Tikhonov regularization method for a system of equilibrium problems in Banach spaces
The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.
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| author | Dang, Thi Hai Ha Nguen, Byong Данг, Тхі Хай Ха Нгуєн, Бионг |
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| description | The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given. |
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UDC 517.9
Nguyen Buong (Vietnam. Acad. Sci. and Technol. Inst. Inform. Technol., Hanoi, Vietnam),
Dang Thi Hai Ha (Vietnam. Forestry Univ.)
TIKHONOV REGULARIZATION METHOD FOR SYSTEM
OF EQUILIBRIUM PROBLEMS IN BANACH SPACES*
МЕТОД РЕГУЛЯРИЗАЦIЇ ТIХОНОВА ДЛЯ СИСТЕМИ ЗАДАЧ
ПРО РIВНОВАГУ В БАНАХОВИХ ПРОСТОРАХ
The purpose of the paper is to investigate the Tikhonov regularization method for solving a system of ill-posed
equilibrium problems in Banach spaces with a posteriori regularization parameter choice. An application to
convex minimization problems with coupled constraints is also given.
Метою роботи є дослiдження методу регуляризацiї Тiхонова для розв’язку системи некоректних задач
про рiвновагу в банахових просторах з апостерiорним вибором параметра регуляризацiї. Наведено
застосування методу до задач опуклої мiнiмiзацiї iз зчепленими обмеженнями.
1. Introduction. LetX be a real reflexive Banach space,X∗ be its dual space which both
are assumed to be strictly convex, and letK be a nonempty closed (in the strong topology)
and convex subset of X. For the sake of simplicity norms of X and X∗ are denoted by
the symbol ‖.‖. Assume that the space X possesses the property: weak convergence and
convergence in norm for any sequence in X follow its strong convergence. The symbol〈
x∗, x
〉
denotes the value of the linear and continuous functional x∗ ∈ X∗ at the point
x ∈ X. Let Us, s ≥ 2, be the generalized duality mapping of the space X, i.e., Us is
the mapping from X onto X∗ satisfying the condition
〈Us(x), x〉 = ‖Us(x)‖‖x‖, ‖Us(x)‖ = ‖x‖s−1.
Concerning Us, assume that〈
Us(x)− Us(y), x− y
〉
≥ ms‖x− y‖2,
where ms is some positive number.
Let Fj , j = 1, . . . , N, be a family of bifunctions from K ×K to (−∞,+∞), i.e.,
Fj all satisfy the following set of standard properties.
Condition 1. The bifunction F is such that:
(i) F (u, u) = 0 ∀u ∈ K;
(ii) F (u, v) + F (v, u) ≤ 0 ∀(u, v) ∈ K ×K;
(iii) for every u ∈ K, F (u, .) : K → (−∞,+∞) is lower semicontinuous and
convex;
(iv) limt→+0F ((1− t)u+ tz, v) ≤ F (u, v) ∀(u, z, v) ∈ K ×K ×K.
Consider the system of equilibrium problems: find u∗ ∈ K such that
Fj(u∗, v) ≥ 0 ∀v ∈ K, j = 1, . . . , N. (1)
In the case of a single equilibrium, i.e., N = 1, problem (1) was called equilibri-
um problem, and shown in [1 – 3] to cover monotone inclusion problems, saddle point
*This work was supported by the Vietnamese Fundamental Research Program in Natural Sciences
№ 100506.
c© NGUYEN BUONG, DANG THI HAI HA, 2009
1098 ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
TIKHONOV REGULARIZATION METHOD FOR SYSTEM OF EQUILIBRIUM PROBLEMS ... 1099
problems, variational inequality problems, minimization problems, Nash equilibria in
noncooperative games, vector equilibrium problems, as well as certain fixed point
problems (see also [4]). For finding approximative solutions of (1) there exist several
approaches: the regularization approach in [5 – 8], the gap-function approach in [8 –
10], and the dynamical system or iterative procedure approach in [1, 2, 7, 11 – 21]. In
particular, this problem are considered in Banach spaces in [9, 17].
In the case N > 1, we are only aware of result [6] in Hilbert spaces where on base of
constructing the resolvant of bifunction, which is a set-valued operator, P. L. Combettes
and S. A. Hirstoaga study the block-iterative algorithms, and a regularization method
only for the particular case N = 1.
In this paper, on the base of the idea in [22] we present the Tikhonov regularization
method constructing the regularized solution, the posteriori regularization parameter
choice depending on h when Fj are given by the approximations Fhj , h > 0, in the
general case N > 1, and an application for convex minimization problem with coupled
constraints.
Set
Sj =
{
u∗ ∈ K : Fj(u∗, v) ≥ 0 ∀v ∈ H
}
, j = 1, . . . , N, S =
N⋂
j=1
Sj .
From now on, suppose that S 6= ∅. In addition, we assume that Fj all are hemicontinuous
in the variable u for each fixed v ∈ K and weakly lower semicontinuous in the variable
v for each fixed u ∈ K instead of (iv) and (iii) in condition 1, respectively.
The strong and weak convergences of any sequence are denoted by → and ⇀,
respectively.
2. Main results. First, we formulate the following facts in [1, 3] which are necessary
in the proof of our results.
Proposition 1. (i) If F (., v) is hemicontinuous for each v ∈ K and F is monotone,
i.e., satisfies (ii) in condition 1, then U∗ = V∗, where
U∗ is the solution set of F (u∗, v) ≥ 0 ∀v ∈ K,
V∗ is the solution set of F (u, v∗) ≤ 0 ∀u ∈ K, and it is convex and closed.
(ii) If F (., v) is hemicontinuous for each v ∈ K and F is strongly monotone, i.e.,
there exists a positive constant τ such that
F (u, v) + F (v, u) ≤ −τ‖u− v‖2,
then U∗ contains a unique element.
Each set Sj is closed convex (Proposition 1 (i)). Hence, S is closed convex, too.
We construct the Tikhonov regularization solution uα by solving the single equili-
brium problem
Fα(uα, v) ≥ 0 ∀v ∈ K, uα ∈ K,
Fα(u, v) :=
N∑
j=1
αµjFj(u, v) + α
〈
Us(u), v − u
〉
, α > 0, (2)
µ1 = 0 < µj < µj+1 < 1, j = 1, 2, . . . , N − 1,
and α is the regularization parameter.
We have the following results.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1100 NGUYEN BUONG, DANG THI HAI HA
Theorem 1. (i) For each α > 0, problem (2) has a unique solution uα.
(ii) limα→+0 uα = u∗, u∗ ∈ S, ‖u∗‖ ≤ ‖y‖ ∀y ∈ S.
(iii) If Fj(u, v) are bounded, i.e., there exists a positive constant C such that
|Fj(u, v)| ≤ C ∀u, v ∈ U with ‖u‖, ‖v‖ ≤ C̃, that also is a positive constant, then
we have
‖uα − uβ‖ ≤
|α− β|
2msα
‖u∗‖s−1+
+
√
|α− β|
2msα
√
|α− β|‖u∗‖2(s−1) + 4msαC(N − 1), α, β > 0.
Proof. It is easy to verify that Fα(u, v) is a bifunction, i.e., Fα(u, v) satisfies
condition 1, and strongly monotone with constant msα > 0. Therefore, (2) has a unique
solution uα for each α > 0.
Now we shall prove that
‖uα‖ ≤ ‖y‖ ∀y ∈ S. (3)
Since y ∈ S, then Fj(y, uα) ≥ 0, j = 1, . . . , N. Consequently,
N∑
j=1
αµjFj(y, uα) ≥ 0 ∀y ∈ S. (4)
This fact, uα is the solution of (2) and property (ii) in condition 1 of Fj give〈
Us(uα), y − uα
〉
≥ 0 ∀y ∈ S,
that implies (3). It means that {uα} is bounded. Let uαk
⇀ u∗ ∈ X, as k → +∞.
First, note that u∗ ∈ K, because K also is weakly closed in X. We prove that u∗ ∈ S1.
Indeed, from (ii) in condition 1 and (2) we have
F1(v, uαk
) +
N∑
j=2
α
µj
k Fj(v, uαk
) ≤ αk〈Us(uαk
), v − uαk
〉 ≤
≤ αk〈Us(v), v − uαk
〉 ∀y ∈ K.
By virtue of weak lower semicontinuous property of the bifunction Fj(u, v) in the
variable v we obtain F1(v, u∗) ≤ 0 ∀v ∈ U, i.e., u∗ ∈ S1. Now, we shall prove that
u∗ ∈ Sj , j = 2, . . . , N. From (2) and property (ii) in condition 1 of the bifunction F1 it
implies that
F2(y, uαk
) +
N∑
j=3
α
µj−µ2
k Fj(y, uαk
) ≤ αk1−µ2〈Us(y), y − uαk
〉 ∀y ∈ S1.
Tending k →∞, we have got
F2(y, u∗) ≤ 0 ∀y ∈ S1.
Therefore, F2(u∗, y) ≥ 0 ∀v ∈ S1, i.e., u∗ is a minimizer of the convex functional
F2(v, u∗) on the set S1. Since S1 ∩ S2 6= ∅, then
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
TIKHONOV REGULARIZATION METHOD FOR SYSTEM OF EQUILIBRIUM PROBLEMS ... 1101
u∗ ∈ arg min
v∈K
F2(u∗, v),
i.e., F2(u∗, y) ≥ 0 ∀y ∈ K.
Set S̃i =
i⋂
k=1
Sk. Then, S̃i is also closed convex, and S̃i 6= ∅.
Now, suppose that we have proved that u∗ ∈ S̃i, and need to show that u∗ belongs
to Si+1. Again, by virtue of (2) for y ∈ S̃i we can write
Fi+1(y, uαk
) +
N∑
j=i+2
α
µj−µi+1
k Fj(y, uαk
) ≤ αk1−µi+1〈Us(y), y − uαk
〉 ∀y ∈ S̃i.
After passing k →∞, we obtain
Fi+1(y, u∗) ≤ 0 ∀y ∈ S̃i.
Since S̃i ∩ Si+1 6= ∅, then u∗ also is an element of Si+1, i.e., Fi+1(u∗, y) ≥ 0
∀y ∈ K. Inequality (3) and the weak convergence of {uαk
} to u∗ ∈ S, which is a closed
convex subset in the strictly convex space X, give the strong convergence of {uαk
} to
u∗ : ‖u∗‖ ≤ ‖y‖ ∀y ∈ S.
Let uβ be a solution of (2) when α is replaced by β. By virtue of (ii) in condition 1
we have Fj(uα, uβ) + Fj(uβ , uα) ≤ 0. Therefore, from (2) it follows
N∑
j=1
(αµj − βµj )Fj(uα, uβ) + α〈Us(uα), uβ − uα〉+ β〈Us(uβ), uα − uβ〉 ≥ 0
or
msα‖uα − uβ‖2 ≤ |α− β|‖uβ‖s−1‖uα − uβ‖+
N∑
j=1
|αµj − βµj ||Fj(uα, uβ)|.
Using (3), the boundedness of Fj and the Lagrange’s mean-value theorem for the function
α(t) = t−µ, 0 < µ < 1, t ∈ [1,+∞), on [α, β] if α < β or [β, α] if β < α we have
conclusion (iii).
Theorem is proved.
Remark . Obviously, if uαk
→ ũ where uαk
is the solution of (2) with α = αk → 0,
as k → +∞, then S 6= ∅.
Let Fhj be the approximation bifunctions for Fj satisfy the condition
‖Fj(u, v)− Fhj (u, v)‖ ≤ hg(‖u‖)‖u− v‖, (5)
whith the bounded (image of bounded set is bounded) nonegative function g(t), t ≥ 0.
Note that condition (5) was used in the regularizing the variational inequality
〈A(x∗), x− x∗〉 ≥ 0 ∀x ∈ K, x∗ ∈ K,
where A is a hemicontinuous monotone from X into X∗, and is given approximatively
by the hemicontinuous monotone operators Ah also from X into X∗ such that∥∥Ah(x)−A(x)
∥∥ ≤ hg(‖u‖).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1102 NGUYEN BUONG, DANG THI HAI HA
By setting F̃ (u, v) = 〈A(u), v− u〉 and F̃h(u, v) = 〈Ah(u), v− u〉 we see that F̃ (u, v)
and F̃h(u, v) are the bifunctions satisfying condition (5).
Since Fhj are also the bifunctions, then the following single equilibrium problem:
Fhα (uhα, v) ≥ 0 ∀v ∈ K, uhα ∈ K,
Fhα (u, v) :=
N∑
j=1
αµjFhj (u, v) + α〈Us(u), v − u〉, α > 0,
(6)
has a unique solution denoted by uhα for each α, h > 0. As well as for the variational
inequalities [23, 24] or the operator equation of Hammerstein type [25, 26], we have the
following conclusion.
Theorem 2. If h/α→ 0 as h, α→ 0, then uhα → u∗.
Proof. From (4) with that uα is replaced by uhα, (5), (6) and the properties of the
bifunctions Fhj it follows
N∑
j=1
αµj
[
Fj(y, uhα)− Fhj (y, uhα)
]
+ α〈Us(uhα), y − uhα〉 ≥ 0 ∀y ∈ S.
Therefore,
ms‖y − uhα‖2 ≤ 〈Us(y), y − uhα〉+
1
α
N∑
j=1
αµj
∣∣Fhj (y, uhα)− Fj(y, uhα)
∣∣ =
=
〈
Us(y), y − uhα
〉
+
h
α
(N − 1)g(‖y‖)‖y − uhα‖,
for α ≤ 1. Thus,
‖y − uhα‖ ≤
1
ms
[
‖y‖s−1 +
(N − 1)h
α
g(‖y‖)
]
. (7)
It means that {uhα} is bounded, when h, α, h/α → 0. Since X is reflexive, then there
exist a subsequence {uk := uhk
αk
} ⊂ {uhα} and an element x̃ ∈ X such that uk ⇀ x̃ as
k → +∞, and K is also weak closed. Hence, the element x̃ is an element of K. By
repeating the proof in Theorem 1 we obtain that x̃ ∈ S and uhα → x̃ = u∗.
Theorem is proved.
Now, we study the problem of choosing α = α(h). For this purpose, consider the
function ρ(α) := α(a0 + t(α)), where t(α) = ‖uhα‖ for each fixed h > 0. Obviously,
from (5), (6) and property (ii) in condition 1 of Fhj it implies that
msα0‖uhα1
− uhα2
‖2 ≤ |α1 − α2|‖uhα2
‖s−1‖uhα1
− uhα2
‖+
+
N∑
j=1
|αµj
2 − α
µj
1 | |Fhj (uhα2
, uhα1
)|
for αi ∈ [α0,+∞), i = 1, 2, and α0 > 0, where∣∣Fhj (uhα2
, uhα1
)
∣∣ ≤ ∣∣Fhj (uhα2
, uhα1
)− Fj(uhα2
, uhα1
)
∣∣+ ∣∣Fj(uhα2
, uhα1
)
∣∣.
Therefore, if Fj(u, v) all satisfy condition (iii) in Theorem 1, then
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
TIKHONOV REGULARIZATION METHOD FOR SYSTEM OF EQUILIBRIUM PROBLEMS ... 1103
msα0‖uhα1
− uhα2
‖2 ≤
|α1 − α2|‖uhα2
‖s−1 + hg(‖uhα2
‖)
N∑
j=1
|αµj
2 − α
µj
1 |
×
×‖uhα1
− uhα2
‖+ C
N∑
j=1
|αµj
2 − α
µj
1 |.
Hence,
‖uhα1
− uhα2
‖ ≤ c̃,
c̃ =
d
2msα0
+
1
2msα0
√
d2 + 4msα0C(N − 1)|α1 − α2|,
d =
[
‖uhα2
‖s−1 + h(N − 1)g(‖uhα2
‖)
]
|α1 − α2|.
Thus, uhα1
→ uhα2
as α1 → α2. It means that t(α) is continuous on [α0,+∞). So, is the
function ρ(α). We shall choose α̃ = α(h) satisfying the following equation:
ρ(α) = hpα−q, p, q > 0. (8)
Theorem 3. Assume that Fj(u, v) all satisfy condition (iii) in Theorem 1. Then,
we have:
(i) for each fixed h > 0 there exists at least a value α̃ = α(h) satisfying (8),
(ii) limh→0 α(h) = 0, and
(iii) if q ≥ p, then limh→0 h/α(h) = 0.
Proof. First, from (7) we can obtain the following inequality:
αqρ(α) ≤ α1+q
[
a0 + ‖y‖+
1
ms
‖y‖s−1
]
+ αq
(N − 1)h
ms
g
(
‖y
∥∥)
for a fixed element y ∈ S. Therefore,
lim
α→+0
αqρ(α) = 0.
On the other hand,
lim
α→+∞
α1+qρ(α) ≥ a0 lim
α→+∞
αq+1 = +∞.
The intermidiate value theorem gives (i).
The second conclusion is proved by using the inequality
0 ≤ α(h) ≤ a−1/(1+q)
0 hp/(1+q)
that is followed from α1+q(h)
[
a0 + t(α(h))
]
= hp.
Since [
h
α(h)
]p
= [hpα−q(h)]αq−p(h) = ρ(α(h))αq−p(h) =
= α(h)[a0 + t(α(h))]αq−p(h) ≤
≤ [a0 + ‖y‖+
1
ms
‖y‖s−1]α1+q−p(h) + αq−p(h)
(N − 1)h
ms
g(‖y‖),
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
1104 NGUYEN BUONG, DANG THI HAI HA
then
lim
h→0
h/α(h) = 0.
Theorem is proved.
3. Application. We consider the following convex minimization problems with
coupling constraints: find u∗ ∈ K such that
ϕ(u∗) = min
u∈S
ϕ(u),
S =
{
ũ ∈ K : Fj(ũ, v) ≥ 0 ∀v ∈ K, j = 1, . . . , N
}
,
(9)
where ϕ is a weak continuous convex functional on X, and Fj all are the bifunctions.
In addition, assume that ϕ(u) ≥ 0 for each u ∈ X and is Gateau differentiable with
the derivative A. Then, u∗ solves (9) iff it solves the following variational inequality
problem:
〈A(u∗), v − u∗〉 ≥ 0 ∀v ∈ K, Fj(u∗, v) ≥ 0, j = 1, . . . N,
that is studied in [27] and [28] in the finite-dimensional Hilbert space Rn. The presence
of the functional constraints Fj(u∗, v), which couple the parameters and the variables
of the problem, is the basic distintion of this statement from the standard one. Set
FN+1(u, v) = ϕ(v)− ϕ(u).
It is easy to verify that FN+1(u, v) is a bifunction. The regularized solution of problem (9)
can be constructed by solving the single equlibrium problem
Fα(uα, v) ≥ 0 ∀v ∈ K, uα ∈ K,
Fα(u, v) :=
N+1∑
j=1
αµjFj(u, v) + α〈Us(u), v − u〉, α > 0,
µ1 = 0 < µj < µj+1 < 1, j = 2, 3, . . . , N,
and α is the regularization parameter.
Note that the nonegative property of ϕ permits to obtain the estimate (3). From the
proof of Theorem 1 it implies that ϕ(v) ≥ ϕ(u∗) ∀v ∈ S =
N⋂
j=1
Sj .
In particular, if the bifunctions Fj all are defined on the whole space X, then we
introduce additionally the bifunction F0(u, v) := dis(v,K)− dis(u,K), where
dis(x,K) = min
y∈K
‖x− y‖.
Then, we have the following single equilibrium:
Fα(uα, v) ≥ 0 ∀v ∈ X, uα ∈ X,
Fα(u, v) :=
N+1∑
j=0
αµjFj(u, v) + α〈Us(u), v − u〉, α > 0,
µ0 = 0 < µj < µj+1 < 1, j = 2, 3, . . . , N,
and α is the regularization parameter.
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TIKHONOV REGULARIZATION METHOD FOR SYSTEM OF EQUILIBRIUM PROBLEMS ... 1105
1. Bianchi M., Schaible S. Generalized monotone bifunctions and equilibrium problems // J. Optimiz.
Theory and Appl. – 1996. – 90. – P. 31 – 43.
2. Blum E., Oettli W. From optimization and variational inequalities to equilibrium problems // Math. Stud.
– 1994. – 63. – P. 123 – 145.
3. Oettli W. A remark on vector-valued equilibria and generalized monotonicity // Acta Math. Vietnam. –
1997. – 22. – P. 215 – 221.
4. Göpfert A., Riahi H., Tammer C., Zalinescu C. Variational methods in partially ordered spaces. – New
York: Springer, 2003.
5. Chadli O., Schaible S., Yao J. C. Regularized equilibrium problems with an application to noncoercive
hemivariational inequalities // J. Optimiz. Theory and Appl. – 2004. – 121. – P. 571 – 596.
6. Combettes P. L., Hirstoaga S. A. Equilibrium programming in Hilbert spaces // J. Nonlinear and Convex
Anal. – 2005. – 6, № 1. – P. 117 – 136.
7. Konnov I. V., Pinyagina O. V. D-gap functions for a class of monotone equilibrium problems in Banach
spaces // Comput. Methods Appl. Math. – 2003. – 3, № 2. – P. 274 – 286.
8. Stukalov A. C. Regularization extragradient method for solving equilibrium programming problems in
Hilbert spaces // Zh. Vych. Mat. i Mat. Fiz. – 2005. – 45, № 9. – P. 1538 – 1554.
9. Konnov I. V., Pinyagina O. V. D-gap functions and descent methods for a class of monotone equilibrium
problems // Lobachevskii J. Math. – 2003. – 13. – P. 57 – 65.
10. Mastroeni G. Gap functions for equilibrium problems // J. Global Optimiz. – 2003. – 27. – P. 411 – 426.
11. Antipin A. S. Equilibrium programming: gradient methods // Automat. and Remote Control. – 1997. –
58, № 8. – P. 1337 – 1347.
12. Antipin A. S. Equilibrium programming: Proximal methods // Zh. Vych. Mat. i Mat. Fiz. – 1997. – 37,
№ 11. – P. 1327 – 1339 (Comput. Math. and Math. Phys. – 1997. – 37, № 11. – P. 1285 – 1296.
13. Bounkhel M., Al-Senan B. R. An iterative method for nonconvex equilibrium problems // J. Inequalit.
Pure and Appl. Math. – 2006. – 7, Issue 2, Article 75.
14. Chadli O., Konnov I. V., Yao J. C. Descent methods for equilibrium problems in Banach spaces //
Comput. and Math. Appl. – 2004. – 48. – P. 609 – 616.
15. Konnov I. V. Application of the proximal point method to nonomonotone equilibrium problems // J.
Optimiz. Theory and Appl. – 2005. – 126. – P. 309 – 322.
16. Konnov I. V., Schaible S., Yao J. C. Combined relaxation method for mixed equilibrium problems // Ibid.
– 2003. – 119. – P. 317 – 333.
17. Mastroeni G. On auxiliary principle for equilibrium problems // Techn. Rept Dep. Math. Pisa Univ. –
2000. – № 3. – P. 244 – 258.
18. Moudafi A. Second-order differential proximal methods for equilibrium problems // J. Inequal. Pure and
Appl. Math. – 2003. – 4, Issue 1, Article 18.
19. Moudafi A., Théra M. Proximal and dynamical approaches to equilibrium problems // Lect. Notes Econ.
and Math. Syst. – 1999. – 477. – P. 187 – 201.
20. Noor M. A. Auxiliary principle technique for equilibrium problems // J. Optimiz. Theory and Appl. –
2004. – 122. – P. 371 – 386.
21. Noor M. A., Noor K. I. On equilibrium problems // Appl. Math. E-N7otes (AMEN). – 2004. – 4.
22. Buong Ng. Regularization for unconstrained vector optimization of convex functionals in Banach spaces
// Zh. Vych. Mat. i Mat. Fiz. – 2006. – 46, № 3. – P. 372 – 378.
23. Buong Ng. Convergence rates and finite-dimensional approximations for a class of ill-posed variational
inequalities // Ukr. Math. J. – 1996. – 48, № 9. – P. 1 – 9.
24. Buong Ng. On ill-posed problems in Banach spaces // South. Asian Bull. Math. – 1997. – 21. – P. 95 – 193.
25. Buong Ng. Operator equations of Hammerstein type under monotone perturbations // Proc. NCST
Vietnam. – 1999. – 11, № 2. – P. 3 – 7.
26. Buong Ng. Convergence rates in regularization for the case of monotone pertubations // Ukr. Math. J. –
2000. – 52, № 2. – P. 285 – 293.
27. Antipin A. S. Solution methods for variational inequalities with coupled constraints // Comput. Math.
and Math Phys. – 2000. – 40, № 9. – P. 1239 – 1254 (Transl. from Zh. Vych. Mat. i Mat. Fiz. – 2000. –
40, № 9. – P. 1291 – 1307).
28. Antipin A. S. Solving variational inequalities with coupling constraints with the use of differential
equations // Different. Equat. – 2000. – 36, № 11. – P. 1587 – 1596 (Transl. from Differents. Uravneniya.
– 2000. – 36, № 11. – P. 1443 – 1451).
Received 16.07.07
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 8
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| id | umjimathkievua-article-3083 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:55Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fa/0366643a0be1da907691035350460bfa.pdf |
| spelling | umjimathkievua-article-30832020-03-18T19:44:57Z Tikhonov regularization method for a system of equilibrium problems in Banach spaces Метод регуляризації Тіхонова для системи задач про рівновагу в банахових просторах Dang, Thi Hai Ha Nguen, Byong Данг, Тхі Хай Ха Нгуєн, Бионг The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given. Метою роботи є дослідження методу регуляризації Тіхоновадля розв'язку системи некоректних задач про рівновагу в банахових просторах з апостеріорним вибором параметра регуляризації. Наведено застосування методу до задач опуклої мiнiмiзaцiї із зчепленими обмеженнями. Institute of Mathematics, NAS of Ukraine 2009-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3083 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 8 (2009); 1098-1105 Український математичний журнал; Том 61 № 8 (2009); 1098-1105 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3083/2915 https://umj.imath.kiev.ua/index.php/umj/article/view/3083/2916 Copyright (c) 2009 Dang Thi Hai Ha; Nguen Byong |
| spellingShingle | Dang, Thi Hai Ha Nguen, Byong Данг, Тхі Хай Ха Нгуєн, Бионг Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title | Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title_alt | Метод регуляризації Тіхонова для системи задач про рівновагу в банахових просторах |
| title_full | Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title_fullStr | Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title_full_unstemmed | Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title_short | Tikhonov regularization method for a system of equilibrium problems in Banach spaces |
| title_sort | tikhonov regularization method for a system of equilibrium problems in banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3083 |
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