# A Hybrid Gradient-Projection Algorithm for Averaged Mappings in Hilbert Spaces

Authors: Ming Tian; Min-Min Li
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

LetH be a real Hilbert space and C a nonempty closed and convex subset of H. Consider the following constrained convex minimization problem:
(1.1)minimizex∈Cf(x),
where f:C→ℝ is a real-valued convex and continuously Fréchet differentiable function. The gradient ∇f satisfies the following Lipschitz condition:
(1.2)‖∇f(x)-∇f(y)‖≤L‖x-y‖,∀x,y∈C,
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 references therein). It has recently been applied to solve split feasibility problems [6–10]. Levitin and Polyak [1] consider the following gradient-projection algorithm:
(1.3)xn+1:=ProjC(xn-λn∇f(xn)),n≥0.
Let {λn}n=0∞ satisfy
(1.4)0<liminf⁡n→∞λn≤limsup⁡n→∞λn<2L.
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:
(1.5)xn+1=(1-αn)xn+αnProjC(xn-λn∇f(xn)),n≥0,
converges weakly to a minimizer of (1.1) [11].Since the Lipschitz continuity of the gradient off 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,Tx-Ty〉≥ν∥Tx-Ty∥2forallx,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}i=1N 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:
(1.6)xn+1=θnh(xn)+(1-θn)ProjC(xn-λn∇f(xn)).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
(1.7)x*∈S,〈(I-h)x*,x-x*〉≥0,x∈S.On the other hand, Ming Tian [16] introduced the following general iterative algorithm for solving the variational inequality
(1.8)xn+1=αnγf(xn)+(I-μαnF)Txn,n≥0,
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
(1.9)〈(μF-γf)x~,x~-z〉≤0,z∈Fix⁡(T).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:
(1.10)xn+1=θnγh(xn)+(I-μθnF)ProjC(xn-λn∇f(xn)),n≥0.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 (VI)(1.11)x*∈S,〈(μF-γh)x*,x-x*〉≥0,∀x∈S,
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 writexn⇀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}n=1∞.Lemma 2.1 (see [17]).
Assume that{an}n=0∞ is a sequence of nonnegative real numbers such that
(2.1)an+1≤(1-γn)an+γnδn+βn,n≥0,
where {γn}n=0∞ and {βn}n=0∞ are sequences in [0,1] and {δn}n=0∞ is a sequence in ℝ such that(i)
∑n=0∞‍γn=∞;(ii)
eitherlimsupn→∞δn≤0 or ∑n=0∞‍γn|δn|<∞;(iii)
∑n=0∞‍βn<∞. Then limn→∞an=0.Lemma 2.2 (see [18]).
LetC be a closed and convex subset of a Hilbert space H, and let T:C→C be a nonexpansive mapping with FixT≠∅. If {xn}n=1∞ is a sequence in C weakly converging to x and if {(I-T)xn}n=1∞ converges strongly to y, then (I-T)x=y.Lemma 2.3.
LetH 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,
(2.2)〈x-y,(μF-γh)x-(μF-γh)y〉≥(μη-γρ)‖x-y‖2,∀x,y∈C.
That is, μF-γh is strongly monotone with coefficient μη-γρ.Lemma 2.4.
LetC be a closed subset of a real Hilbert space H, given x∈H and y∈C. Then, y=PCx if and only if there holds the inequality
(2.3)〈x-y,y-z〉≥0,∀z∈C.


## 3. Main Results

LetH 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 ∥h(x)-h(y)∥≤ρ∥x-y∥,forallx,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,
(3.1)Π={h:hisacontractionon⁡C}.
Now given h∈Π with 0<ρ<1, s∈(0,1). Let 0<μ<2η/κ2,   0<γ<μ(η-(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
(3.2)Xs(x)=sγh(x)+(I-sμF)ProjC(I-λs∇f)(x),x∈C.
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
(3.3)Xs(x)=sγh(x)+(I-sμF)Vs(x).
First observe that for s∈(0,1), we can get
(3.4)‖(I-sμF)Vs(x)-(I-sμF)Vs(y)‖2=‖Vs(x)-Vs(y)-sμ(FVs(x)-FVs(y))‖2=‖Vs(x)-Vs(y)‖2-2sμ〈Vs(x)-Vs(y),FVs(x)-FVs(y)〉+s2μ2‖FVs(x)-FVs(y)‖2≤‖x-y‖2-2sμη‖Vs(x)-Vs(y)‖2+s2μ2κ2‖Vs(x)-Vs(y)‖2≤(1-sμ(2η-sμκ2))‖x-y‖2≤(1-sμ(2η-sμκ2)2)2‖x-y‖2≤(1-sμ(η-μκ22))2‖x-y‖2=(1-sτ)2‖x-y‖2.
Indeed, we have
(3.5)‖Xs(x)-Xs(y)‖=‖sγh(x)+(I-sμF)Vs(x)-sγh(y)-(I-sμF)Vs(y)‖≤sγ‖h(x)-h(y)‖+‖(I-sμF)Vs(x)-(I-sμF)Vs(y)‖≤sγρ‖x-y‖+(1-sτ)‖x-y‖=(1-s(τ-γρ))‖x-y‖.
Hence, Xs has a unique fixed point, denoted xs, which uniquely solves the fixed-point equation
(3.6)xs=sγh(xs)+(I-sμF)Vs(xs).
The next proposition summarizes the properties of {xs}.Proposition 3.1.
Letxs be defined by (3.6).(i)
{xs}isboundedfors∈(0,(1/τ)).(ii)
lims→0∥xs-ProjC(I-λs∇f)(xs)∥=0.(iii)
xsdefinesacontinuouscurvefrom(0,1/τ)intoH.Proof.
(i) Take ax¯∈S, then we have
(3.7)‖xs-x¯‖=‖sγh(xs)+(I-sμF)ProjC(I-λs∇f)(xs)-x¯‖=‖(I-sμF)ProjC(I-λs∇f)(xs)-(I-sμF)ProjC(I-λs∇f)(x¯)+s(γh(xs)-μFProjC(I-λs∇f)(x¯))‖≤(1-sτ)‖xs-x¯‖+s‖γh(xs)-μF(x¯)‖≤(1-sτ)‖xs-x¯‖+sγρ‖xs-x¯‖+s‖γh(x¯)-μF(x¯)‖.
It follows that
(3.8)‖xs-x¯‖≤‖γh(x¯)-μF(x¯)‖τ-γρ.
Hence, {xs} is bounded.
(ii) By the definition of{xs}, we have
(3.9)‖xs-ProjC(I-λs∇f)(xs)‖=‖sγh(xs)+(I-sμF)ProjC(I-λs∇f)(xs)-ProjC(I-λs∇f)(xs)‖=s‖γh(xs)-μFProjC(I-λs∇f)(xs)‖⟶0,{xs} is bounded, so are {h(xs)} and {FProjC(I-λs∇f)(xs)}.
(iii) Takes, s0∈(0,1/τ), and we have
(3.10)‖xs-xs0‖=‖sγh(xs)+(I-sμF)ProjC(I-λs∇f)(xs)-s0γh(xs0)-(I-s0μF)ProjC(I-λs0∇f)(xs0)‖≤‖(s-s0)γh(xs)+s0γ(h(xs)-h(xs0))‖+‖(I-s0μF)ProjC(I-λs0∇f)(xs)-(I-s0μF)ProjC(I-λs0∇f)(xs0)‖+‖(I-sμF)ProjC(I-λs∇f)(xs)-(I-s0μF)ProjC(I-λs0∇f)(xs)‖≤‖(s-s0)γh(xs)+s0γ(h(xs)-h(xs0))‖+‖(I-s0μF)ProjC(I-λs0∇f)(xs)-(I-s0μF)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-s0μF)ProjC(I-λs0∇f)(xs)‖≤|s-s0|γ‖h(xs)‖+s0γρ‖xs-xs0‖+(1-s0τ)‖xs-xs0‖+|λs-λs0|‖∇f(xs)‖+‖sμFProjC(I-λs0∇f)(xs)-s0μFProjC(I-λs0∇f)(xs)‖=|s-s0|γ‖h(xs)‖+s0γρ‖xs-xs0‖+(1-s0τ)‖xs-xs0‖+|λs-λs0|‖∇f(xs)‖+|s-s0|‖μFProjC(I-λs0∇f)(xs)‖=(γ‖h(xs)‖+μ‖FProjC(I-λs0∇f)(xs)‖)|s-s0|+s0γρ‖xs-xs0‖+(1-s0τ)‖xs-xs0‖+|λs-λs0|‖∇f(xs)‖.
Therefore,
(3.11)‖xs-xs0‖≤γ‖h(xs)‖+μ‖FProjC(I-λs0∇f)(xs)‖s0(τ-γρ)|s-s0|+‖∇f(xs)‖s0(τ-γρ)|λs-λs0|.
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
(3.12)〈(μF-γh)x*,x~-x*〉≥0,∀x~∈S.
Equivalently, we have Projs(I-(μF-γh))x*=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 thatxs→x*(s→0), we write, for a given x~∈S,
(3.13)xs-x~=sγh(xs)+(I-sμF)ProjC(I-λs∇f)(xs)-x~=s(γh(xs)-μFx~)+(I-sμF)ProjC(I-λs∇f)(xs)-(I-sμF)ProjC(I-λs∇f)(x~).
It follows that
(3.14)‖xs-x~‖2=s〈γh(xs)-μFx~,xs-x~〉+〈(I-sμF)ProjC(I-λs∇f)(xs)-(I-sμF)ProjC(I-λs∇f)(x~),xs-x~〉≤(1-sτ)‖xs-x~‖2+s〈γh(xs)-μFx~,xs-x~〉.
Hence,
(3.15)‖xs-x~‖2≤1τ〈γh(xs)-μFx~,xs-x~〉≤1τ{γρ‖xs-x~‖2+〈γh(x~)-μFx~,xs-x~〉}.
To derive that
(3.16)‖xs-x~‖2≤1τ-γρ〈γh(x~)-μFx~,xs-x~〉.
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
(3.17)‖xsn-ProjC(I-λ∇f)xsn‖≤‖xsn-ProjC(I-λsn∇f)xsn‖+‖ProjC(I-λsn∇f)xsn-ProjC(I-λ∇f)xsn‖≤‖xsn-ProjC(I-λsn∇f)xsn‖+‖(λ-λsn)∇f(xsn)‖=‖xsn-ProjC(I-λsn∇f)xsn‖+|λ-λsn|‖∇f(xsn)‖.
From the boundedness of {xs} and lims→0∥ProjC(I-λs∇f)xs-xs∥=0, we conclude that
(3.18)lim⁡n→∞⁡‖xsn-ProjC(I-λ∇f)xsn‖=0.
Since xsn⇀x¯, by Lemma 2.2, we obtain
(3.19)x¯=ProjC(I-λ∇f)x¯.
This shows that x¯∈S.
We next prove thatx¯ is a solution of the variational inequality (3.12). Since
(3.20)xs=sγh(xs)+(I-sμF)ProjC(I-λs∇f)(xs),
we can derive that
(3.21)(μF-γh)(xs)=-1s(I-ProjC(I-λs∇f)(xs)+μ(F(xs)-FProjC(I-λs∇f)(xs)).
Therefore, for x~∈S,
(3.22)〈(μF-γh)(xs),xs-x~〉=-1s〈(I-ProjC(I-λs∇f))(xs)-(I-ProjC(I-λs∇f))(x~),xs-x~〉+μ〈F(xs)-FProjC(I-λs∇f)(xs),xs-x~〉≤μ〈F(xs)-FProjC(I-λs∇f)(xs),xs-x~〉.
Since ProjC(I-λs∇f) is nonexpansive, we obtain that I-ProjC(I-λs∇f) is monotone, that is,
(3.23)〈(I-ProjC(I-λs∇f))(xs)-(I-ProjC(I-λs∇f))(x~),xs-x~〉≥0.
Taking the limit through s=sn→0 ensures that x¯ is a solution to (3.12). That is to say
(3.24)〈(μF-γh)(x¯),x¯-x~〉≤0.
Hence x¯=x* by uniqueness. Therefore, xs→x* as s→0. The variational inequality (3.12) can be written as
(3.25)〈(I-μF+γh)x*-x*,x~-x*〉≤0,∀x~∈S.
So, by Lemma 2.4, it is equivalent to the fixed-point equation
(3.26)PS(I-μF+γh)x*=x*.TakingF=A, μ=1 in Theorem 3.2, we get the followingCorollary 3.3.
We have that{xs} converges in norm as s→0 to a minimizer of (1.1) which solves the variational inequality
(3.27)〈(A-γh)x*,x~-x*〉≥0,∀x~∈S.
Equivalently, we have Projs(I-(A-γh))x*=x*.TakingF=I, μ=1, γ=1 in Theorem 3.2, we get the following.Corollary 3.4.
Letzs∈H be the unique fixed point of the contraction z↦sh(z)+(1-s)ProjC(I-λs∇f)(z). Then, {zs} converges in norm as s→0 to the unique solution of the variational inequality
(3.28)〈(I-h)x*,x~-x*〉≥0,∀x~∈S.Finally, we consider the following hybrid gradient-projection algorithm,(3.29){x0∈Carbitrarily,xn+1=θnγh(xn)+(I-μθnF)ProjC(xn-λn∇f(xn)),∀n≥0.
Assume that the sequence {λn}n=0∞ satisfies the condition (1.4) and, in addition, that the following conditions are satisfied for {λn}n=0∞ and {θn}n=0∞⊂[0,1]:(i)
θn→0;(ii)
∑n=0∞‍θn=∞;(iii)
∑n=0∞‍|θn+1-θn|<∞;(iv)
∑n=0∞‍|λn+1-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}n=0∞ is bounded. Setting
(3.30)Vn:=ProjC(I-λn∇f).
Indeed, we have, for x¯∈S,
(3.31)‖xn+1-x¯‖=‖θnγh(xn)+(I-μθnF)Vnxn-x¯‖=‖θn(γh(xn)-μF(x¯))+(I-μθnF)Vnxn-(I-μθnF)Vnx¯‖≤(1-θnτ)‖xn-x¯‖+θnργ‖xn-x¯‖+θn‖γh(x¯)-μF(x¯)‖=(1-θn(τ-γρ))‖xn-x¯‖+θn‖γh(x¯)-μF(x¯)‖≤max⁡{‖xn-x¯‖,1τ-γρ‖γh(x¯)-μF(x¯)‖},∀n≥0.
By induction,
(3.32)‖xn-x¯‖≤max⁡{‖x0-x¯‖,‖γh(x¯)-μF(x¯)‖τ-γρ}.
In particular, {xn}n=0∞ is bounded.
(2) We prove that ∥xn+1-xn∥→0 as n→∞. Let M be a constant such that
(3.33)M>max⁡{sup⁡n≥0γ‖h(xn)‖,sup⁡κ,n≥0μ‖FVκxn‖,sup⁡n≥0‖∇f(xn)‖}.
We compute
(3.34)‖xn+1-xn‖=‖θnγh(xn)+(I-μθnF)Vnxn-θn-1γh(xn-1)-(I-μθn-1F)Vn-1xn-1‖=‖θnγ(h(xn)-h(xn-1))+γ(θn-θn-1)h(xn-1)+(I-μθnF)Vnxn-(I-μθnF)Vnxn-1+(I-μθnF)Vnxn-1-(I-μθn-1F)Vn-1xn-1‖=‖θnγ(h(xn)-h(xn-1))+γ(θn-θn-1)h(xn-1)+(I-μθnF)Vnxn-(I-μθnF)Vnxn-1+(I-μθnF)Vnxn-1-(I-μθnF)Vn-1xn-1+(I-μθnF)Vn-1xn-1-(I-μθn-1F)Vn-1xn-1‖≤θnγρ‖xn-xn-1‖+γ|θn-θn-1|‖h(xn-1)‖+(1-θnτ)‖xn-xn-1‖+‖Vnxn-1-Vn-1xn-1‖+μ|θn-θn-1|‖FVn-1xn-1‖≤θnγρ‖xn-xn-1‖+M|θn-θn-1|+(1-θnτ)‖xn-xn-1‖+‖Vnxn-1-Vn-1xn-1‖+M|θn-θn-1|=(1-θn(τ-γρ))‖xn-xn-1‖+2M|θn-θn-1|+‖Vnxn-1-Vn-1xn-1‖,(3.35)‖Vnxn-1-Vn-1xn-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‖=|λn-λn-1|‖∇f(xn-1)‖≤M|λn-λn-1|.
Combining (3.34) and (3.35), we can obtain
(3.36)‖xn+1-xn‖≤(1-(τ-γρ)θn)‖xn-xn-1‖+2M(|θn-θn-1|+|λn-λn-1|).
Apply Lemma 2.1 to (3.36) to conclude that ∥xn+1-xn∥→0 as n→∞.
(3) We prove that ωw(xn)⊂S. Let x^∈ωw(xn), and assume that xnj⇀x^ for some subsequence {xnj}j=1∞ of {xn}n=0∞. 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 FixV=S. It turns out that
(3.37)‖xnj-Vxnj‖≤‖xnj-Vnjxnj‖+‖Vnjxnj-Vxnj‖≤‖xnj-xnj+1‖+‖xnj+1-Vnjxnj‖+‖Vnjxnj-Vxnj‖≤‖xnj-xnj+1‖+θnj‖γh(xnj)-μFVnjxnj‖+‖ProjC(I-λnj∇f)xnj-ProjC(I-λ∇f)xnj‖≤‖xnj-xnj+1‖+θnj‖γh(xnj)-μFVnjxnj‖+|λ-λnj|‖∇f(xnj)‖≤‖xnj-xnj+1‖+2M(θnj+|λ-λnj|)⟶0asj⟶∞.
So Lemma 2.2 guarantees that ωw(xn)⊂FixV=S.
(4) We prove that xn→x* as n→∞, where x* is the unique solution of the VI (3.12). First observe that there is some x^∈ωw(xn)⊂S Such that
(3.38)limsup⁡n→∞⁡〈(μF-γh)x*,xn-x*〉=〈(μF-γh)x*,x^-x*〉≥0.
We now compute(3.39)‖xn+1-x*‖2=‖θnγh(xn)+(I-μθnF)ProjC(I-λn∇f)(xn)-x*‖2=‖θn(γh(xn)-μFx*)+(I-μθnF)Vn(xn)-(I-μθnF)Vnx*‖2=‖θnγ(h(xn)-h(x*))+(I-μθnF)Vn(xn)-(I-μθnF)Vnx*+θn(γh(x*)-μFx*)‖2≤‖θnγ(h(xn)-h(x*))+(I-μθnF)Vn(xn)-(I-μθnF)Vnx*‖2+2θn〈(γh-μF)x*,xn+1-x*〉=‖θnγ(h(xn)-h(x*))‖2+‖(I-μθnF)Vn(xn)-(I-μθnF)Vnx*‖2+2θnγ〈h(xn)-h(x*),(I-μθnF)Vn(xn)-(I-μθnF)Vnx*〉+2θn〈(γh-μF)x*,xn+1-x*〉≤θn2γ2ρ2‖xn-x*‖2+(1-θnτ)2‖xn-x*‖2+2θnγρ(1-θnτ)‖xn-x*‖2+2θn〈(γh-μF)x*,xn+1-x*〉=(θn2γ2ρ2+(1-θnτ)2+2θnγρ(1-θnτ))‖xn-x*‖2+2θn〈(γh-μF)x*,xn+1-x*〉≤(θnγ2ρ2+1-2θnτ+θnτ2+2θnγρ)‖xn-x*‖2+2θn〈(γh-μF)x*,xn+1-x*〉=(1-θn(2τ-γ2ρ2-τ2-2γρ))‖xn-x*‖2+2θn〈(γh-μF)x*,xn+1-x*〉.
Applying Lemma 2.1 to the inequality (3.39), together with (3.38), we get ∥xn-x*∥→0 as n→∞.Corollary 3.6 (see [11]).
Let{xn} be generated by the following algorithm:
(3.40)xn+1=θnh(xn)+(1-θn)ProjC(xn-λn∇f(xn)),∀n≥0.
Assume that the sequence {λn}n=0∞ 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:
(3.41)xn+1=θnγh(xn)+(I-θnA)ProjC(xn-λn∇f(xn)),∀n≥0.
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.



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*Source: 782960-2012-07-25.xml*