Volume 2010, Article ID 657192, 20 pagesdoi:10.1155/2010/657192 Research Article A New Method for Solving Monotone Generalized Variational Inequalities Pham Ngoc Anh and Jong Kyu Kim Dep
Trang 1Volume 2010, Article ID 657192, 20 pages
doi:10.1155/2010/657192
Research Article
A New Method for Solving Monotone Generalized Variational Inequalities
Pham Ngoc Anh and Jong Kyu Kim
Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, Republic of Korea
Correspondence should be addressed to Jong Kyu Kim,jongkyuk@kyungnam.ac.kr
Received 11 May 2010; Revised 27 August 2010; Accepted 4 October 2010
Academic Editor: Siegfried Carl
Copyrightq 2010 P N Anh and J K Kim 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
We suggest new dual algorithms and iterative methods for solving monotone generalized variational inequalities Instead of working on the primal space, this method performs a dual step on the dual space by using the dual gap function Under the suitable conditions, we prove
the convergence of the proposed algorithms and estimate their complexity to reach an ε-solution.
Some preliminary computational results are reported
1 Introduction
Let C be a convex subset of the real Euclidean spaceRn , F be a continuous mapping from C
intoRn , and ϕ be a lower semicontinuous convex function from C intoR We say that a point
x∗is a solution of the following generalized variational inequality if it satisfies
Fx∗, x − x∗ ϕx − ϕx∗ ≥ 0, ∀x ∈ C, GVI
where·, · denotes the standard dot product in R n
Associated with the problemGVI, the dual form of this is expressed as following
which is to find y∗∈ C such that
F x, x − y∗
ϕx − ϕy∗
≥ 0, ∀x ∈ C. DGVI
In recent years, this generalized variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, transportations, economics, and nonlinear analysissee 1 9
Trang 2It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems see 10–16 Recently these techniques have been used to develop proximal iterative algorithm for variational inequalitiessee 17–
22
In addition Nesterov 23 introduced a dual extrapolation method for solving variational inequalities Instead of working on the primal space, this method performs a dual step on the dual space
In this paper we extend results in 23 to the generalized variational inequality problemGVI in the dual space In the first approach, a gap function gx is constructed such that gx ≥ 0, for all x∗ ∈ C and gx∗ 0 if and only if x∗solvesGVI Namely, we first develop a convergent algorithm forGVI with F being monotone function satisfying a certain Lipschitz type condition on C Next, in order to avoid the Lipschitz condition we will show how to find a regularization parameter at every iteration k such that the sequence x k
converges to a solution ofGVI
The remaining part of the paper is organized as follows In Section 2, we present two convergent algorithms for monotone and generalized variational inequality problems with Lipschitzian condition and without Lipschitzian condition.Section 3deals with some preliminary results of the proposed methods
2 Preliminaries
First, let us recall the well-known concepts of monotonicity that will be used in the sequel
see 24
Definition 2.1 Let C be a convex set inRn , and F : C → Rn The function F is said to be
i pseudomonotone on C if
F
y
, x − y≥ 0 ⇒F x, x − y≥ 0, ∀x, y ∈ C, 2.1
ii monotone on C if for each x, y ∈ C,
F x − Fy
iii strongly monotone on C with constant β > 0 if for each x, y ∈ C,
F x − Fy
, x − y≥ βx − y2
iv Lipschitz with constant L > 0 on C shortly L-Lipschitz, if
F x − F
y ≤ Lx − y, ∀x,y ∈ C. 2.4
Note that when ϕ is differentiable on some open set containing C, then, since ϕ is lower
semicontinuous proper convex, the generalized variational inequalityGVI is equivalent to the following variational inequalitiessee 25,26:
Trang 3Find x∗∈ C such that
Fx∗ ∇ϕx∗, x − x∗ ≥ 0, ∀x ∈ C. 2.5 Throughout this paper, we assume that:
A1 the interior set of C, int C is nonempty,
A2 the set C is bounded,
A3 F is upper semicontinuous on C, and ϕ is proper, closed convex and subdifferen-tiable on C,
A4 F is monotone on C.
In special case ϕ 0, problem GVI can be written by the following
Find x∗∈ C such that
Fx∗, x − x∗ ≥ 0, ∀x ∈ C. VI
It is well known that the problemVI can be formulated as finding the zero points of the
operator Tx Fx N C x, where
N C x
⎧
⎨
⎩
y ∈ C :y, z − x≤ 0, ∀z ∈ C , if x ∈ C,
The dual gap function of problemGVI is defined as follows:
g x : sup F
y
, x − y ϕx − ϕy
| y ∈ C . 2.7
The following lemma gives two basic properties of the dual gap function2.7 whose proof can be found, for instance, in6
Lemma 2.2 The function g is a gap function of GVI , that is,
i gx ≥ 0 for all x ∈ C,
ii x∗ ∈ C and gx∗ 0 if and only if x∗ is a solution to DGVI Moreover, if F is
pseudomonotone then x∗is a solution toDGVI if and only if it is a solution to GVI.
The problem sup{Fy, x − y ϕx − ϕy | y ∈ C} may not be solvable and the
dual gap function g may not be well-defined Instead of using gap function g, we consider a truncated dual gap function g R Suppose that x ∈ int C fixed and R > 0 The truncated dual
gap function is defined as follows:
g R x : max F
y
, x − y ϕx − ϕy
| y ∈ C,y − x ≤ R 2.8
For the following consideration, we define B R x : {y ∈ R n
inRn centered at x and radius R, and C R : C ∩ BR x The following lemma gives some properties for g R
Trang 4Lemma 2.3 Under assumptions (A1)–(A4), the following properties hold.
i The function g R · is well-defined and convex on C.
ii If a point x∗∈ C ∩ B R x is a solution to DGVI then g R x∗ 0.
iii If there exists x0 ∈ C such that g R x0 0
then x0is a solution toDGVI (and also GVI).
Proof i Note that Fy, x − y ϕx − ϕy is upper semicontinuous on C for x ∈ C and B R x is bounded Therefore, the supremum exists which means that g Ris well-defined
Moreover, since ϕ is convex on C and g is the supremum of a parametric family of convex
functionswhich depends on the parameter x, then g R is convex on C
ii By definition, it is easy to see that g R x ≥ 0 for all x ∈ C ∩ B R x Let x∗ be a solution ofDGVI and x∗∈ B R x Then we have
F
y
, x∗− y ϕx∗ − ϕy
≤ 0 ∀y ∈ C. 2.9
In particular, we have
F
y
, x∗− y ϕy
− ϕx∗ ≤ 0 2.10
for all y ∈ C ∩ B R x Thus
g R x∗ sup F
y
, x∗− y ϕx∗ − ϕy
| y ∈ C ∩ B R x ≤ 0, 2.11
this implies g R x∗ 0
iii For some x0 ∈ C ∩ int B R x, g R x0 0 means that x is a solution to DGVI
restricted to C ∩ int B R x Since F is pseudomonotone, x0is also a solution toGVI restricted
to C ∩ B R x Since x0∈ int B R x, for any y ∈ C, we can choose λ > 0 sufficiently small such
that
y λ: x0 λ y − x0 ∈ C ∩ B R x, 2.12
0≤F
x0 , y λ − x0
ϕy λ
− ϕ x0
F
x0 , x0 λ y − x0 − x0
ϕ x0 λ y − x0 − ϕ x0
≤ λF
x0
, y − x0
λϕy
1 − λϕx0
− ϕx0
λ F
x0 , y − x0
ϕy
− ϕ x0 ,
2.13
where2.13 follows from the convexity of ϕ· Since λ > 0, dividing this inequality by λ, we obtain that x0is a solution toGVI on C Since F is pseudomonotone, x0is also a solution to
DGVI
Trang 5Let C ⊆ Rn be a nonempty, closed convex set and x ∈ Rn Let us denote d C x the Euclidean distance from x to C and P r C x the point attained this distance, that is,
d C x : min
y ∈Cy − x, P r C x : arg min
y ∈Cy − x. 2.14
As usual, P r C is referred to the Euclidean projection onto the convex set C It is well-known that P r C is a nonexpansive and co-coercive operator on Csee 27,28
The following lemma gives a tool for the next discussion
Lemma 2.4 For any x, y, z ∈ R n and for any β > 0, the function d C and the mapping P r C defined
by2.14 satisfy
P r C x − x, y − Pr C x≥ 0, ∀y ∈ C, 2.15
d2C
x y≥ d2
C x d2
C
P r C x y− 2y, P r C x − x, 2.16
x − Pr C
x 1
β y
2≤ 1
β2y2− d2
C
x1
β y
, ∀x ∈ C. 2.17
Proof Inequality2.15 is obvious from the property of the projection Pr C see 27 Now,
we prove the inequality2.16 For any v ∈ C, applying 2.15 we have
v − x y2 v − Pr C x y Pr C x − x2
v − Pr C x y2 2v−P r C x y, P r C x − x C 2
v − Pr C x y2
2Pr C x − x, v − Pr C x
≥v − Pr C x y2− 2y, P r C x − x C 2.
2.18
Using the definition of d C · and noting that d2
2 and taking minimum
with respect to v ∈ C in 2.18, then we have
d2C
x y≥ d2
C
P r C x y d2
C x − 2y, P r C x − x, 2.19 which proves2.16
Trang 6From the definition of d C, we have
d C2
x 1
β y
Pr C
x 1
β y
− x −1
β y
2
1
β2
y2 −x 1
β y − Pr C
x 1
β y
−
x − Pr C
x 1
β y
2
Pr C
x 1
β y
− x −1
β y
2
1
β2
y2 −x − Pr
C
x 1
β y
2
2
x1
β y − Pr C
x 1
β y
, x − Pr C
x 1
β y
.
2.20
Since x ∈ C, applying 2.15 with Pr C x 1/βy instead of Pr C x and y x for 2.20, we obtain the last inequality inLemma 2.4
For a given integer number m ≥ 0, we consider a finite sequence of arbitrary points
{x k}m
k0⊂ C, a finite sequence of arbitrary points {w k}m
k0⊂ Rnand a finite positive sequence
{λ k}m
k0⊆ 0, ∞ Let us define
w mm
k0
λ k w k , λ mm
k0
λ k , x m 1
λ m
m
k0
λ k x k 2.21
Then upper bound of the dual gap function g Ris estimated in the following lemma
Lemma 2.5 Suppose that Assumptions (A1)–(A4) are satisfied and
w k ∈ −F x k − ∂ϕ x k 2.22
Then, for any β > 0,
i max{w, y −x | y ∈ C R 2−β/2d2
C x1/βwβR2/2, for all x ∈ C,
w∈ Rn
ii g R x m ≤ 1/λ mm
k0λ k w k , x − x k m 2− β/2d2
C x 1/βw m
βR2/2 .
Trang 7Proof i We define Lx, ρ w, y − x ρ/2R2 2 as the Lagrange function of the maximizing problem max{w, y − x | y ∈ CR} Using duality theory in convex optimization, then we have
max w, y − x| y ∈ C R
max
w, y − x| y ∈ C,y − x2≤ R2
max
y ∈C min
ρ≥0
w, y − x ρ R2−y − x2
min
ρ≥0
max
y ∈C
w, y − x−ρ
2y − x2
ρ
2R 2
min
ρ≥0
1
2ρmaxy ∈C
2− ρ2
y − x − ρ1w
2
ρ
2R 2
≤ 1
2β
2− β2min
y ∈C
y − x − β1w
2
βR2 2
1
2β
2−β
2d
2
C
x 1
β w
βR2
2 .
2.23
ii From the monotonicity of F and 2.22, we have
m
k0
λ k
F
y
, x k − y ϕ x k − ϕy ≤ −m
k0
λ k
F
x k , y − x k
ϕy
− ϕ x k
≤m
k0
λ k
w k , y − x k
≤m
k0
λ k w k , y − x m
k0
λ k
w k , x − x k
w m , y − xm
k0
λ k
w k , x − x k
.
2.24
Combining2.24,Lemma 2.5i and
g R
x m
max F
y
, x m − y ϕx m
− ϕy
| y ∈ C R
max
F
y
, 1
λ m
m
k0
λ k x k − y
ϕ 1
λ m
m
k0
λ k x k
!
− ϕy
| y ∈ C R
≤ max
1
λ m
m
k0
λ k
F
y
, x k − y ϕ x k − ϕy | y ∈ C R
1
λ m
max
m
k0
λ k
F
y
, x k − y ϕ x k − ϕy | y ∈ C R
,
2.25
Trang 8we get
g R
x m
≤ 1
λ m
max w m , y − x| y ∈ C R
m
k0
λ k
w k , x − x k
≤ 1
λ m
1
2βw m2−β
2d
2
C
x1
β w
m
βR2
2 m
k0
λ k
w k , x − x k!
.
2.26
3 Dual Algorithms
Now, we are going to build the dual interior proximal step for solvingGVI The main idea
is to construct a sequence{x k } such that the sequence g R x k tends to 0 as k → ∞ By virtue
ofLemma 2.5, we can check whether x k is an ε-solution toGVI or not
The dual interior proximal stepu k , x k , w k , w k at the iteration k ≥ 0 is generated by
using the following scheme:
u k: PrC
x 1
β w
k−1
,
x k: arg min
F
u k , y − u k
ϕy
− ϕ u k βρ k
2
y − u k2
| y ∈ C
,
w k: wk−1 1
ρ k w
k ,
3.1
where ρ k > 0 and β > 0 are given parameters, w k∈ Rnis the solution to2.22
The following lemma shows an important property of the sequenceu k , x k , s k , w k
Lemma 3.1 The sequence u k , x k , w k , w k generated by scheme 3.1 satisfies
d C2
x 1
β w
k
≥ d2
C
x 1
β w
k−1
x k − u k2
π k
C − x k2
− 2
βρ k
π C k − x k , ξ k w k
1
β2ρ2
k
w k2
2
βρ k
w k , x − x k 1
β w
k−1
,
3.2
where η k ∈ ∂ϕx k , ξ k η k Fu k and π k
C Pr C x k 1/βρ k ξ k w k As a consequence, we
have
d C2
x 1
β w
k
− d2
C
x1
β w
k−1
≥ 2
βρ k
w k , x − x k
1
β2
w k2
− 1
β2
w k−12
− 1
β2ρ2
k
ξ k w k2
.
3.3
Trang 9Proof We replace x by x 1/βy and y by 1/βz into 2.16 to obtain
d2C
x 1
β
y z≥ d2
C
x 1
β y
d2
C
P r C
x1
β y
1
β z
− 2
β
z, P r C
x1
β y
−
x1
β y
.
3.4
Using the inequality3.4 with x x, y w k−1, z 1/ρ k w k and noting that u k Pr C x
1/βw k−1, we get
d2
C
x 1
β w
k−1 1
βρ k
w k
≥ d2
C
x1
β w
k−1
d2
C
P r C
x1
β w
k−1
1
βρ k
w k
− 2
βρ k
w k , P r C
x 1
β w
k−1
− x −1
β w
k−1
.
3.5
This implies that
d2
C
x 1
β w
k
≥ d2
C
x 1
β w
k−1
d2
C
u k 1
βρ k
w k
− 2
βρ k
w k , u k − x −1
β w
k−1
. 3.6
From the subdifferentiability of the convex function ϕ to scheme 3.1, using the first-order necessary optimality condition, we have
F
u k η k βρ k
x k − u k , v − x k
≥ 0, ∀v ∈ C, 3.7
for all η k ∈ ∂ϕx k This inequality implies that
x k Pr C
u k− 1
βρ k
ξ k
where ξ k η k Fu k
Trang 10We apply inequality3.4 with x u k , y −1/ρ k ξ k and z 1/ρ k ξ k w k and using3.8 to obtain
d2C
u k 1
βρ k w k
≥ d2
C
u k− 1
βρ k ξ k
d2
C
x k 1
βρ k
ξ k w k
− 2
βρ k
ξ k w k , x k − u k 1
βρ k
ξ k
Pr C
u k− 1
βρ k ξ
k
− u k 1
βρ k ξ
k
2
d2
C
x k 1
βρ k
ξ k w k − 2
βρ k
ξ k w k , x k − u k 1
βρ k ξ
k
x k − u k 1
βρ k ξ k
2 d2
C
x k 1
βρ k
ξ k w k
2
βρ k
ξ k w k , u k− 1
βρ k
ξ k − x k
x k − u k2
1
β2ρ2k
ξ k2
2
βρ k
ξ k , x k − u k
d2
C
x k 1
βρ k
ξ k w k 2
βρ k
ξ k w k , u k− 1
βρ k ξ k − x k
.
3.9
Combine this inequality and3.6, we get
d2C
x 1
β w
k
≥ d2
C
x 1
β w
k−1
− 2
βρ k
w k , u k − x −1
β w
k−1
x k − u k2
1
β2ρ2
k
ξ k2
2
βρ k
ξ k , x k − u k
d2
C
x k 1
βρ k
ξ k w k 2
βρ k
ξ k w k , u k− 1
βρ k ξ
k − x k
.
3.10
On the other hand, if we denote π k
C Pr C x k 1/βρ k ξ k w k, then it follows that
d C2
x k 1
βρ k
ξ k w k
π k
C − x k− 1
βρ k
ξ k w k 2
π k
C − x k2
− 2
βρ k
π C k − x k , ξ k w k
1
β2ρ2
k
ξ k w k2
.
3.11
Trang 11Combine3.10 and 3.11, we get
d C2
x 1
β w
k
≥ d2
C
x 1
β w
k−1
x k − u k2
π k
C − x k2
− 2
βρ k
π C k − x k , ξ k w k
1
β2ρ2
k
w k2
2
βρ k
w k , x − x k 1
β w
k−1
,
3.12
which proves3.2
On the other hand, from3.9 we have
d2C
u k 1
βρ k w
k
≥x k − u k2
1
β2ρ k2
ξ k2
2
βρ k
ξ k , x k − u k
2
βρ k
ξ k w k , u k− 1
βρ k ξ k − x k
.
3.13
Then the inequality3.3 is deduced from this inequality and 3.6
The dual algorithm is an iterative method which generates a sequenceu k , x k , w k , w k based on scheme3.1 The algorithm is presented in detail as follows:
Algorithm 3.2 One has the following.
Initialization:
Given a tolerance ε > 0, fix an arbitrary point x
C } Take w−1: 0 and k : −1
Iterations:
For each k 0, 1, 2, , k ε, execute four steps below
Step 1 Compute a projection point u kby taking
u k: PrC
x 1
β w
k−1
Step 2 Solve the strongly convex programming problem
min
F
u k , y − u k
ϕy
β 2
y − u k2
| y ∈ C
3.15
to get the unique solution x k
... ξ k η k Fu k Trang 10We apply inequality3.4... x 1/βw m
βR2/2 .
Trang 7Proof i... k w k2
.
3.11
Trang 11Combine3.10