Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 782960, 14 pages
doi:10.1155/2012/782960
Research Article
A Hybrid Gradient-Projection Algorithm for
Averaged Mappings in Hilbert Spaces
Ming Tian and Min-Min Li
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Ming Tian, tianming1963@126.com
Received 18 April 2012; Accepted 21 June 2012
Academic Editor: Hong-Kun Xu
Copyright q 2012 M. Tian and M.-M. Li. 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.
It is well known that the gradient-projection algorithm GPA is very useful in solving constrained
convex minimization problems. In this paper, we combine a general iterative method with the
gradient-projection algorithm to propose a hybrid gradient-projection algorithm and prove that
the sequence generated by the hybrid gradient-projection algorithm converges in norm to a
minimizer of constrained convex minimization problems which solves a variational inequality.
1. Introduction
Let H be a real Hilbert space and C a nonempty closed and convex subset of H. Consider the
following constrained convex minimization problem:
minimizex∈C fx,
1.1
where f : C → R is a real-valued convex and continuously Fréchet differentiable function.
The gradient ∇f satisfies the following Lipschitz condition:
∇fx − ∇f y ≤ Lx − y,
∀x, y ∈ C,
1.2
where L > 0. Assume that the minimization problem 1.1 is consistent, and let S denote its
solution set.
It is well known that the gradient-projection algorithm is very useful in dealing with
constrained convex minimization problems and has extensively been studied 1–5 and the
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references therein. It has recently been applied to solve split feasibility problems 6–10.
Levitin and Polyak 1 consider the following gradient-projection algorithm:
xn1 : ProjC xn − λn ∇fxn ,
n ≥ 0.
1.3
2
.
L
1.4
Let {λn }∞
n0 satisfy
0 < lim inf λn ≤ lim sup λn <
n→∞
n→∞
It is proved that the sequence {xn } generated by 1.3 converges weakly to a minimizer of
1.1.
Xu proved that under certain appropriate conditions on {αn } and {λn } the sequence
{xn } defined by the following relaxed gradient-projection algorithm:
xn1 1 − αn xn αn ProjC xn − λn ∇fxn ,
n ≥ 0,
1.5
converges weakly to a minimizer of 1.1 11.
Since the Lipschitz continuity of the gradient of f implies that it is indeed inverse
strongly monotone ism 12, 13, its complement can be an averaged mapping. Recall that
a mapping T is nonexpansive if and only if it is Lipschitz with Lipschitz constant not more
than one, that a mapping is an averaged mapping if and only if it can be expressed as a
proper convex combination of the identity mapping and a nonexpansive mapping, and that
a mapping T is said to be ν-inverse strongly monotone if and only if x−y, T x−T y
≥ νT x−
T y2 for all x, y ∈ H, where the number ν > 0. Recall also that the composite of finitely many
averaged mappings is averaged. That is, if each of the mappings {Ti }N
i1 is averaged, then so is
the composite T1 · · · TN 14. In particular, an averaged mapping is a nonexpansive mapping
15. As a result, the GPA can be rewritten as the composite of a projection and an averaged
mapping which is again an averaged mapping.
Generally speaking, in infinite-dimensional Hilbert spaces, GPA has only weak
convergence. Xu 11 provided a modification of GPA so that strong convergence is
guaranteed. He considered the following hybrid gradient-projection algorithm:
xn1 θn hxn 1 − θn ProjC xn − λn ∇fxn .
1.6
It is proved that if the sequences {θn } and {λn } satisfy appropriate conditions, the
sequence {xn } generated by 1.6 converges in norm to a minimizer of 1.1 which solves the
variational inequality
x∗ ∈ S,
I − hx∗ , x − x∗ ≥ 0,
x ∈ S.
1.7
On the other hand, Ming Tian 16 introduced the following general iterative
algorithm for solving the variational inequality
xn1 αn γfxn I − μαn F T xn ,
n ≥ 0,
1.8
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3
where F is a κ-Lipschitzian and η-strongly monotone operator with κ > 0, η > 0 and
f is a contraction with coefficient 0 < α < 1. Then, he proved that if {αn } satisfying
appropriate conditions, the {xn } generated by 1.8 converges strongly to the unique solution
of variational inequality
μF − γf x,
x − z ≤ 0,
z ∈ FixT .
1.9
In this paper, motivated and inspired by the research work in this direction, we will
combine the iterative method 1.8 with the gradient-projection algorithm 1.3 and consider
the following hybrid gradient-projection algorithm:
xn1 θn γhxn I − μθn F ProjC xn − λn ∇fxn ,
n ≥ 0.
1.10
We will prove that if the sequence {θn } of parameters and the sequence {λn } of
parameters satisfy appropriate conditions, then the sequence {xn } generated by 1.10
converges in norm to a minimizer of 1.1 which solves the variational inequality V I
x∗ ∈ S,
μF − γh x∗ , x − x∗ ≥ 0,
∀x ∈ S,
1.11
where S is the solution set of the minimization problem 1.1.
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the
next section. Some of them are known; others are not hard to derive.
Throughout this paper, we write xn x to indicate that the sequence {xn } converges
weakly to x, xn → x implies that {xn } converges strongly to x. ωw xn : {x : ∃xnj x} is
the weak ω-limit set of the sequence {xn }∞
n1 .
Lemma 2.1 see 17. Assume that {an }∞
n0 is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an γn δn βn ,
n ≥ 0,
2.1
∞
∞
where {γn }∞
n0 and {βn }n0 are sequences in 0, 1 and {δn }n0 is a sequence in R such that
i ∞
n0 γn ∞;
ii either lim supn → ∞ δn ≤ 0 or ∞
n0 γn |δn | < ∞;
∞
iii n0 βn < ∞.
Then limn → ∞ an 0.
Lemma 2.2 see 18. Let C be a closed and convex subset of a Hilbert space H, and let T : C → C
be a nonexpansive mapping with Fix T /
∅. If {xn }∞
n1 is a sequence in C weakly converging to x and
∞
if {I − T xn }n1 converges strongly to y, then I − Tx y.
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Lemma 2.3. Let H be a Hilbert space, and let C be a nonempty closed and convex subset of H.
h : C → C a contraction with coefficient 0 < ρ < 1, and F : C → C a κ-Lipschitzian continuous
operator and η-strongly monotone operator with κ, η > 0. Then, for 0 < γ < μη/ρ,
2
x − y, μF − γh x − μF − γh y ≥ μη − γρ x − y ,
∀x, y ∈ C.
2.2
That is, μF − γh is strongly monotone with coefficient μη − γρ.
Lemma 2.4. Let C be a closed subset of a real Hilbert space H, given x ∈ H and y ∈ C. Then,
y PC x if and only if there holds the inequality
x − y, y − z ≥ 0,
∀z ∈ C.
2.3
3. Main Results
Let H be a real Hilbert space, and let C be a nonempty closed and convex subset of H such
that C ± C ⊂ C. Assume that the minimization problem 1.1 is consistent, and let S denote its
solution set. Assume that the gradient ∇f satisfies the Lipschitz condition 1.2. Since S is a
closed convex subset, the nearest point projection from H onto S is well defined. Recall also
that a contraction on C is a self-mapping of C such that hx − hy ≤ ρx − y, for all x, y ∈
C, where ρ ∈ 0, 1 is a constant. Let F be a κ-Lipschitzian and η-strongly monotone operator
on C with κ, η > 0. Denote by Π the collection of all contractions on C, namely,
Π {h : h is a contraction on C}.
3.1
Now given h ∈ Π with 0 < ρ < 1, s ∈ 0, 1. Let 0 < μ < 2η/κ2 , 0 < γ < μη − μκ2 /2/ρ τ/ρ. Assume that λs with respect to s is continuous and, in addition, λs ∈ a, b ⊂ 0, 2/L.
Consider a mapping Xs on C defined by
Xs x sγhx I − sμF ProjC I − λs ∇f x,
x ∈ C.
3.2
It is easy to see that Xs is a contraction. Setting Vs : ProjC I − λs ∇f. It is obvious that Vs is a
nonexpansive mapping. We can rewrite Xs x as
Xs x sγhx I − sμF Vs x.
3.3
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First observe that for s ∈ 0, 1, we can get
I − sμF Vs x − I − sμF Vs y 2
2
Vs x − Vs y − sμ FVs x − FVs y 2
Vs x − Vs y − 2sμ Vs x − Vs y , FVs x − FVs y
2
s2 μ2 FVs x − FVs y 2
2
2
≤ x − y − 2sμηVs x − Vs y s2 μ2 κ2 Vs x − Vs y 2
≤ 1 − sμ 2η − sμκ2 x − y
3.4
2
sμ 2η − sμκ2
x − y2
1−
2
≤
≤
μκ2
1 − sμ η −
2
2
x − y2
2
1 − sτ2 x − y .
Indeed, we have
Xs x − Xs y sγhx I − sμF Vs x − sγh y − I − sμF Vs y ≤ sγ hx − h y I − sμF Vs x − I − sμF Vs y ≤ sγρx − y 1 − sτx − y
1 − s τ − γρ x − y.
3.5
Hence, Xs has a unique fixed point, denoted xs , which uniquely solves the fixed-point
equation
xs sγhxs I − sμF Vs xs .
The next proposition summarizes the properties of {xs }.
Proposition 3.1. Let xs be defined by 3.6.
i {xs } is bounded for s ∈ 0, 1/τ.
ii lims → 0 xs − ProjC I − λs ∇fxs 0.
iii xs defines a continuous curve from 0, 1/τ into H.
3.6
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Proof. i Take a x ∈ S, then we have
xs − x sγhxs I − sμF ProjC I − λs ∇f xs − x
I − sμF ProjC I − λs ∇f xs − I − sμF ProjC I − λs ∇f x
s γhxs − μF ProjC I − λs ∇f x ≤ 1 − sτxs − x sγhxs − μFx
≤ 1 − sτxs − x sγρxs − x sγhx − μFx.
3.7
It follows that
γhx − μFx
.
xs − x ≤
τ − γρ
3.8
Hence, {xs } is bounded.
ii By the definition of {xs }, we have
xs − Proj I − λs ∇f xs sγhxs I − sμF Proj I − λs ∇f xs C
C
−ProjC I − λs ∇f xs sγhxs − μF Proj I − λs ∇f xs −→ 0,
C
{xs } is bounded, so are {hxs } and {F ProjC I − λs ∇fxs }.
iii Take s, s0 ∈ 0, 1/τ, and we have
xs − xs0 sγhxs I − sμF ProjC I − λs ∇f xs − s0 γhxs0 − I − s0 μF ProjC I − λs0 ∇f xs0 ≤ s − s0 γhxs s0 γhxs − hxs0 I − s0 μF ProjC I − λs0 ∇f xs − I − s0 μF ProjC I − λs0 ∇f xs0 I − sμF ProjC I − λs ∇f xs − I − s0 μF ProjC I − λs0 ∇f xs ≤ s − s0 γhxs s0 γhxs − hxs0 I − s0 μF ProjC I − λs0 ∇f xs − I − s0 μF ProjC I − λs0 ∇f xs0 I − sμF ProjC I − λs ∇f xs − I − sμF ProjC I − λs0 ∇f xs I − sμF ProjC I − λs0 ∇f xs − I − s0 μF ProjC I − λs0 ∇f xs ≤ |s − s0 |γhxs s0 γρxs − xs0 1 − s0 τxs − xs0 |λs − λs0 |∇fxs sμF ProjC I − λs0 ∇f xs − s0 μF ProjC I − λs0 ∇f xs 3.9
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|s − s0 |γhxs s0 γρxs − xs0 1 − s0 τxs − xs0 |λs − λs0 |∇fxs |s − s0 |μF ProjC I − λs0 ∇f xs γhxs μF ProjC I − λs0 ∇f xs |s − s0 |
s0 γρxs − xs0 1 − s0 τxs − xs0 |λs − λs0 |∇fxs .
3.10
Therefore,
γhxs μF ProjC I − λs0 ∇f xs |s − s0 |
xs − xs0 ≤
s0 τ − γρ
∇fxs |λs − λs0 |.
s0 τ − γρ
3.11
Therefore, xs → xs0 as s → s0 . This means xs is continuous.
Our main result in the following shows that {xs } converges in norm to a minimizer of
1.1 which solves some variational inequality.
Theorem 3.2. Assume that {xs } is defined by 3.6, then xs converges in norm as s → 0 to a
minimizer of 1.1 which solves the variational inequality
μF − γh x∗ , x − x∗ ≥ 0,
∀x ∈ S.
3.12
Equivalently, we have Projs I − μF − γhx∗ x∗ .
Proof. It is easy to see that the uniqueness of a solution of the variational inequality 3.12.
By Lemma 2.3, μF − γh is strongly monotone, so the variational inequality 3.12 has only one
solution. Let x∗ ∈ S denote the unique solution of 3.12.
To prove that xs → x∗ s → 0, we write, for a given x ∈ S,
xs − x sγhxs I − sμF ProjC I − λs ∇f xs − x
s γhxs − μF x I − sμF ProjC I − λs ∇f xs − I − sμF ProjC I − λs ∇f x.
3.13
It follows that
2 s γhxs − μF x,
xs − x
xs − x
I − sμF ProjC I − λs ∇f xs − I − sμF ProjC I − λs ∇f x,
xs − x
≤ 1 − sτxs − x
2 s γhxs − μF x,
xs − x .
3.14
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Hence,
1
xs − x
γhxs − μF x,
τ
1
γρxs − x
≤
2 γhx
− μF x,
xs − x .
τ
2≤
xs − x
3.15
To derive that
2≤
xs − x
1 γhx
− μF x,
xs − x .
τ − γρ
3.16
Since {xs } is bounded as s → 0, we see that if {sn } is a sequence in 0,1 such that sn → 0
and xsn x, then by 3.16, xsn → x. We may further assume that λsn → λ ∈ 0, 2/L due to
condition 1.4. Notice that ProjC I − λ∇f is nonexpansive. It turns out that
xs − Proj I − λ∇f xs n
n
C
≤ xsn − ProjC I − λsn ∇f xsn ProjC I − λsn ∇f xsn − ProjC I − λ∇f xsn ≤ xsn − ProjC I − λsn ∇f xsn λ − λsn ∇fxsn xsn − ProjC I − λsn ∇f xsn |λ − λsn |∇fxsn .
3.17
From the boundedness of {xs } and lims → 0 ProjC I − λs ∇fxs − xs 0, we conclude that
lim xsn − ProjC I − λ∇f xsn 0.
n→∞
3.18
Since xsn x, by Lemma 2.2, we obtain
x ProjC I − λ∇f x.
3.19
This shows that x ∈ S.
We next prove that x is a solution of the variational inequality 3.12. Since
xs sγhxs I − sμF ProjC I − λs ∇f xs ,
3.20
we can derive that
μF − γh xs −
1
I − ProjC I − λs ∇f xs μ Fxs − F ProjC I − λs ∇f xs .
s
3.21
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Therefore, for x ∈ S,
μF − γh xs , xs − x
1 xs − x
I − ProjC I − λs ∇f xs − I − ProjC I − λs ∇f x,
s
μ Fxs − F ProjC I − λs ∇f xs , xs − x
≤ μ Fxs − F ProjC I − λs ∇f xs , xs − x .
−
3.22
Since ProjC I − λs ∇f is nonexpansive, we obtain that I − ProjC I − λs ∇f is monotone, that
is,
xs − x ≥ 0.
I − ProjC I − λs ∇f xs − I − ProjC I − λs ∇f x,
3.23
Taking the limit through s sn → 0 ensures that x is a solution to 3.12. That is to say
μF − γh x, x − x ≤ 0.
3.24
Hence x x∗ by uniqueness. Therefore, xs → x∗ as s → 0. The variational inequality 3.12
can be written as
I − μF γh x∗ − x∗ , x − x∗ ≤ 0,
∀x ∈ S.
3.25
So, by Lemma 2.4, it is equivalent to the fixed-point equation
PS I − μF γh x∗ x∗ .
3.26
Taking F A, μ 1 in Theorem 3.2, we get the following
Corollary 3.3. We have that {xs } converges in norm as s → 0 to a minimizer of 1.1 which solves
the variational inequality
A − γh x∗ , x − x∗ ≥ 0,
∀x ∈ S.
3.27
Equivalently, we have Projs I − A − γhx∗ x∗ .
Taking F I, μ 1, γ 1 in Theorem 3.2, we get the following.
Corollary 3.4. Let zs ∈ H be the unique fixed point of the contraction z → shz 1 − sProjC I −
λs ∇fz. Then, {zs } converges in norm as s → 0 to the unique solution of the variational inequality
I − hx∗ , x − x∗ ≥ 0,
∀x ∈ S.
3.28
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Finally, we consider the following hybrid gradient-projection algorithm,
x0 ∈ Carbitrarily,
xn1 θn γhxn I − μθn F ProjC xn − λn ∇fxn , ∀n ≥ 0.
3.29
Assume that the sequence {λn }∞
n0 satisfies the condition 1.4 and, in addition, that the
∞
following conditions are satisfied for {λn }∞
n0 and {θn }n0 ⊂ 0, 1:
i θn → 0;
ii ∞
n0 θn ∞;
∞
iii n0 |θn1 − θn | < ∞;
iv ∞
n0 |λn1 − λn | < ∞.
Theorem 3.5. Assume that the minimization problem 1.1 is consistent and the gradient ∇f satisfies
the Lipschitz condition 1.2. Let {xn } be generated by algorithm 3.29 with the sequences {θn } and
{λn } satisfying the above conditions. Then, the sequence {xn } converges in norm to x∗ that is obtained
in Theorem 3.2.
Proof. 1 The sequence {xn }∞
n0 is bounded. Setting
Vn : ProjC I − λn ∇f .
3.30
Indeed, we have, for x ∈ S,
xn1 − x θn γhxn I − μθn F Vn xn − x
θn γhxn − μFx I − μθn F Vn xn − I − μθn F Vn x
≤ 1 − θn τxn − x θn ργxn − x θn γhx − μFx
1 − θn τ − γρ xn − x θn γhx − μFx
≤ max xn − x,
1 γhx − μFx ,
τ − γρ
3.31
∀n ≥ 0.
By induction,
γhx − μFx
.
xn − x ≤ max x0 − x,
τ − γρ
3.32
In particular, {xn }∞
n0 is bounded.
2 We prove that xn1 − xn → 0 as n → ∞. Let M be a constant such that
M > max sup γhxn , sup μFVκ xn , sup ∇fxn .
n≥0
κ,n≥0
n≥0
3.33
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We compute
xn1 − xn θn γhxn I − μθn F Vn xn − θn−1 γhxn−1 − I − μθn−1 F Vn−1 xn−1 θn γhxn − hxn−1 γθn − θn−1 hxn−1 I − μθn F Vn xn
− I − μθn F Vn xn−1 I − μθn F Vn xn−1 − I − μθn−1 F Vn−1 xn−1 θn γhxn − hxn−1 γθn − θn−1 hxn−1 I − μθn F Vn xn
− I − μθn F Vn xn−1 I − μθn F Vn xn−1 − I − μθn F Vn−1 xn−1
I − μθn F Vn−1 xn−1 − I − μθn−1 F Vn−1 xn−1 ≤ θn γρxn − xn−1 γ|θn − θn−1 |hxn−1 1 − θn τxn − xn−1 Vn xn−1 − Vn−1 xn−1 μ|θn − θn−1 |FVn−1 xn−1 ≤ θn γρxn − xn−1 M|θn − θn−1 | 1 − θn τxn − xn−1 Vn xn−1 − Vn−1 xn−1 M|θn − θn−1 |
1 − θn τ − γρ xn − xn−1 2M|θn − θn−1 | Vn xn−1 − Vn−1 xn−1 ,
3.34
Vn xn−1 − Vn−1 xn−1 ProjC I − λn ∇f xn−1 − ProjC I − λn−1 ∇f xn−1 ≤ I − λn ∇f xn−1 − I − λn−1 ∇f xn−1 3.35
|λn − λn−1 |∇fxn−1 ≤ M|λn − λn−1 |.
Combining 3.34 and 3.35, we can obtain
xn1 − xn ≤ 1 − τ − γρ θn xn − xn−1 2M|θn − θn−1 | |λn − λn−1 |.
3.36
Apply Lemma 2.1 to 3.36 to conclude that xn1 − xn → 0 as n → ∞.
3 We prove that ωw xn ⊂ S. Let x ∈ ωw xn , and assume that xnj x for some
∞
subsequence {xnj }∞
j1 of {xn }n0 . We may further assume that λnj → λ ∈ 0, 2/L due to
condition 1.4. Set V : ProjC I − λ∇f. Notice that V is nonexpansive and Fix V S. It
turns out that
xnj − V xnj ≤ xnj − Vnj xnj Vnj xnj − V xnj ≤ xnj − xnj 1 xnj 1 − Vnj xnj Vnj xnj − V xnj ≤ xnj − xnj 1 θnj γh xnj − μFVnj xnj ProjC I − λnj ∇f xnj − ProjC I − λ∇f xnj 12
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≤ xnj − xnj 1 θnj γh xnj − μFVnj xnj λ − λnj ∇f xnj ≤ xnj − xnj 1 2M θnj λ − λnj −→ 0 as j −→ ∞.
3.37
So Lemma 2.2 guarantees that ωw xn ⊂ Fix V S.
4 We prove that xn → x∗ as n → ∞, where x∗ is the unique solution of the V I 3.12.
First observe that there is some x ∈ ωw xn ⊂ S Such that
lim sup μF − γh x∗ , xn − x∗ μF − γh x∗ , x − x∗ ≥ 0.
n→∞
3.38
We now compute
2
xn1 − x∗ 2 θn γhxn I − μθn F ProjC I − λn ∇f xn − x∗ 2
θn γhxn − μFx∗ I − μθn F Vn xn − I − μθn F Vn x∗ 2
θn γhxn − hx∗ I − μθn F Vn xn − I − μθn FVn x∗ θn γhx∗ − μFx∗ 2
≤ θn γhxn − hx∗ I − μθn F Vn xn − I − μθn F Vn x∗ 2θn γh − μF x∗ , xn1 − x∗
2
2 θn γhxn − hx∗ I − μθn F Vn xn − I − μθn F Vn x∗ 2θn γhxn − hx∗ , I − μθn F Vn xn − I − μθn F Vn x∗ 2θn γh − μF x∗ , xn1 − x∗ ≤ θn2 γ 2 ρ2 xn − x∗ 2 1 − θn τ2 xn − x∗ 2 2θn γρ1 − θn τxn − x∗ 2
2θn γh − μF x∗ , xn1 − x∗
θn2 γ 2 ρ2 1 − θn τ2 2θn γρ1 − θn τ xn − x∗ 2
2θn γh − μF x∗ , xn1 − x∗
≤ θn γ 2 ρ2 1 − 2θn τ θn τ 2 2θn γρ xn − x∗ 2
2θn γh − μF x∗ , xn1 − x∗
1 − θn 2τ − γ 2 ρ2 − τ 2 − 2γρ xn − x∗ 2 2θn γh − μF x∗ , xn1 − x∗ .
3.39
Applying Lemma 2.1 to the inequality 3.39, together with 3.38, we get xn − x∗ → 0 as
n → ∞.
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13
Corollary 3.6 see 11. Let {xn } be generated by the following algorithm:
xn1 θn hxn 1 − θn ProjC xn − λn ∇fxn ,
∀n ≥ 0.
3.40
Assume that the sequence {λn }∞
n0 satisfies the conditions 1.4 and (iv) and that {θn } ⊂ 0, 1 satisfies
the conditions (i)–(iii). Then {xn } converges in norm to x∗ obtained in Corollary 3.4.
Corollary 3.7. Let {xn } be generated by the following algorithm:
xn1 θn γhxn I − θn AProjC xn − λn ∇fxn ,
∀n ≥ 0.
3.41
Assume that the sequences {θn } and {λn } satisfy the conditions contained in Theorem 3.5, then {xn }
converges in norm to x∗ obtained in Corollary 3.3.
Acknowledgments
Ming Tian is Supported in part by The Fundamental Research Funds for the Central
Universities the Special Fund of Science in Civil Aviation University of China: No.
ZXH2012 K001 and by the Science Research Foundation of Civil Aviation University of
China No. 2012KYM03.
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