A NEW EXTRAGRADIENT ITERATION ALGORITHM FORBILEVEL VARIATIONAL INEQUALITIES PHAM NGOC ANH Abstract.. In this paper, we introduce an approximation extragradient itera-tion method for sol
Trang 1A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR
BILEVEL VARIATIONAL INEQUALITIES
PHAM NGOC ANH
Abstract In this paper, we introduce an approximation extragradient
itera-tion method for solving bilevel variaitera-tional inequalities involving two variaitera-tional
inequalities and we show that these problems can be solved by projection
se-quences and fixed point techniques We obtain a strong convergence of three
iteration sequences generated by this method in a real Hilbert space.
1 Introduction Let H be a real Hilbert space with an inner product h·, ·i and the induced norm
k · k, and let C be a nonempty closed convex subset of H We consider the bilevel variational inequalities(shortly BV I):
Find x∗
∈ Sol(G, C) such that hF (x∗
), x − x∗
i ≥ 0 ∀x ∈ Sol(G, C), where G : H → H, Sol(G, C) denotes the set of solutions of the following varia-tional inequalities:
Find y∗
∈ C such that hG(y∗
), y − y∗
i ≥ 0 ∀y ∈ C, and F : C → H We denote by Sol(BV I) the set of solutions of (BV I)
The problems (BV I) are also called to be quasivariational inequalities (see [8, 9, 10]) There problems are very interesting because they cover a class of mathematical programs with equilibrum constraints (see [12]), bilevel minimiza-tion problems (see [16]), variaminimiza-tional inequalities and complementarity problems (see [1, 2, 5, 7, 13])
If F ≡ 0, then the bilevel variational inequalities (BV I) become the following variational inequalities shortly V I(G, C):
Find x∗
∈ C such that hG(x∗
), x − x∗
i ≥ 0 ∀x ∈ C
Suppose that f : H → R It is well-known in convex programming that if f is convex and differentiable on Sol(G, C) then x∗
is a solution to min{f (x) | x ∈ Sol(G, C)}
Received November 19, 2010; in revised form July 7, 2011.
2010 Mathematics Subject Classification 65K10, 90C25.
Key words and phrases Bilevel variational inequalities, monotonicity, Lipschitz continuous, extragradient algorithm.
This work is supported by the Vietnam National Foundation for Science Technology Devel-opment (NAFOSTED).
Trang 2if and only if x∗
is the solution to the variational inequalities V I ∇f, Sol(G, C), where ∇f is the differentiation of f Then the bilevel variational inequalities (BV I) are written by a form of mathematical programs with equilibrum con-straints:
( min f (x)
x∈ {y∗ | hG(y∗
), z − y∗i ≥ 0 ∀z ∈ C}
If f, g are two convex and differentiable functions, then the problems (BV I) (where F := ∇f and G := ∇g) become the following bilevel minimization prob-lem (see [16]):
( min f (x)
x∈ argmin{g(x) | x ∈ C}
In recent years, variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, trans-portations, economics, nonlinear analysis (see [6, 9, 19]) Methods for solving vari-ational inequalities have been studied extensively The extragradient algorithm for solving the variational inequalities V I(G, C) was introduced by Korpelevich
in [11], where the iteration sequence {xk} is defined by
x0 ∈ C,
yk = P rC xk− ckG(xk),
xk+1= P rC xk− ckG(yk), and extended by many other authors (see [5, 9, 14, 18]) One of the main con-ditions ensures the convergence result of this method is that the cost mapping enjoys the Lipschitzian continuity property However, such a condition is rather restrictive In order to avoid it, the following Armijo-backtracking linesearch has been used to construct a hyperplane separating xk from the solution set Then the new iterate xk+1 is the projection of xk onto this hyperplane Recently, Anh and Kuno in [4] extended these results to generalized monotone nonlipschitzian multivalued variational inequalities Precisely, the authors first used the interior proximal function to develop a convergent algorithm for the multivalued varia-tional inequalities V I(F, C), where F is a generalized monotone multifunction Next the authors constructed an appropriate hyperplane which separates the cur-rent iterative point from the solution set Then the next iterate is the projection
of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set
Note that since the constraint set Sol(G, C) being the solution set of the prob-lem VI(G, C) is not explicitly given, the existing algorithms for variational in-equalities can not be directly applied because the subproblems can not be im-plemented by the available algorithms of convex programming In this paper
we extend results in [3] to the bilevel variational inequalities (BV I), but in a real Hilbert space We are interested in finding a solution to bilevel variational inequalities (BV I) where the functions F and G satisfy the following usual con-ditions:
Trang 3(A1) G is monotone on C and F is β-strongly monotone on C,
(A2) F is L1-Lipschitz continuous on C,
(A3) G is L2-Lipschitz continuous on C,
(A4) The solution set of (BV I) denoted by Sol(BV I) is nonempty
In the next section, we give a new approximation extragradient algorithm for solving problems (BV I)
2 Preliminaries
We list some known definitions and properties of the projection under the Euclidean norm which will be required in our following analysis
Definition 2.1 Let C be a nonempty closed convex subset in a real Hilbert space H We denote the projection on C by P rC(·) with images
P rC(x) = {y ∈ C | ky − xk = min
v∈Ckv − xk} ∀x ∈ H
The function ϕ : C → H is said to be
(i) γ-strongly monotone on C if for any x, y ∈ C, we have
hϕ(x) − ϕ(y), x − yi ≥ γkx − yk2, (ii) monotone on C if for any x, y ∈ C, we have
hϕ(x) − ϕ(y), x − yi ≥ 0, (iii) Lipschitz on C with constant L > 0 (shortly L-Lipschitz) if for any x, y ∈
C, we have
kϕ(x) − ϕ(y)k ≤ Lkx − yk
If ϕ : C → C and L = 1 then ϕ is called nonexpansive on C
The projection P rC(·) has the following basic properties:
(P roj1) kP rC(x) − P rC(y)k ≤ kx − yk ∀x, y ∈ H
(P roj2) kP rC(x) − P rC(y)k2 ≤ hP rC(x) − P rC(y), x − yi ∀x, y ∈ H
(P roj3) hx − P rC(x), y − P rC(y)i ≤ 0 ∀y ∈ C, x ∈ H
(P roj4) kP rC(x) − yk2 ≤ kx − yk2− kP rC(x) − xk2 ∀y ∈ C, x ∈ H
(P roj5) kP rC(x) − P rC(y)k2 ≤ kx − yk2− kP rC(x) − x + y − P rC(y)k2 ∀x, y ∈ H Now we are in a position to propose a new extragradient-type algorithm for (BV I)
Algorithm 2.2 Initialization Choose k = 0, x0 ∈ H, 0 < λ ≤ 2βL2, positive sequences {k}, {βk}, {γk}, {δk}, {λk}, {αk} and {¯k} such that
{αk} ⊂ [m, n] for some m, n ∈ (0, 1), λk≤ L1
2 ∀k ≥ 0, lim
k→∞δk = 0,
∞
P
k=0
¯
k<∞, 0 < lim inf
k→∞ βk <lim sup
k→∞
βk<1,
k+ βk+ γk= 1 ∀k ≥ 0, lim
k→∞k= 0,
∞
P
k=0
k= ∞
Trang 4Step 1 If xk ∈ Sol(BV I), then stop Otherwise compute yk = P rC xk −
λkG(xk) and zk= P rC xk− λkG(yk)
Step 2 Inner iterations j = 0, 1, · · · Compute
xk,0 = zk− λF (zk),
yk,j = P rC xk,j− δjG(xk,j),
xk,j+1 = jxk,0+ βjxk,j+ γjP rC xk,j− δjG(yk,j)
Find hksuch that khk− lim
j→∞xk,jk ≤ ¯kand set xk+1= αkxk+(1−αk)hk Step 3 Increase k by 1 and go to Step 1
Remark 2.3 If xk+1= αkxk+(1−αk)hkis substituted for xk+1 = ¯αku+ ¯βkxk+
¯
γkhk, where ¯αk, ¯βk,γ¯k ∈ [0, 1] for all k ≥ 0, u ∈ Rn and ¯αk+ ¯βk+ ¯γk = 1, then Algorithm 2.2 becomes Algorithm 2.1 in Rnproposed by Anh et al in [3] Using this fixed point technique allows us to extend the result from a finite-dimensional space Rn to a real Hilbert space H
Remark 2.4 Suppose that αk = δk = λ = 0 Then we can choose hk = zk
and it is easy to see that the sequence {xk} in Algorithm 2.2 is the well-known extragradient iteration sequence which was first introduced by Korpelevich in [11]
3 Convergence results Let C be a nonempty closed convex subset of H, G : H → H be monotone and L2-Lipschitz on C, and S : C → C be a nonexpansive mapping such that Sol(G, C) ∩ F ix(S) 6= ∅, where F ix(S) := {x ∈ C | S(x) = x} is the set of fixed points of S Let the sequences {xk} and {yk} be generated by
x0 ∈ H,
yk= P rC xk− δkG(xk),
xk+1 = kx0+ βkxk+ γkSP rC xk− δkG(yk)
∀k ≥ 0, where {k}, {βk}, {γk} and {δk} satisfy the following conditions:
δk>0 ∀k ≥ 0, lim
k→∞δk= 0,
k+ βk+ γk = 1 ∀k ≥ 0,
∞
P
k=1
k= ∞, lim
k→∞k= 0,
0 < lim inf
k→∞ βk<lim sup
k→∞
βk <1
Under these conditions, Yao et al showed that the sequences {xk} and {yk} converge strongly to the same point P rSol(G,C)∩F ix(S)(x0) in [18]
Apply these iteration sequences with S being the identity mapping, we have the following lemma
Trang 5Lemma 3.1 Suppose that the assumptions(A1) − (A4) hold Then the sequence {xk,j} generated by Algorithm 2.2 converges strongly to the point P rSol(G,C) zk−
λF(zk) as j → ∞ Consequently, we have
khk− P rSol(G,C) zk− λF (zk)k ≤ ¯k ∀k ≥ 0
Lemma 3.2 Let sequences {xk} and {zk} be generated by Algorithm 2.2, G be
L2-Lipschitz and monotone on C, and x∗
∈ Sol(G, C) Then, we have
(3.1) kzk− x∗
k2 ≤ kxk− x∗
k2− (1 − λkL2)kxk− ykk2− (1 − λkL2)kyk− zkk2 Proof Let x∗
be a solution to probems V I(G, C), x∗ ∈ C and
hG(x∗
), x − x∗
i ≥ 0 ∀x ∈ C
Then, for each λk>0, x∗
is a fixed point of mapping T (x) = P rC x− λkG(x)
on C (see [9]), i.e.,
x∗
= P rC x∗
− λkG(x∗
)
Substituting x by xk− λkG(yk) and y by x∗
into (P roj4), we get
kzk− x∗
k2≤kxk− λkG(yk) − x∗
k2− kxk− λkG(yk) − zkk2
=kxk− x∗
k2− 2λkhG(yk), xk− x∗
i + λ2kkG(yk)k2− kxk− zkk2
− λ2kkG(ykk2+ 2λkhG(yk), xk− zki
=kxk− x∗
k2− kxk− zkk2+ 2λkhG(yk), x∗
− zki
=kxk− x∗
k2− kxk− zkk2+ 2λkhG(yk) − G(x∗
), x∗
− yki + 2λkhG(x∗
), x∗
− yki + 2λkhG(yk), yk− zki
≤kxk− x∗
k2− kxk− zkk2+ 2λkhG(yk), yk− zki
(3.2)
The last inequality holds because yk ∈ C, x∗ ∈ Sol(G, C) and G is monotone on C
Substituting x by xk− λkG(xk) and y by zk into (P roj3), we have
hxk− λkG(xk) − yk, zk− yki ≤ 0
Trang 6Combining this with (3.2) and the Lipchitzian continuity of G on C with constant
L2, we obtain
kzk− x∗
k2 ≤kxk− x∗
k2− k(xk− yk) + (yk− zk)k2+ 2λkhG(yk), yk− zki
=kxk− x∗
k2− kxk− ykk2− kyk− zkk2− 2hxk− yk, yk− zki + 2λkhG(yk), yk− zki
=kxk− x∗
k2− kxk− ykk2− kyk− zkk2− 2hxk− λkG(yk) − yk, yk− zki
=kxk− x∗
k2− kxk− ykk2− kyk− zkk2− 2hxk− λkG(xk) − yk, yk− zki + 2λkhG(xk) − G(yk), zk− yki
≤kxk− x∗
k2− kxk− ykk2− kyk− zkk2+ 2λkhG(xk) − G(yk), zk− yki
≤kxk− x∗
k2− kxk− ykk2− kyk− zkk2+ 2λkkG(xk) − G(yk)kkzk− ykk
≤kxk− x∗
k2− kxk− ykk2− kyk− zkk2+ 2λkL2kxk− ykkkzk− ykk
≤kxk− x∗
k2− kxk− ykk2− kyk− zkk2+ λkL2 kxk− ykk2+ kzk− ykk2
≤kxk− x∗
k2− (1 − λkL2)kxk− ykk2− (1 − λkL2)kyk− zkk2
Lemma 3.3 Suppose that Assumptions (A1) − (A4) hold Then, the sequence {xk} generated by Algorithm 2.2 is bounded
Proof Suppose that x∗
is a solution to problems (BV I),
hF (x∗
), x − x∗
i ≥ 0 ∀x ∈ Sol(G, C),
we have
x∗
= P rSol(G,C) x∗
− λF (x∗)
Then, it follows from (P roj1), β-strongly monotonicity and L1-Lipschitz conti-nuity of F , and 0 < λ ≤ 2βL2 that
kP rSol(G,C) zk− λF (zk) − x∗
k2
= kP rSol(G,C) zk− λF (zk) − P rSol(G,C) x∗
− λF (x∗
)k2
≤ kzk− λF (zk) − x∗
+ λF (x∗
)k2
≤ kzk− x∗
k2− 2λhF (zk) − F (x∗
), zk− x∗
i + λ2kF (zk) − F (x∗
)k2
≤ (1 − 2βλ + λ2L21)kzk− x∗
k2
≤ kzk− x∗
k2 (3.3)
Trang 7It follows from λk ≤ L1
2 and (3.1) that kzk− x∗
k ≤ kxk− x∗
k Combining this with (3.3) and Assumptions 0 < λ ≤ L2β
1,
∞
P
k=0
¯
k<+∞, we have
kxk+1− x∗
k = kαkxk+ (1 − αk)hk− x∗
k
≤αkkxk− x∗
k + (1 − αk)khk− x∗
k
≤αkkxk− x∗
k + (1 − αk)khk− P rSol(G,C) zk− λF (zk)k + (1 − αk)kP rSol(G,C) zk− λF (zk) − x∗
k
≤αkkxk− x∗
k + (1 − αk)¯k+ (1 − αk)kzk− x∗
k
≤αkkxk− x∗
k + (1 − αk)¯k+ (1 − αk)kxk− x∗
k
=kxk− x∗
k + (1 − αk)¯k (3.4)
<kxk− x∗
k + ¯k
≤kx0− x∗
k +
+∞
X
k=0
¯
k
<+ ∞
Lemma 3.4 (see [9]) Let {ak} and {bk} be two positive real sequences such that
ak+1≤ ak+ bk ∀k ≥ 0 and
∞
X
k=0
bk<+∞
Then there exists lim
k→∞ak= c
Lemma 3.5 Suppose that Assumptions (A1) − (A4) hold and the sequences {xk} and {zk} are generated by Algorithm 2.2 Then, we have
kxk+1− x∗
k2 ≤kxk− x∗
k2+ 2(1 − αk)¯kkzk− x∗
k + (1 − αk)¯2k
− (1 − αk)(1 − λkL2)kxk− ykk2− (1 − αk)(1 − λkL2)kyk− zkk2 (3.5)
Consequently, we have
lim
k→∞kxk− ykk = lim
k→∞kyk− zkk = lim
k→∞kxk− zkk = 0
Proof For each k ≥ 0, Lemma 3.1 shows that there exists
lim
j→∞xk,j= P rSol(G,C) zk− λF (zk)
Trang 8Combining this with 0 < λ ≤ 2βL2, (3.1), Lemma 3.1 and x∗
∈ Sol(BV I), for
k≥ 0 we have
kxk+1− x∗
k2 =kαkxk+ (1 − αk)hk− x∗
k2
≤αkkxk− x∗
k2+ (1 − αk)khk− x∗
k2
≤αkkxk− x∗
k2+ (1 − αk)kP rSol(G,C) zk− λF (zk) − x∗
k + ¯k2
=αkkxk− x∗
k2+ (1 − αk)
× {kP rSol(G,C) zk− λF (zk) − P rSol(G,C) x∗
− λF (x∗
)k + ¯k}2
≤αkkxk− x∗
k2+ (1 − αk)q1 − 2ηλ + λ2L21kzk− x∗
k + ¯k
2
≤αkkxk− x∗
k2+ (1 − αk)kzk− x∗
k + ¯k
2
=αkkxk− x∗
k2+ (1 − αk)kzk− x∗
k2 + 2(1 − αk)¯kkzk− x∗
k + (1 − αk)¯2k
≤αkkxk− x∗
k2+ 2(1 − αk)¯kkzk− x∗
k + (1 − αk)¯2k+ (1 − αk)
×kxk− x∗
k2− (1 − λkL2)kxk− ykk2− (1 − λkL2)kyk− zkk2
=kxk− x∗
k2+ 2(1 − αk)¯kkzk− x∗
k + (1 − αk)¯2k
− (1 − αk)(1 − λkL2)kxk− ykk2− (1 − αk)(1 − λkL2)kyk− zkk2 This implies (3.5) It follows from (3.4) that
kxk+1− x∗
k ≤ kxk− x∗
k + ¯k Combining this,
∞
P
k=0
¯
k<+∞ and Lemma 3.4, there exists
k→∞kxk− x∗
k = c
Hence by (3.5), we have kxk− ykk → 0 as k → ∞ Since λk ≤ 1
L2, (3.5), (3.6) and {αk} ⊂ [m, n] for some m, n ∈ (0, 1), we obtain
(1 − αk)(1 − λkL2)kzk− ykk2 ≤kxk− x∗
k2+ 2(1 − αk)¯kkzk− x∗
k + (1 − αk)¯2k
− kxk+1− x∗
k2, and hence kzk− ykk → 0 as k → ∞ Consequently,
kxk− zkk ≤ kxk− ykk + kyk− zkk ⇒ lim
k→∞kxk− zkk = 0
2 Lemma 3.6 (see [15]) Let H be a real Hilbert space, {αk} be a sequence of real numbers such that 0 < a ≤ αk ≤ b < 1 for all k ≥ 0, and two sequences
Trang 9{xk}, {yk} in H such that
lim sup
k→∞
kxkk ≤ c, lim sup
k→∞
kykk ≤ c, lim
k→∞kαkxk+ (1 − αk)ykk = c
Then, lim
k→∞kxk− ykk = 0
Lemma 3.7 (see [17]) Let H be a real Hilbert space and C be a nonempty closed convex subset of H Let {xk} be a sequence in H Suppose that, for all x∗∈ C,
kxk+1− x∗
k ≤ kxk− x∗
k ∀k ≥ 0
Then, the sequence {P rC(xk)} converges strongly to some ¯x∈ C
Theorem 3.8 Suppose that Assumptions (A1) − (A4) hold Then three se-quences {xk}, {yk} and {zk} generated by Algorithm 2.2 converge strongly to a solution x∗
of problems (BV I) Moreover, we have
x∗
= lim
k→∞P rSol(G,C)(xk)
Proof It follows from (3.1), (3.3) and (3.6) that
lim sup
k→∞
khk− x∗
k ≤ lim sup
k→∞
{khk− P rSol(G,C) zk− λF (zk)k + kP rSol(G,C) zk− λF (zk) − x∗
k}
≤ lim sup
k→∞
{¯k+ kzk− x∗
k}
≤ lim sup
k→∞
{¯k+ kxk− x∗
k}
= c
(3.7)
Using xk+1 = αkxk+ (1 − αk)hk and {αk} ⊂ [m, n] ⊂ (0, 1), we have
k→∞kαk(xk− x∗
) + (1 − αk)(hk− x∗
)k = lim
k→∞kxk+1− x∗
k = c
Combining Lemma 3.6, (3.7) and (3.8), we have
lim
k→∞khk− xkk = 0
Consequently, we get
k→∞kxk+1− xkk = lim
k→∞(1 − αk)khk− xkk = 0
Trang 10From (P roj1), it follows that
kP rSol(G,C) yk− λF (yk)−xk+1k
≤kP rSol(G,C) yk− λF (yk) − P rSol(G,C) zk− λF (zk)k + kP rSol(G,C) zk− λF (zk) − hkk + khk− xk+1k
≤(1 + λL1)kyk− zkk + ¯k+ khk− xk+1k
=(1 + λL1)kyk− zkk + ¯k+ αk
1 − αk
kxk− xk+1k
Then, we have
kP rSol(G,C) xk− λF (xk) − xkk
≤kP rSol(G,C) xk− λF (xk) − P rSol(G,C) zk− λF (zk)k + kxk+1− xkk + kP rSol(G,C) yk− λF (yk) − xk+1k
+ kP rSol(G,C) yk− λF (yk) − P rSol(G,C) zk− λF (zk)k
≤(1 + λL1)kxk− zkk + (1 + λL1)kyk− zkk + kxk+1− xkk
+ kP rSol(G,C) yk− λF (yk) − xk+1k
≤(1 + λL1)kxk− zkk + (1 + λL1)kyk− zkk + kxk+1− xkk
(1 + λL1)kyk− zkk + ¯k+ αk
1 − αkkxk− xk+1k
≤(1 + λL1)kxk− zkk + 2(1 + λL1)kyk− zkk + ¯k+ 1
1 − αkkxk− xk+1k (3.10)
It follows from (3.9), (3.10) and Lemma 3.5 that
k→∞kP rSol(G,C) xk− λF (xk) − xkk = 0
Lemma 3.3 shows that the sequence {xk} is bounded Then, there exists M > 0 such that
(3.12) kP rSol(G,C)(xk− λF (xk)) − x∗
k ≤ M ∀k ≥ 0
Since (P roj1), F is β-strongly monotone and L1-Lipschitz continuous, we have
kP rSol(G,C)(xk− λF (xk)) − x∗
k
=kP rSol(G,C)(xk− λF (xk)) − P rSol(G,C)(x∗
− λF (x∗
))k2
≤kxk− λF (xk) − (x∗
− λF (x∗
))k2
=kxk− x∗
k2− 2λhF (xk) − F (x∗
), xk− x∗
i + λ2kF (xk) − F (x∗
)k2
≤kxk− x∗
k2− 2λβkxk− x∗
k2+ λ2L21kxk− x∗
k2