The mapping H involved in the GNCP is said to be α-strongly monotone in Rn
if there are positive constants σ > 0, α > 1 such that
1α
x − y, Hx − H y ≥ σ x − y ,
∀x, y ∈ Rn .
2.20
Base on Theorem 2.5, we are at the position to state our main results in the following.
Theorem 2.7. Suppose that the hypotheses of Theorem 2.5 hold, H is α-strongly monotone, and the
set Ω : {x ∈ Rn | Hx 0} is convex. Then, there exists a constant ρ > 0, such that, for any
x ∈ Rn , there exists x∗ ∈ X ∗ such that
x − x∗ ≤ ρ Hx1/α Rx ,
∀x ∈ Rn ,
2.21
Journal of Applied Mathematics
7
where
⎧ 1/1γ−min{v1 ,v2 } ⎫
⎪
⎪
L1 L2 β 2L1 βL2 L1 L2 β − min{v1 ,v2 }
1
⎪
⎪
⎪
⎪
⎪
, max
,
σ
,⎪
⎪
⎪
⎪
⎪
σ
βμ
βμ
⎪
⎪
⎪
⎪ ⎪
1/1γ−min{v1 ,v2 } ⎪
⎪
⎪
⎪
⎪
⎪
⎪
L1 L2 β 2L1 βL2 L1 L2 β − max{v1 ,v2 }
⎪
⎪
⎪
⎪
max
,
σ
,
⎪
⎪
⎨
⎬
βμ
βμ
ρ max
,
1/1γ−max{v1 ,v2 }
⎪
⎪
⎪
⎪
L1 L2 β 2L1 βL2 L1 L2 β − min{v1 ,v2 }
⎪
⎪
⎪
max
,
σ
, ⎪
⎪
⎪
⎪
⎪
⎪
⎪
βμ
βμ
⎪
⎪
⎪
⎪
⎪
1/1γ−max{v1 ,v2 } ⎪
⎪
⎪
⎪
⎪
βL
L
β
2L
L
⎪
⎪
L1 L 2 β
1
2
1
2
⎪
⎪
−
max{v
,v
}
1 2
⎪
⎪
max
,
σ
,
⎩
⎭
βμ
βμ
2.22
⎫
⎧
1/1γ−min{v1 ,v2 }
⎪
⎪
⎪
⎪
rx Hxmin{v1 ,v2 }/α
⎪
⎪
⎪
⎪
⎪
⎪
1/1γ−min{v
,v
}
⎪
⎪
1
2
⎪
⎪
⎬
⎨ rx Hxmax{v1 ,v2 }/α
2.23
Rx max 1/1γ−max{v1 ,v2 } .
⎪
⎪
min{v1 ,v2 }/α
⎪
⎪
rx
Hx
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩ rx Hxmax{v1 ,v2 }/α 1/1γ−max{v1 ,v2 } ⎪
Proof. For given x ∈ Rn , we only need to first project x to Ω : {x ∈ Rn | Hx 0}, that
is, there exists a vector x ∈ Ω such that x − x distx, Ω. By Definition 2.6, there exist
constants σ > 0, α > 0 such that
x − x1α ≤
1
x − x, Hx − Hx
σ
≤
1
x − xHx − Hx
σ
1
x − xHx,
σ
that is, distx, Ω x − x ≤ 1/σHx1/α .
Since
rx − rx ≤ ex − ex
Fx − PK Fx − βGx − Fx − PK Fx − βGx ≤ Fx − Fx PK Fx − βGx − PK Fx − βGx ≤ Fx − Fx Fx − βGx − Fx − βGx ≤ 2Fx − Fx βGx − Gx
2.24
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Journal of Applied Mathematics
≤ 2L1 x − xν1 βL2 x − xν2
2L1 βL2 x − xmin{v1 ,v2 } , if
≤ 2L1 βL2 x − xmax{v1 ,v2 } , if
2L1 βL2 distx, Ωmin{v1 ,v2 } ,
2L1 βL2 distx, Ωmax{v1 ,v2 } ,
x − x < 1
x − x ≥ 1
if x − x ≤ 1
if x − x ≥ 1,
2.25
Combining this, we have
2L1 βL2 distx, Ωmin{v1 ,v2 } , if x − x < 1
rx ≤ rx 2L1 βL2 distx, Ωmax{v1 ,v2 } , if x − x ≥ 1.
2.26
Combining 2.26 with Theorem 2.5, we have the following results.
Case 1 if rx ≤ βμ/L1 L2 β and x − x ≤ 1. Combining 2.4 with the first inequality in
2.26, we can obtain that
x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β
rx
βμ
1/1γ−min{v1 ,v2 }
≤ distx, Ω
1/1γ−min{v1 ,v2 }
2L1 βL2 L1 L2 β
L 1 L2 β
min{v1 ,v2 }
rx distx, Ω
βμ
βμ
1/1γ−min{v1 ,v2 }
1
Hx1/α η1 rx Hxmin{v1 ,v2 }/α
σ
1/1γ−min{v1 ,v2 } 1/α
min{v1 ,v2 }/α
≤ ρ1 Hx rx Hx
,
≤
2.27
where η1 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − min{v1 ,v2 } }
ρ1 max{1/σ, η1 }.
1/1γ−min{v1 ,v2 }
,
Journal of Applied Mathematics
9
Case 2 If rx ≤ βμ/L1 L2 β and x − x ≥ 1. Combining 2.4 with the second inequality
in 2.26, we can also obtain that
x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β
rx
βμ
1/1γ−min{v1 ,v2 }
≤ distx, Ω
1/1γ−min{v1 ,v2 }
2L1 βL2 L1 L2 β
L 1 L2 β
max{v1 ,v2 }
rx distx, Ω
βμ
βμ
1/1γ−min{v1 ,v2 }
1
Hx1/α η2 rx Hxmax{v1 ,v2 }/α
σ
1/1γ−min{v1 ,v2 } ≤ ρ2 Hx1/α rx Hxmax{v1 ,v2 }/α
,
≤
2.28
1/1γ−min{v1 ,v2 }
where η2 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − max{v1 ,v2 } }
,
ρ2 max{1/σ, η2 }.
Case 3 if rx > βμ/L1 L2 β and x −x ≤ 1. Combining 2.5-2.6 with the first inequality
in 2.26, we can obtain that
x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β
rx
βμ
1/1γ−max{v1 ,v2 }
≤ distx, Ω
1/1γ−max{v1 ,v2 }
2L1 βL2 L1 L2 β
L 1 L2 β
min{v1 ,v2 }
rx distx, Ω
βμ
βμ
1/1γ−max{v1 ,v2 }
1
Hx1/α η3 rx Hxmin{v1 ,v2 }/α
σ
1/1γ−max{v1 ,v2 } ≤ ρ1 Hx1/α rx Hxmin{v1 ,v2 }/α
,
≤
2.29
where η3 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − min{v1 ,v2 } }
ρ3 max{1/σ, η3 }.
1/1γ−max{v1 ,v2 }
,
10
Journal of Applied Mathematics
Case 4 If rx > βμ/L1 L2 β and x − x ≥ 1. Combining 2.5-2.6 with the second
inequality in 2.26, we can also obtain that
x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β
rx
βμ
1/1γ−max{v1 ,v2 }
≤ distx, Ω
1/1γ−max{v1 ,v2 }
2L1 βL2 L1 L2 β
L 1 L2 β
max{v1 ,v2 }
rx distx, Ω
βμ
βμ
1/1γ−max{v1 ,v2 }
1
Hx1/α η4 rx Hxmax{v1 ,v2 }/α
σ
1/1γ−max{v1 ,v2 } 1/α
max{v1 ,v2 }/α
≤ ρ4 Hx rx Hx
,
≤
2.30
where η4 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − max{v1 ,v2 } }
ρ4 max{1/σ, η4 }.
By 2.27–2.30, we can deduce that 2.21 holds.
1/1γ−max{v1 ,v2 }
,
Based on Theorem 2.7, we can further establish a global error bound for the GNCP.
First, we give that the needed result from 12 mainly discusses the error bound for a
polyhedral cone to reach our claims.
Lemma 2.8. For polyhedral cone P {x ∈ Rn | D1 x d1 , D2 x ≤ d2 } with D1 ∈ Rl×n , D2 ∈
Rm×n , d1 ∈ Rl , and d2 ∈ Rm , there exists a constant c1 > 0 such that
distx, P ≤ c1 D1 x − d1 D2 x − d2 ∀x ∈ Rn .
2.31
Theorem 2.9. Suppose that the hypotheses of Theorem 2.5 hold, and H is linear mapping. Then, there
exists a constant μ > 0, such that, for any x ∈ Rn , there exists x∗ ∈ X ∗ such that
x − x∗ ≤ μ{Hx Rx},
∀x ∈ Rn ,
2.32
where
⎧
1/1γ−min{v1 ,v2 } ⎫
⎪
⎪
⎪
⎪
rx Hxmin{v1 ,v2 }
⎪
⎪
⎪
⎪
⎪
1/1γ−min{v1 ,v2 } ⎪
⎪
⎪
⎪
⎪
⎨ rx Hxmax{v1 ,v2 }
⎬
Rx max 1/1γ−max{v1 ,v2 } .
⎪
⎪
⎪
⎪
rx Hxmin{v1 ,v2 }
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1/1γ−max{v
,v
}
⎪
1 2 ⎪
⎩ rx Hxmax{v1 ,v2 }
⎭
2.33
Journal of Applied Mathematics
11
Proof. For given x ∈ Rn , we only need to first project x to Ω, that is, there exists a vector
x ∈ Ω such that x − x distx, Ω. By Lemma 2.8, there exists a constant τ > 0 such that
distx, Ω x − x ≤ τHx. In the following, the proof is similar to that of Theorem 2.7,
and we can deduce that 2.32 holds.
Remark 2.10. If the constraint condition Hx 0 is removed in 1.1, F is strongly monotone
with respect to G i.e., γ 1, and both F and G are Lipschitz continuous i.e., v1 v2 1,
the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in 9.
If the constraint condition Hx 0 is removed in 1.1 and Fx x, Gx is strongly
monotone i.e., γ 1 and Lipschitz continuous i.e., v1 v2 1, the error bound in
Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in 8.
If the constraint condition Hx 0 is removed in 1.1 and Fx x, Gx is γstrongly monotone in set {x ∈ Rn | x − x∗ ≤ 1} and H ölder continuous, the error bound in
Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 2 in 6.
Acknowledgments
The authors wish to give their sincere thanks to the anonymous referees for their valuable
suggestions and helpful comments which improved the presentation of the paper. This
work was supported by the Natural Science Foundation of China Grant nos. 11171180,
11101303, Specialized Research Fund for the Doctoral Program of Chinese Higher Education
20113705110002, and Shandong Provincial Natural Science Foundation ZR2010AL005,
ZR2011FL017.
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Probability and Statistics
Volume 2014
The Scientific
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International Journal of
Differential Equations
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International Journal of
Advances in
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Mathematical Physics
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Journal of
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International
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Stochastic Analysis
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International Journal of
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Discrete Dynamics in
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Applied Mathematics
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Optimization
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