Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems
The purpose of this paper is to investigate an iterative regularization method of proximal point type for solving ill posed vector convex optimization problems in Hilbert spaces. Applications to the convex feasibility problems and the problem of common fixed points for nonexpansive potential mapping...
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| author | Nguen, Byong Нгуєн, Бионг |
| author_facet | Nguen, Byong Нгуєн, Бионг |
| author_sort | Nguen, Byong |
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| description | The purpose of this paper is to investigate an iterative regularization method of proximal point type for solving ill posed vector convex optimization problems in Hilbert spaces. Applications to the convex feasibility problems and the problem of common fixed points for nonexpansive potential mappings are also given. |
| first_indexed | 2026-03-24T02:38:48Z |
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UDC 517.9
Nguyen Buong (Vietnam. Acad. Sci. and Technol., Inst. Inform. Technol.)
REGULARIZATION INERTIAL PROXIMAL POINT
ALGORITHM FOR UNCONSTRAINED VECTOR CONVEX
OPTIMIZATION PROBLEMS
∗∗∗∗
REHULQRYZACIJNYJ INERCIAL|NYJ ALHORYTM
TYPU PROKSYMAL|NO} TOÇKY DLQ VEKTORNYX
OPUKLYX ZADAÇ OPTYMIZACI} BEZ OBMEÛEN|
The purpose of the paper is to investigate an iterative regularization method of proximal point type for
solving ill-posed vector convex optimization problems in Hilbert spaces. The application to the convex
feasibility problems and the common fixed points for nonexpansive potential mappings is also given.
DoslidΩeno iteratyvnyj metod rehulqryzaci] typu proksymal\no] toçky dlq rozv’qzku ne-
korektnyx vektornyx opuklyx zadaç optymizaci] u hil\bertovyx prostorax. Navedeno takoΩ
zastosuvannq metodu do zadaç opuklo] prypustymosti ta do zadaçi pro spil\ni neruxomi toçky
dlq nerozßyrnyx vidobraΩen\ potenciala.
1. Introduction. Let H be a real Hilbert space with the scalar product and the norm
denoted by the symbols 〈⋅ ⋅〉, and ⋅ , respectively.
Consider the problem of unconstrained vector convex optimization: find an ele-
ment x0 ∈ H sych that
ϕ j x( )0 = inf ( )
x H
j x
∈
ϕ ∀ j = 0, 1, … , N, (1.1)
where ϕ j are the weakly lower semi-continuous and proper convex functionals on H.
Set
Sj = x H x xj
x H
j∈ =
∈
: ( ) inf ( )ϕ ϕ , j = 0, 1, … , N, S =
Sj
j
N
=0
∩ .
Here, we suppose that S ≠ ∅, and θ ∉S, where θ is the zero element of H .
It is well known that Sj coincides with the set of solutions of the following in-
clusion:
∂ϕ j x( ) ' θ , (1.2)
and is a closed convex subset in H, where ∂ϕ j x( ) is the subdifferential of ϕ j at the
point x ∈ H and assummed to be bounded in the sense
y ≤ d1
∀ ∈ ∂
∈
y xj
x B
ϕ ( )∪ , B = x H x d∈ ≤{ }: 0
in this paper, where d0, d1 are some positive constants.
Without additional conditions on ∂ϕ j such as the strongly or uniformly monotone
property each inclusion (1.2) is ill-posed. By this we mean that the solution set Sj do-
es not depend continuously on the data ∂ϕ j . Therefore, problem (1.1) is also ill-po-
sed. To solve (1.1), in [1] when ϕ j are Gateau differentiable with the derivative Aj
( = ∂ϕ j ) , we have proposed an operator method of regularization describing by the
∗
This work was supported by the Vietnamese Fundamental Research Program in Natural Sciences
#121104.
© NGUYEN BUONG, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9 1275
1276 NGUYEN BUONG
operator equation
α αµ j A x U xj
n
j
N
( ) ( )
=
∑ +
0
= θ , (1.3)
µ0 = 0 < µj < µ j+1 < 1, j = 1, 2, … , N – 1,
depending on the regularization parameter α, where Aj
n are the monotone hemi-con-
tinuous approximations for Aj in the sense
A x A xj j
n( ) ( )− ≤ h g xn ( ),
with hn → 0, as n → + ∞ , g ( t ) is a positive bounded (image of bounded set is bo-
unded) function, and U is the normalized duality mapping of Banach space which is
the identity operator I in H . Equation (1.3) has a unique solution xn
α for each fixed
α > 0, and the sequence { }xn
α converges strongly to the solution x0 with
x0 = min
x S
x
∈
,
as hn /α , α → 0, n → + ∞ .
In this paper, we consider the regularization inertial proximal point algorithm,
where zn+1 is defined by
c A z z z zn n
j
j
n
n
j
N
n
N
n n nα α( )+
=
+
+ +∑ +
+ −1
0
1
1 1 ' γ n n nz z( )− −1 , z0, z1 ∈ H, (1.4)
where { }cn and { }γ n are the sequences of positive numbers, and Aj
n are the maxi-
mal monotone approximations for ∂ϕ j in the sense
ρ ϕ( )( ), ( )A x xj
n
j∂ ≤ h g xn ( ), (1.5)
where ρ ( P, Q ) is the Hausdorff metric for the set P and Q.
Since Aj
n are maximal monotone, then the operators in (1.4) are maximal mo-
notone (see [2]) and coercive. Hence, (1.4) has a unique solution denoted by zn+1 for
n ≥ 1.
To solve the inclusion A ( x ) ' f involving the maximal monotone operator A in
H, in [3] the proximal point algorithm
c A z f zn n n( )( )0 1 1+ +− + ' zn , z0 ∈ H , (1.6)
where cn > c0 > 0, is studied. Under some conditions { }zn converges weakly to a
solution of (1.1), if this solution is unique. R. T. Rockafellar in [3] posed an open qu-
estion whether (or not) the proximal algorithm (1.6) always converges strongly. This
question was resolved in the negative by O. Güler [4] and after by H. H. Bauschke et
al. in [5]. To obtain the strong convergence M. V. Solodov and B. F. Svaiter in [6]
have combined the proximal algorithm with simple projection step onto intersection of
two halfspaces containing solution set. Recently, to obtain the strong convergence
I. P. Ryazantseva in [7] has combined the proximal point algorithm with Tikhonov re-
gularization in the form
c A z U z f U zn
n
n n n n n( )( ) ( ) ( )+ + ++ − +1 1 1α ' U zn( ) , z0 ∈ X ,
for the case of reflexive Banach space X , where f fn − ≤ δn → 0, as n → + ∞ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
REGULARIZATION INERTIAL PROXIMAL POINT ALGORITHM … 1277
The strong convergence of { }zn of this algorithm is guaranteed by it boundedness
which is followed from the same property of the solution set of A ( x ) ' f (see [7]).
There is an open question for the case where the solution set is not bounded. For
example, the system of linear algebraic equations A x = b with a nonnegative matrix
A , det A = 0, and r ( A ) = r ( [ A , b ] ) has the unbounded solution set.
Notice that for the simple case N = 0, the algorithm (1.4) without the regularizati-
on term was proposed to solve the monotone inclusions in [8] when An
0 ≡ ∂ϕ0. Fur-
ther, this algorithm was generalized for the case An
0 = A n
0
ε , the enlargement of the
operator ∂ϕ0 in [9, 10].
In this paper, for the more general case N ≥ 0, in Sectoin 2 we shall show that the
boundedness of the sequence { }zn is automatically confirmed by combiniting the in-
ertial proximal algorithm with regularization in form (1.4). An application for the con-
vex feasibility problems and the problem of common fixed points for nonexpansive po-
tential operators is given in Section 3.
Above and below, the symbols Æ and → denote the weak convergence and
convergence in the norm, respectively.
2. Main result. First, consider the inclusion
α αn
j
j
n
j
N
n
NA x x( )
=
+∑ +
0
1 ' θ . (2.1)
Since Aj
n are the maximal monotone operators defined on H , then the operator
α αn
j
j
n
j
N
n
NA I=
+∑ +
0
1 is maximal monotone (see [3] and coercive. Hence, (2.1) has a
unique solution denoted by xn .
We have a result.
Theorem 2.1. If 0 < αn ≤ 1, hn n
N/α +1, α n → 0, as n → + ∞ , then
limn nx→+∞ = x0 ∈ S with
x xn n+ −1 = O
h hn n
n
N
n n
n
N
+
+
+
+
+
+
+ + −
1
1
1
1
1
1α
α α
α
.
Proof. When ∂ϕ j = Aj and Aj
n are hemi-continuous monotone operators, the
proof of the first part is given in [1]. For convenience, we do it here again.
From (2.1) it follows
α αn
j
j
n
n n
j
N
n
N
n nA x x x x x x( ), ,− + −
=
+∑
0
1 = 0 ∀ x ∈ S . (2.2)
On the base of (1.2), (1.5) and the monotone property of Aj
n we obtain
xn ≤ x
h
g x Nn
n
N+ ( ) ++α 1 1( ). (2.3)
Hence, { }xn is bounded. Let xnk
Æ x H∈ as k → + ∞ . First, we prove that
x S∈ 0. Indeed, by virtue of the monotone property of Ank
0 and (2.1) we can write
A x x xn
n
k
k0 ( ), − ≥ A x x xn
n n
k
k k0 ( ), − ≥
≥ α αn
j
j
n
n n
j
N
n
N
n nk
k
k k k k k
A x x x x x x( ), ,− + −
=
+∑
1
1 ≥
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1278 NGUYEN BUONG
≥ α αn
j
j
n
n
j
N
n
N
nk
k
k k k
A x x x x x x( ), ,− + −
=
+∑
1
1 ∀ x ∈ H .
By tending k → + ∞ in the last inequality we have
∂ −ϕ0( ),x x x ≥ 0 ∀ x ∈ H .
Since ∂ϕ0 is maximal monotone, then x S∈ 0. Now, we shall prove that x Sj∈ , j =
= 1, 2, … , N . Indeed, from (1.2), (2.1) and the monotone property of Ank
0 it implies
that
A x x x A x x xn
n n n
j
j
n
n n
j
N
k
k k k
k
k k1
1
2
( ), ( ),− + −−
=
∑α +
+ αn
N
n nk k k
x x x, − ≤ 0 ∀ x ∈ S0
or
A x x x A x x x x x xn
n n
j
j
n
n
j
N
n
N
n
k
k k
k
k k k1
1
2
( ), ( ), ,− + − + −−
=
∑α α ≤ 0.
After passing k → + ∞ , it gives
∂ −ϕ1( ),x x x ≤ 0 ∀ x ∈ S0 .
Thus, x is a local minimizer for ϕ1 on S0 . Since S S0 1∩ ≠ ∅, then x is also a
global minimizer for ϕ1, i.e., x S∈ 1.
Set S̃i =
Sll
i
=0∩ . Then, S̃i is also closed convex, and S̃i ≠ ∅.
Now, suppose that we have proved x Si∈ ˜ , and need to show that x belongs to
Si+1. Again, by virtue of (2.1) for x Si∈ ˜ we can write
A x x x A x x xi
n
n n n
j i
j
n
n n
j i
N
k
k k k
k
k k+
− +
= +
− + −∑1
1
2
( ), ( ),( )α + αn
N i
n nk k k
x x x− −, ≤ 0,
or
A x x x A x x x x x xi
n
n n
j i
j
n
n
j i
N
n
N i
n
k
k k
k
k k k+
− +
= +
−− + − + −∑1
1
2
( ), ( ), ,( )α α ≤ 0.
After passing k → + ∞ , it is clear that
∂ −+ϕi x x x1( ), ≤ 0 ∀ ∈x Si
˜ .
So, x Si∈ +1. It means that x S∈ . S is a closed convex subset in H , because each
Sj is closed convex. Hence, from (2.3) and
x xnk
Æ it deduces that x is the mini-
mal norm element of S . This element is unique. Consequently, all sequence { }xn
converges weakly to x . Again, from (2.2), x S∈ , and the weakly convergent
property of { }xn , we have x x≤ ∀ ∈x S . Therefore, limn nx→+∞ = x , and
x x= 0.
Now, because of (2.1),
α αn
N
n
Ny x y x+
+ +− −1
1 1 , = α α αn
N
n
N
n
Ny x x y x+
+
+
+ +− + − 〈 − 〉1
1 2
1
1 1( ) , ,
α αn
j
j
n
n
j
j
nA y A x y x+
+ − −1
1( ) ( ), = αn
j
j
n
j
nA y A x y x+
+ +− −1
1 1( ) ( ), +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
REGULARIZATION INERTIAL PROXIMAL POINT ALGORITHM … 1279
+ α α αn
j
j
n
j
n
n
j
n
j
j
nA x A x y x A x y x+
+
+− − + − −1
1
1( ) ( ), ( ),( ) ,
A x A x y xj
n
j
n+ − −1( ) ( ), ≤ ( )h h g x y xn n+ + ( ) −1
for x d≤ 0, and
a bj j− = ( )( )a b a a b ab bj j j j− + +…+ +− − − −1 2 2 1 ,
we obtain the estimation
x xn n+ −1 ≤
d
n
N n
j
n
j
j
N
1
1
1 1
1α
α α
+
+ +
=
−∑ +
+ d
h h
g xn
N
n
N
n
N
n n
n
N n
j
j
N
0
1
1 1
1
1
1
1
1 1
0
α α
α α
α+
+ +
+
+
+
+
+ +
=
−
+ + ( )∑ ≤
≤ M̃
h hn n
n
N
n n
n
N
+
+
+
+
+
+
+ + −
1
1
1
1
1
1α
α α
α
: = D̃n ,
and M̃ is some positive constant.
The theorem is proved.
Theorem 2.2. Assume that the parameters ck , γ k and αk are chosen such
that
(i) 0 0 0< < <c c Cn , 0 10≤ < <γ γn , αn ↘ 0,
(ii) α̃nn=
∞∑ 1
= + ∞ , α̃n = c cn n
N
n n
Nα α+ ++1 11( ),
(iii) γ nn n nz z=
∞
−∑ −
1 1 < + ∞ ,
(iv) lim ˜n n nD→+∞ α = lim ˜n n n n nz z→+∞ −−γ α1 = 0 where
Dn =
h hn n
n
N
n n
n
N
+
+
+
+
+
+
+ + −1
1
1
1
1
1α
α α
α
.
Then, the sequence { }zn defined by (1.5) converges strongly to the element x0 ,
as n → + ∞ .
Proof. From (1.5) and (2.1), it follows
µ αn n
j
j
n
n
j
N
nA z z( )+
=
+∑ +1
0
1 ' β β γn n n n n nz z z+ − −( )1 ,
µ αn n
j
j
n
n
j
N
nA x x( )
=
∑ +
0
' βn nx ,
µn = cn nβ , βn = 1 1 1( )+ +cn n
Nα .
Hence,
µ αn n
j
j
N
j
n
n j
n
n n n n n n nA z A x z x z x z x
=
+ + + +∑ − − + − −
0
1 1 1 1( ) ( ), , =
= β β γn n n n n n n n n n nz x z x z z z x− − + − −+ − +, ,1 1 1 .
Therefore,
z xn n+ −1 ≤ β β γn n n n n n nz x z z− + − −1 .
Consequently,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
1280 NGUYEN BUONG
z xn n+ +−1 1 ≤ z x x xn n n n+ +− + −1 1 ≤
≤ β γn n n n n n nz x z z D− + − +−1
˜ ≤ ( ˜ ) ˜1− − +αn n n nz x d ,
since βn < 1, where d̃n = γ n n n nz z D− +−1
˜ . Therefore, z xn n+ +−1 1 → 0, as
n → + ∞ is followed from the lemma.
Lemma. Let { }un , { }an , { }bn be the sequences of positive numbers satisfying
the conditions
(i) un+1 ≤ ( )1− +a u bn n n , 0 ≤ an ≤ 1,
(ii) ann=
∞∑ 0
= + ∞ , lim
n
n
n
b
a→+∞
= 0.
Then, lim
n
nu
→+∞
= 0.
On the other hand, x xn − 0 → 0, as n → + ∞ . In final, we have xn → x0 , as
n → + ∞ .
The theorem is proved.
Remark. The sequences { }αk and { }γ k which are defined by
hn = ( )1+ −n h , αn = ( )1+ −n p , 0 < 2 1p N( )+ < h < 1,
γn =
( ) , ,
, ,
1
1
0
0 0
1
1
2 1
1
+ −
+ −
− ≠
− =
− −
−
−
−
n
z z
z z
z z
z z
n n
n n
n n
n n
τ if
if
with τ > 1 1+ +p N( ) satisfy all conditions in Theorem 2.2.
3. Application. Given a finite family of weakly lower semi-continuous convex
functionals fj , j = 0, 1, … , N, find an x0 ∈ H such that
f xj ( )0 ≤ 0, j = 0, 1, … , N.
Denote by Cj = x f xj: ( ) ≤{ }0 , j = 0, 1, … , N. Then, Cj are closed convex. The
problem of finding x Cjj
N
0 0
∈ =∩ is the convex feasibility one. It is intensively
studied for the last time (see [11 – 13] and references therein), and can be rewritten in
the form of unconstrained vector convex optimization as follows. Define
ϕ j x( ) = max , ( )0 f xj{ }.
Then Cj is coincided with the set Sj .
It is easy to see that every convex program with the objective function f and the
constrain described by the functions fj , j = 0, … , N – 1, can be also rewritten in the
form of unconstrained vector optimization with fN = f .
The problem of common fixed point is formulated as follows. Find x0 ∈ C =
= Cjj
N
=0∩ , where Cj = F Tj( ), j = 0, … , N, where F Tj( ) is the fixed point set of
the nonexpansive operator Tj . It is intensively studied in recent under condition
C = F T T TN N( )− …1 0 = F T T TN N( )− …1 0 = … = F T T T TN N( )0 1 1… −
(see [14 – 16]). After, this results are generalized to Banach spaces in [17 – 19]. Evi-
dently, this condition can be replaced by the potential property of Tj , i.e., there exists
a functional f xj ( ) such that ′f xj ( ) = T xj ( ) for each j . Then, ϕ j x( ) = x 2 2 –
– f xj ( ) is convex, since its derivative I Tj− are monotone. Moreover, Sj = Cj ,
and the presented method in this paper can be applied to solve the problems.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
REGULARIZATION INERTIAL PROXIMAL POINT ALGORITHM … 1281
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Received 02.08.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 9
|
| id | umjimathkievua-article-3242 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:38:48Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/1c04da52035d36dc44e3b1e8ae9ec80d.pdf |
| spelling | umjimathkievua-article-32422020-03-18T19:48:57Z Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems Регуляризаційний інерціальний алгоритм типу проксимальної точки для векторних опуклих задач оптимізації без обмежень Nguen, Byong Нгуєн, Бионг The purpose of this paper is to investigate an iterative regularization method of proximal point type for solving ill posed vector convex optimization problems in Hilbert spaces. Applications to the convex feasibility problems and the problem of common fixed points for nonexpansive potential mappings are also given. Досліджено ітеративний метод регуляризації типу проксимальної точки для розв'язку некоректних векторних опуклих задач оптимізації у гільбертових просторах. Наведено також застосування методу до задач опуклої припустимості та до задачі про спільні нерухомі точки для нерозширних відображень потенціала. Institute of Mathematics, NAS of Ukraine 2008-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3242 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 9 (2008); 1275–1281 Український математичний журнал; Том 60 № 9 (2008); 1275–1281 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3242/3230 https://umj.imath.kiev.ua/index.php/umj/article/view/3242/3231 Copyright (c) 2008 Nguen Byong |
| spellingShingle | Nguen, Byong Нгуєн, Бионг Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title | Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title_alt | Регуляризаційний інерціальний алгоритм типу проксимальної точки для векторних опуклих задач оптимізації без обмежень |
| title_full | Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title_fullStr | Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title_full_unstemmed | Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title_short | Regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| title_sort | regularization inertial proximal point algorithm for unconstrained vector convex optimization problems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3242 |
| work_keys_str_mv | AT nguenbyong regularizationinertialproximalpointalgorithmforunconstrainedvectorconvexoptimizationproblems AT nguênbiong regularizationinertialproximalpointalgorithmforunconstrainedvectorconvexoptimizationproblems AT nguenbyong regulârizacíjnijínercíalʹnijalgoritmtipuproksimalʹnoítočkidlâvektornihopuklihzadačoptimízacííbezobmeženʹ AT nguênbiong regulârizacíjnijínercíalʹnijalgoritmtipuproksimalʹnoítočkidlâvektornihopuklihzadačoptimízacííbezobmeženʹ |