A new hybrid method for solving variational inequalities
We introduce a new method for solving variational inequalities with monotone and Lipschitzcontinuous operators acting in a Hilbert space. The iterative process based on the well-known
 projection method and the hybrid (or outer approximations) method. However, we do not use
 an extra...
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| citation_txt | A new hybrid method for solving variational inequalities / Yu.V. Malitsky, V.V. Semenov // Доповiдi Нацiональної академiї наук України. — 2014. — № 4. — С. 49-55. — Бібліогр.: 12 назв. — англ. |
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| description | We introduce a new method for solving variational inequalities with monotone and Lipschitzcontinuous operators acting in a Hilbert space. The iterative process based on the well-known
projection method and the hybrid (or outer approximations) method. However, we do not use
an extrapolation step in the projection method. The absence of one projection in our method
is explained by a slightly different choice of sets in the hybrid method. We prove the strong
convergence of the sequences generated by our method.
Запропоновано новий гiбридний метод для розв’язання варiацiйних нерiвностей з монотонними i лiпшицевими операторами, що дiють у гiльбертовому просторi. Iтерацiйний процес базується на двох добре вiдомих методах: проективному та гiбридному (або зовнiшнiх
апроксимацiй). Причому не використовується екстраполяцiйний крок у проективному методi. Вiдсутнiсть однiєї проекцiї досягається шляхом iншого вибору наборiв множин у гiбридному методi. Доведено сильну збiжнiсть породжених методом послiдовностей.
Предложен новый гибридный метод для решения вариационных неравенств с монотонными
и липшицевыми операторами, действующими в гильбертовом пространстве. Итерационный процесс основан на двух хорошо известных методах: проективном и гибридном (или
внешних аппроксимаций). Причем не используется экстраполяционный шаг в проективном
методе. Отсутствие одной проекции достигается путем иного выбора наборов множеств
в гибридном методе. Доказана сильная сходимость порожденных методом последовательностей.
|
| first_indexed | 2025-12-07T16:37:24Z |
| format | Article |
| fulltext |
UDC 517.9
Yu.V. Malitsky, V. V. Semenov
A new hybrid method for solving variational inequalities
(Presented by Corresponding Member of the NAS of Ukraine S. I. Lyashko)
We introduce a new method for solving variational inequalities with monotone and Lipschitz-
continuous operators acting in a Hilbert space. The iterative process based on the well-known
projection method and the hybrid (or outer approximations) method. However, we do not use
an extrapolation step in the projection method. The absence of one projection in our method
is explained by a slightly different choice of sets in the hybrid method. We prove the strong
convergence of the sequences generated by our method.
Introduction. Variational inequality theory is an important tool in studying a wide class of
obstacle, unilateral, and equilibrium problems arising in several branches of pure and applied
sciences in a unified general framework. This field is dynamical and is experiencing an explosive
growth in both theory and applications. Several numerical methods have been developed for sol-
ving variational inequalities and related optimization problems (see [1, 2] and references therein).
We consider the classical variational inequality problem, which is to find a point x∗ ∈ C
such that
(Ax∗, x− x∗) > 0 ∀x ∈ C, (1)
where C is a closed convex set in the Hilbert space H, (·, ·) denotes the inner product in H, and
A : H → H is some mapping. We assume that the following conditions hold:
[(C1)] The solution set of (1), denoted by S, is nonempty.
[(C2)] The mapping A is monotone on C, i. e., (Ax − Ay, x − y) > 0 ∀x, y ∈ C.
[(C3)] The mapping A is Lipschitz-continuous on C with constant L > 0, i. e., there exists
L > 0 such that ‖Ax − Ay‖ 6 L‖x − y‖ ∀x, y ∈ C.
In order to construct an algorithm which provides the strong convergence to a solution of
(1), we propose the following method:
x0, z0 ∈ C,
zn+1 = PC(xn − λAzn),
xn+1 = PCn∩Qn
x0.
(2)
Here, PM denotes the metric projection on the set M , λ ∈ (0, 1/L), and the sets Cn and Qn are
some half-spaces which will be defined in what follows.
The oldest algorithm that provides the convergence of a generated sequence under the above
assumptions is the extragradient method proposed by G.M. Korpelevich in [3]. At present,
there exist many efficient modifications of the extragradient method [4–9]. The natural question
that arises in the case of an infinite-dimensional Hilbert space is how to construct a modified
© Yu.V. Malitsky, V.V. Semenov, 2014
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №4 49
Korpelevich’s extragradient algorithm, which will provide the strong convergence. To answer this
question, Nadezhkina and Takahashi [5] introduced the following method:
x0 ∈ C,
yn = PC(xn − λnAxn),
zn = PC(xn − λnAyn),
Cn = {w ∈ C : ‖zn − w‖ 6 ‖xn − w‖},
Qn = {w ∈ C : (xn −w, x0 − xn) > 0},
xn+1 = PCn∩Qn
x0,
(3)
where λn ∈ [a, b] ⊆ (0, 1/L). Under the above assumptions (C1)–(C3), they proved that the
sequence (xn) generated by (3) converges strongly to PSx0. Their method is based on the
extragradient method and on the hybrid method proposed in [10]. The computational complexity
of (3) on every step is three computations of a metric projection and two computations of A.
Inspired by this scheme, Censor, Gibali, and Reich [11, 12] presented the following algorithm:
x0 ∈ H,
yn = PC(xn − λAxn),
Tn = {w ∈ H : (xn − λAxn − yn, w − yn) 6 0},
zn = αnxn + (1− αn)PTn
(xn − λAyn),
Cn = {w ∈ H : ‖zn −w‖ 6 ‖xn − w‖},
Qn = {w ∈ H : (xn − w, x0 − xn) > 0},
xn+1 = PCn∩Qn
x0.
(4)
In contrast to (3), the sets Cn and Qn are half-spaces. Hence, it is much more simplier to
calculate P
Cn
⋂
Qn
x0 than that on the general convex set C. Therefore, we will not take into
consideration this projection in the next schemes. On the second step, only the projection onto
the half-space Tn, rather than onto the set C like in (3), is calculated. However, on every step
of (4), we need to calculate A at two points, as well as in (3).
In this work, we show that, with some other choice of sets Cn, it is possible to throw out the
step of extrapolation in (3) or in (4), which consists in yn = PC(xn − λAxn). It is easy to see
that our method (2) on every iteration needs only one computation of the projection (as in (4))
and only one computation of A.
Preliminaries. In order to prove our main result, we need the following statements (see [2]).
At first, the following well-known properties of the projection mapping will be used throughout
this paper.
Lemma 1. Let M be a nonempty closed convex set in H, x ∈ H. Then
i) (PMx − x, y − PMx) > 0 ∀ y ∈ M ;
ii) ‖PMx − y‖2 6 ‖x − y‖2 − ‖x − PMx‖2 ∀ y ∈ M .
Two next lemmas are also well-known.
50 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №4
Lemma 2. Assume that A : C → H is a continuous and monotone mapping. Then x∗ is a
solution of (1) iff x∗ is a solution of the following problem:
findx ∈ C, such that (Ay, y − x) > 0 ∀ y ∈ C.
Remark 1. The solution set S of the variational inequality (1) is closed and convex.
We write xn ⇀ x to indicate that the sequence (xn) converges weakly to x, and xn → x
implies that (xn) converges strongly to x.
Lemma 3 (Kadec–Klee property of a Hilbert space). Let (xn) be a sequence in H. Then it
follows from ‖xn‖ → ‖x‖ and xn ⇀ x that xn → x.
At last, we need the following result.
Lemma 4. Let (an), (bn), (cn) be nonnegative real sequences, α, β ∈ R, and let, for all n ∈ N,
the inequality an 6 bn − αcn+1 + βcn hold. If
∞
∑
n=1
bn < +∞ and α > β > 0, then lim
n→∞
an = 0.
3. Algorithm and its convergence. We now formally state our algorithm.
Algorithm 1 (Hybrid algorithm without extrapolation step).
1. Choose x0, z0 ∈ C and two parameters k > 0 and λ > 0.
2. Given the current iterate xn and zn, compute
zn+1 = PC(xn − λAzn). (5)
If zn+1 = xn = zn, then stop. Otherwise, construct sets Cn and Qn as
C0 = H,
Cn =
{
w ∈ H : ‖zn+1 − w‖2 6 ‖xn − w‖2 + k‖xn − xn−1‖
2 −
−
(
1−
1
k
− λL
)
‖zn+1 − zn‖
2 + λL‖zn − zn−1‖
2
}
, n > 1,
Q0 = H,
Qn = {w ∈ H : (xn − w, x0 − xn) > 0}, n > 1,
(6)
and calculate
xn+1 = PCn∩Qn
x0.
3. Set n ← n + 1 and return to step 2.
We remark that the sets Cn look like slightly complicated in contrast to (4). However, it
is only for superficial examination; for a computation, it does not matter. In (6) and in (3),
both Cn are some half-spaces.
First, we note that the stopping criterion in Algorithm 1 is valid.
Lemma 5. If zn+1 = xn = zn in Algorithm 1, then xn ∈ S.
The next lemma is central to our proof of the convergence theorem.
Lemma 6. Let (xn) and (zn) be two sequences generated by Algorithm 1, and let z ∈ S. Then
‖zn+1− z‖2 6 ‖xn− z‖2+ k‖xn− xn−1‖
2−
(
1−
1
k
− λL
)
‖zn+1− zn‖
2 + λL‖zn − zn−1‖
2.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №4 51
Proof. By Lemma 1 we have
‖zn+1 − z‖2 6 ‖xn − λAzn − z‖2 − ‖xn − λAzn − zn+1‖
2 =
= ‖xn − z‖2 − ‖xn − zn+1‖
2 − 2λ(Azn, zn+1 − z). (7)
Since A is monotone and z ∈ S, we see that (Azn, zn − z) > 0. Thus, adding 2λ(Azn, zn − z)
to the right-hand side of (7), we get
‖zn+1 − z‖2 6 ‖xn − z‖2 − ‖xn − zn+1‖
2 − 2λ(Azn, zn+1 − zn) =
= ‖xn − z‖2 − ‖xn − xn−1‖
2 − 2(xn − xn−1, xn−1 − zn+1)−
− ‖xn−1 − zn+1‖
2 − 2λ(Azn, zn+1 − zn) = ‖xn − z‖2 − ‖xn − xn−1‖
2 −
− 2(xn − xn−1, xn−1 − zn+1)− ‖xn−1 − zn‖
2 − ‖zn − zn+1‖
2 −
− 2λ(Azn −Azn−1, zn+1 − zn) + 2(xn−1 − λAzn−1 − zn, zn+1 − zn). (8)
As zn = PC(xn−1 − λAzn−1) and zn+1 ∈ C, we have
(xn−1 − λAzn−1 − zn, zn+1 − zn) 6 0. (9)
Using the triangle, Cauchy–Schwarz, and the Cauchy inequalities, we obtain
2(xn − xn−1, xn−1 − zn+1) 6
6 ‖xn − xn−1‖
2 + ‖xn−1 − zn‖
2 + k‖xn − xn−1‖
2 +
1
k
‖zn+1 − zn‖
2. (10)
Since A is Lipschitz-continuous, we get
2λ(Azn −Azn−1, zn+1 − zn) 6 2λL‖zn − zn−1‖‖zn+1 − zn‖ 6
6 λL(‖zn+1 − zn‖
2 + ‖zn − zn−1‖
2). (11)
Combining inequalities (8)–(11), we see that
‖zn+1− z‖2 6 ‖xn− z‖2+ k‖xn− xn−1‖
2−
(
1−
1
k
− λL
)
‖zn+1− zn‖
2+ λL‖zn− zn−1‖
2,
which completes the proof.
We now can state and prove our main convergence result.
Theorem 1. Assume that (C1)–(C3) hold, and let λ ∈ (0, 1/(2L)), k > 1/(1 − 2λL). Then
the sequences (xn) and (zn) generated by Algorithm 1 converge strongly to PSx0.
Proof. It is evident that the sets Cn and Qn are closed and convex. By Lemma 6, we have
that S ⊆ Cn for all n ∈ Z
+. Let us show by induction that S ⊆ Qn for all n ∈ Z
+. For
n = 0, we have Q0 = H. Suppose that S ⊆ Qn. It is sufficient to show that S ⊆ Qn+1. Since
xn+1 = PCn∩Qn
x0 and S ⊆ Cn
⋂
Qn, it follows that (xn+1 − z, x0 − xn+1) > 0 ∀ z ∈ S. From
this by the definition of Qn, we conclude that z ∈ Qn+1 ∀ z ∈ S. Thus, S ⊆ Qn+1 and, hence,
S ⊆ Cn
⋂
Qn for all n ∈ Z
+. For this reason, the sequence (xn) is defined correctly. Let x̄ = PSx0.
52 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №4
Since xn+1 ∈ Cn
⋂
Qn and x̄ ∈ S ⊆ Cn
⋂
Qn, we have ‖xn+1 − x0‖ 6 ‖x̄− x0‖. Therefore, (xn)
is bounded. From xn+1 ∈ Cn
⋂
Qn ⊆ Qn and xn = PQn
x0, we obtain
‖xn − x0‖ 6 ‖xn+1 − x0‖. (12)
Hence, there exists lim
n→∞
‖xn−x0‖. In addition, since xn = PQn
x and xn+1 ∈ Qn, Lemma 1 yields
‖xn+1 − xn‖
2
6 ‖xn+1 − x0‖
2 − ‖xn − x0‖
2. (13)
From this, it may be concluded that the series
∞
∑
n=1
‖xn+1 − xn‖
2 is convergent. In fact, relati-
ons (13) and (12) yield
∞
∑
n=1
‖xn+1 − xn‖
2
6 ‖x̄− x0‖
2 − ‖x1 − x0‖
2 < +∞.
Since xn+1 ∈ Cn, we see that
‖zn+1 − xn+1‖
2
6 ‖xn+1 − xn‖
2 + k‖xn − xn−1‖
2 −
(
1−
1
k
− λL
)
‖zn+1 − zn‖
2 +
+ λL‖zn − zn−1‖
2.
Set an = ‖zn+1 − xn+1‖
2, bn = ‖xn+1 − xn‖
2 + k‖xn − xn−1‖
2, cn = ‖zn − zn−1‖
2, α = (1 −
− (1/k) − L), β = λL. By Lemma 6, since
∞
∑
n=1
bn < +∞ and α > β,
lim
n→∞
‖zn − xn‖ = 0.
For this reason, (zn) is bounded, and
‖zn+1 − zn‖ 6 ‖zn+1 − xn+1‖+ ‖xn+1 − xn‖+ ‖xn − zn‖ → 0.
As (xn) is bounded, there exists a subsequence (xni
) of (xn) such that (xni
) converges weakly
to some x∗ ∈ H. We will show that x∗ ∈ S. It follows from (5) by Lemma 1 that
(zni+1 − xni
+ λAzni
, y − zni+1) > 0 ∀ y ∈ C.
This is equivalent to
0 6 (zni+1 − zni
+ zni
− xni
, y − zni+1) + λ(Azni
, y − zni
) + λ(Azni
, zni
− zni+1) 6
6 (zni+1 − zni
, y − zni+1) + (zni
− xni
, y − zni+1)λ(Ay, y − zni
) +
+ λ(Azni
, zni
− zni+1) ∀ y ∈ C. (14)
In the last inequality, we used the monotonicity of A. Taking the limit in (14) as i → ∞
and using zni
⇀ x∗ ∈ C, we obtain 0 6 (Ay, y − x∗) ∀ y ∈ C. In view of Lemma 2, this
implies that x∗ ∈ S. Let us show xni
→ x∗. From x̄ = PSx0 and x∗ ∈ S, it follows that
‖x̄− x0‖ 6 ‖x
∗ − x0‖ 6 lim inf
i→∞
‖xni
− x0‖ 6 ‖x̄− x0‖. Thus, lim
i→∞
‖xni
− x0‖ = ‖x
∗ − x0‖. From
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №4 53
this and xni
− x0 ⇀ x∗ − x0 by Lemma 3, we can conclude that xni
− x0 → x∗ − x0. Therefore,
xni
→ x∗. Next, we have
‖xni
− x̄‖2 = (xni
− x0, xni
− x̄) + (x0 − x̄, xni
− x̄) 6 (x0 − x̄, xni
− x̄).
As i → ∞, we obtain ‖x∗ − x̄‖2 6 (x0 − x̄, x∗ − x̄) 6 0. Hence, we have x∗ = x̄. Since the
subsequence (xni
) was arbitrary, we see that xn → x̄. It is clear that zn → x̄.
1. Facchinei F., Pang J.-S. Finite-dimensional variational inequalities and complementarity problem. Vol. 2. –
New York: Springer, 2003. – 666 p.
2. Bauschke H.H., Combettes P. L. Convex analysis and monotone operator theory in Hilbert spaces. – Berlin:
Springer, 2011. – 408 p.
3. Korpelevich G.M. The extragradient method for finding saddle points and other problems // Ekonom. i
Matem. Metody. – 1976. – 12. – P. 747–756.
4. Khobotov E.N. Modification of the extragradient method for solving variational inequalities and certain
optimization problems // USSR Comput. Math. Math. Phys. – 1989. – 27. – P. 120–127.
5. Nadezhkina N., Takahashi W. Strong convergence theorem by a hybrid method for nonexpansive mappings
and Lipschitz-continuous monotone mappings // SIAM J. Optim. – 2006. – 16. – P. 1230–1241.
6. Iusem A.N., Nasri M. Korpelevich’s method for variational inequality problems in Banach spaces // J. of
Global Optim. – 2011. – 50. – P. 59–76.
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Received 04.09. 2013Taras Shevchenko Kiev National University, Ukraine
Ю.В. Малiцький, В.В. Семенов
Новий гiбридний метод для розв’язання варiацiйних нерiвностей
Запропоновано новий гiбридний метод для розв’язання варiацiйних нерiвностей з монотон-
ними i лiпшицевими операторами, що дiють у гiльбертовому просторi. Iтерацiйний про-
цес базується на двох добре вiдомих методах: проективному та гiбридному (або зовнiшнiх
апроксимацiй). Причому не використовується екстраполяцiйний крок у проективному ме-
тодi. Вiдсутнiсть однiєї проекцiї досягається шляхом iншого вибору наборiв множин у гiб-
ридному методi. Доведено сильну збiжнiсть породжених методом послiдовностей.
54 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №4
Ю.В. Малицкий, В.В. Семенов
Новый гибридный метод для решения вариационных неравенств
Предложен новый гибридный метод для решения вариационных неравенств с монотонными
и липшицевыми операторами, действующими в гильбертовом пространстве. Итерацион-
ный процесс основан на двух хорошо известных методах: проективном и гибридном (или
внешних аппроксимаций). Причем не используется экстраполяционный шаг в проективном
методе. Отсутствие одной проекции достигается путем иного выбора наборов множеств
в гибридном методе. Доказана сильная сходимость порожденных методом последователь-
ностей.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №4 55
|
| id | nasplib_isofts_kiev_ua-123456789-87594 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T16:37:24Z |
| publishDate | 2014 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Malitsky, Yu.V. Semenov, V.V. 2015-10-21T17:15:48Z 2015-10-21T17:15:48Z 2014 A new hybrid method for solving variational inequalities / Yu.V. Malitsky, V.V. Semenov // Доповiдi Нацiональної академiї наук України. — 2014. — № 4. — С. 49-55. — Бібліогр.: 12 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/87594 517.9 We introduce a new method for solving variational inequalities with monotone and Lipschitzcontinuous operators acting in a Hilbert space. The iterative process based on the well-known
 projection method and the hybrid (or outer approximations) method. However, we do not use
 an extrapolation step in the projection method. The absence of one projection in our method
 is explained by a slightly different choice of sets in the hybrid method. We prove the strong
 convergence of the sequences generated by our method. Запропоновано новий гiбридний метод для розв’язання варiацiйних нерiвностей з монотонними i лiпшицевими операторами, що дiють у гiльбертовому просторi. Iтерацiйний процес базується на двох добре вiдомих методах: проективному та гiбридному (або зовнiшнiх
 апроксимацiй). Причому не використовується екстраполяцiйний крок у проективному методi. Вiдсутнiсть однiєї проекцiї досягається шляхом iншого вибору наборiв множин у гiбридному методi. Доведено сильну збiжнiсть породжених методом послiдовностей. Предложен новый гибридный метод для решения вариационных неравенств с монотонными
 и липшицевыми операторами, действующими в гильбертовом пространстве. Итерационный процесс основан на двух хорошо известных методах: проективном и гибридном (или
 внешних аппроксимаций). Причем не используется экстраполяционный шаг в проективном
 методе. Отсутствие одной проекции достигается путем иного выбора наборов множеств
 в гибридном методе. Доказана сильная сходимость порожденных методом последовательностей. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Інформатика та кібернетика A new hybrid method for solving variational inequalities Новий гiбридний метод для розв’язання варiацiйних нерiвностей Новый гибридный метод для решения вариационных неравенств Article published earlier |
| spellingShingle | A new hybrid method for solving variational inequalities Malitsky, Yu.V. Semenov, V.V. Інформатика та кібернетика |
| title | A new hybrid method for solving variational inequalities |
| title_alt | Новий гiбридний метод для розв’язання варiацiйних нерiвностей Новый гибридный метод для решения вариационных неравенств |
| title_full | A new hybrid method for solving variational inequalities |
| title_fullStr | A new hybrid method for solving variational inequalities |
| title_full_unstemmed | A new hybrid method for solving variational inequalities |
| title_short | A new hybrid method for solving variational inequalities |
| title_sort | new hybrid method for solving variational inequalities |
| topic | Інформатика та кібернетика |
| topic_facet | Інформатика та кібернетика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/87594 |
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