# On Solving of Constrained Convex Minimize Problem Using Gradient Projection Method

Authors: Taksaporn Sirirut; Pattanapong Tianchai
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

---

## Abstract

LetC and Q be closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let g:C→R be a strictly real-valued convex function such that the gradient ∇g is an 1/L-ism with a constant L>0. In this paper, we introduce an iterative scheme using the gradient projection method, based on Mann’s type approximation scheme for solving the constrained convex minimization problem (CCMP), that is, to find a minimizer q∈C of the function g over set C. As an application, it has been shown that the problem (CCMP) reduces to the split feasibility problem (SFP) which is to find q∈C such that Aq∈Q where A:H1→H2 is a linear bounded operator. We suggest and analyze this iterative scheme under some appropriate conditions imposed on the parameters such that another strong convergence theorems for the CCMP and the SFP are obtained. The results presented in this paper improve and extend the main results of Tian and Zhang (2017) and many others. The data availability for the proposed SFP is shown and the example of this problem is also shown through numerical results.


---

## Body

## 1. Introduction

Throughout this paper, we always assume thatC be a closed convex subset of a real Hilbert space H with inner product and norm are denoted by ·,· and ∥·∥, respectively. Let g:C→R be a strictly real-valued convex function.Consider the following constrained convex minimization problem (CCMP):(1)minx∈Cgx.Assume that (1) is consistent (that is, the CCMP has a solution) and we use U to denote its solution set. If g is Fréchet differentiable, then the gradient projection algorithm (GPA) is usually applied to solving the CCMP (1), which generates a sequence {xn} through the recursion:(2)xn+1=PCI-λ∇gxn,∀n=0,1,2,…,or more generally,(3)xn+1=PCI-λn∇gxn,∀n=0,1,2,…,where the initial guess x0∈C is chosen arbitrarily, the parameters λ or λn are positive real number, and PC is the metric projection from H onto C. It is well known that the convergence of algorithms (2) and (3) depends on the behavior of the gradient ∇g. It is known from Levitin and Polyak [1] that if ∇g is α-strongly monotone and L-Lipschitzian, that is, there exists the constants α>0 and L>0 such that (4)∇gx-∇gy,x-y≥αx-y2,∀x,y∈C,and(5)∇gx-∇gy≤Lx-y,∀x,y∈C,respectively, then, for 0<λ<2α/L2, the operator(6)T=PCI-λ∇gis a contraction; hence, the sequence {xn} defined by the GPA (2) converges in norm to the unique minimizer of the CCMP (1). More generally, for 0<λn<2α/L2 for all n=0,1,2,…, the operator(7)Gn=PCI-λn∇gis a contraction; if the sequence {λn} is chosen satisfying the property (8)0<liminfn→∞⁡λn≤limsupn→∞⁡λn<2αL2,then the sequence {xn} defined by the GPA (3) converges in norm to the unique minimizer of the CCMP (1). However, if the gradient ∇g fails to be α-strongly monotone (it means that the gradient ∇g only satisfies the L-Lipschitzian condition), then the operators T and Gn defined by (6) and (7), respectively, may fail to be contraction; consequently, the sequence {xn} generated by algorithms (2) and (3) may fail to converge strongly (see also Xu [2]) in the setting of infinite-dimensional real Hilbert space, but still converge weakly as the following statement.Theorem 1 (see [1, 2]).
Assume that the CCMP (1) is the unique consistent. Let the gradient ∇g satisfy the L-Lipschitzian condition and the sequence of the parameter {λn} satisfies the following condition:(9)0<a≤λn≤b<2L,for all n=0,1,2,…, where a and b are the constants. Then the sequence generated by the GPA (3) converges weakly to the minimizer of the CCMP (1). Indeed, the results of this theorem still hold on the gradient ∇g which satisfies an 1/L-inverse strongly monotone with L>0 (in brief, we denote 1/L-ism), that is, ∇g(x)-∇g(y),x-y≥1/L∇g(x)-∇g(y)2 for all x,y∈C, because the class of L-Lipschitzian mapping contains the class of 1/L-ism mapping.We observe from Theorem1 that if the parameter {λn} converges to λ∈(0,2/L) such that {λn} satisfies the condition (9) then q∈C solves the CCMP (1) which is the unique consistent if and only if q solves the fixed-point equation (10)q=PCI-λ∇gq.It is well known that the gradient-projection algorithm is very useful in dealing with the CCMP (1) and has extensively been studied (see [1–9] and the references therein). It has recently been applied to solve the split feasibility problems (SFP) (see [10–15]) which find applications in image reconstructions and the intensity modulated radiation therapy (see [13–18]). We now consider the following regularized minimization (that is, the CCMP (1) has the unique minimizer solution) problem: (11)minx∈Cgβnx≔gx+βn2x2,where βn>0 for all n=0,1,2,… and g:C→R is a continuous differentiable function, and we also consider the regularized gradient-projection algorithm which generates a sequence {xn} by the following recursive formula:(12)xn+1=PCI-λ∇gβnxn=PCI-λ∇g+βnIxn,∀n=0,1,2,….Many researchers studied the strong convergency theorems for solving the CCMP (1) using the sequence {xn} which is generated by algorithm (12) for their proposal on the gradient ∇g which is the class of nonexpansive mapping and the class of L-Lipschitzian mapping (see [19–25]) and in case the gradient ∇g is the class of 1/L-ism mapping such that L>0, Xu (2010) introduced the sequence {xn} which is generated by algorithm (12), and he proved that this sequence {xn} converges weakly to the minimizer of the CCMP (1) in the setting of infinite-dimensional real Hilbert space (see [15]) under some appropriate condition.Recently, Tian and Zhang (2017) introduced the sequence{xn} generated by algorithm (12), and they proved that this sequence {xn} converges strongly to the minimizer of the CCMP (1) in the same setting of infinite-dimensional real Hilbert space (see [26]) under the control conditions:(i)
0<λ<2/L+2.(ii)
{βn}⊂(0,1), limn→∞βn=0 and ∑n=0∞‍βn=∞.(iii)
∑n=0∞‍|βn+1-βn|<∞.In this paper, under the motivated and the inspired by above results, we introduce new iterative scheme, based on Mann’s type approximation scheme for solving the CCMP (1) in the case of the gradient ∇g being the class of 1/L-ism mapping such that L>0 as follows: (13)x0∈C,xn+1=αnxn+1-αnPCI-λ∇g+βnIxn,∀n=0,1,2,…,under the mild some appropriate conditions of the parameters {αn},{βn}, and λ, we obtain a strong convergency theorem to solve the CCMP (1), in which condition (iii) ∑n=0∞‍|βn+1-βn|<∞ of Tian and Zhang to be removed. In Section 4 of the applications, it has been shown that the CCMP (1) reduces to the split feasibility problem (SFP) and the data availability for the proposed SFP is shown in Section 5, and the example of this problem is also shown in Section 6 through numerical results.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real Hilbert space H. If g:C→R is a differentiable function, then we denote ∇g the gradient of the function g. We will also use the notation: → to denote the strong convergency, ⇀ to denote the weak convergency, and Fix(T)={x:x=Tx} to denote the fixed point set of the mapping T.Recall that the metric projectionPC:H→C is defined as follows: for each x∈H, PCx is the unique point in C satisfying (14)x-PCx=inf⁡x-y:y∈C. Let g:C→R be a function. Recall that the function g is a strictly real-valued convex function if (15)gλx+1-λy<λgx+1-λgy,∀λ∈0,1,∀x,y∈C, such that x≠y. We collect together some known lemmas and definitions which are our main tool in proving our results.Lemma 2.
LetH be a real Hilbert space. Then, for all x,y∈H, (16)x+y2=x2+2x,y+y2.Lemma 3 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Then, (17)z=PCx⟺x-z,z-y≥0,∀x∈H,y∈C.Definition 4.
LetH be a real Hilbert space. The operator T:H→H is called(i)
L-Lipschitzian with L>0 if (18)Tx-Ty≤Lx-y,∀x,y∈H,(ii)
k-contraction with a positive real number k such that k∈(0,1) if (19)Tx-Ty≤kx-y,∀x,y∈H,(iii)
nonexpansive if(20)Tx-Ty≤x-y,∀x,y∈H,(iv)
monotone if(21)Tx-Ty,x-y≥0,∀x,y∈H,(v)
α-strongly monotone if (22)Tx-Ty,x-y≥αx-y2,∀x,y∈H,(vi)
α-inverse strongly monotone (α-ism) if (23)Tx-Ty,x-y≥αTx-Ty2,∀x,y∈H,(vii)
firmly nonexpansive if(24)Tx-Ty2≤x-y2-I-Tx-I-Ty2,∀x,y∈H.Lemma 5 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Then, for x,y∈C and α∈[0,1], we have (25)αx+1-αy2=αx2+1-αy2-α1-αx-y2.Lemma 6 (see [28]).
LetH be a real Hilbert space and T:H→H be an operator. The following statements are equivalent: (i)
T is firmly nonexpansive,(ii)
Tx-Ty2≤x-y,Tx-Ty, ∀x,y∈H,(iii)
I-T is firmly nonexpansive.Lemma 7 (see [28]).
LetH and K be two real Hilbert spaces and let T:K→K be a firmly nonexpansive mapping such that (I-T)x is a convex function from K to R¯=[-∞,+∞]. Let A:H→K be a bounded linear operator and f(x)=1/2(I-T)Ax2 for all x∈H. Then, (i)
∇f(x)=A∗(I-T)Ax for all x∈H where A∗ is adjoint operator of A,(ii)
∇f is A2-Lipschitzian.Lemma 8 (see [29, 30]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Tn} and φ be two classes of nonexpansive mappings from C into C such that (26)∅≠Fixφ=⋂n=0∞FixTn. Then, for any bounded sequence {zn}⊂C, we have (i)
iflimn→∞zn-Tnzn=0 then limn→∞zn-Tzn=0 for all T∈φ, which is called that the NST-condition(I),(ii)
iflimn→∞zn+1-Tnzn=0 then limn→∞zn-Tmzn=0 for all m∈N∪{0}, which is called that the NST-condition (II).Lemma 9 (see [31] (demiclosedness principle)).
LetC be a nonempty closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping with Fix(S)≠∅. If the sequence {xn}⊂C converges weakly to x and the sequence {(I-S)xn} converges strongly to y. Then, (I-S)x=y, in particular, if y=0 then x∈Fix(S).Lemma 10 (see [32]).
Let{an} be a sequence of nonnegative real number such that (27)an+1≤1-γnan+γnδn,∀n=0,1,2,…, where {γn} is a sequence in (0,1) and {δn} is a sequence in R such that (i)
∑n=0∞‍γn=∞;(ii)
limsupn→∞δn≤0 or ∑n=0∞‍|γnδn|<∞.
Then,limn→∞an=0.


## 3. Main Result

Throughout this paper, we letC be a nonempty closed convex subset of a real Hilbert space H. First, we will show that Gn which is defined by (28)Gnx=αnx+1-αnPCI-λ∇g+βnIx,∀x∈C,n=0,1,2,… has the unique fixed point under the conditions 0<λ<2/L+2, 0≤αn<1 and 0<βn<1 where g:C→R be a strictly real-valued convex function such that ∇g is 1/L-ism with L>0. Since, ∇g is 1/L-ism and the nonexpansiveness of PC. Then, for each x,y∈C, we have(29)PCI-λ∇g+βnIx-PCI-λ∇g+βnIy2≤I-λ∇g+βnIx-I-λ∇g+βnIy2=1-λβnx-y-λ∇gx-∇gy2=1-λβn2x-y2+λ2∇gx-∇gy2-2λ1-λβnx-y,∇gx-∇gy≤1-λβn2x-y2+λ2∇gx-∇gy2-2Lλ1-λβn∇gx-∇gy2=1-λβn2x-y2+λ2-2Lλ1-λβn∇gx-∇gy2≤1-λβn2x-y2+λ2-2Lλ+2Lλ2∇gx-∇gy2=1-λβn2x-y2-λ2L1-λ-λ∇gx-∇gy2≤1-λβn2x-y2.Therefore,(30)PCI-λ∇g+βnIx-PCI-λ∇g+βnIy≤1-λβnx-y.It follows that, for x≠y, by (30) we have(31)Gnx-Gny=αnx+1-αnPCI-λ∇g+βnIx-αny+1-αnPCI-λ∇g+βnIy=αnx-y+1-αnPCI-λ∇g+βnIx-PCI-λ∇g+βnIy≤αnx-y+1-αnPCI-λ∇g+βnIx-PCI-λ∇g+βnIy≤αnx-y+1-αn1-λβnx-y<αnx-y+1-αnx-y=x-y. That is,(32)Gnx-Gny<x-y,∀x≠y. So, Gn is a contraction, therefore, by Banach’s contraction principle, Gn has the unique fixed point. Therefore, Gn is well-defined.LetU be the solution set of the CCMP (1). It is clear that U is a closed and convex sets. We now ready to present my main results as follows.Theorem 11.
LetC be a nonempty closed convex subset of a real Hilbert space H, g:C→R is a strictly real-valued convex function such that the gradient ∇g is 1/L-ism with L>0. Assume that U≠∅ and let {xn}⊂C be a sequence generated by (33)x0∈C,xn+1=αnxn+1-αnPCI-λ∇g+βnIxn,∀n=0,1,2,…, where λ∈(0,2/2+L) and {αn}⊂[0,1),{βn}⊂(0,1) satisfy the following conditions: (i)
αn≤βnm such that m>1 for all n=0,1,2,…,(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞,
then the sequence{xn} converges strongly to q∈U, which is the unique minimizer of the CCMP (1).Proof.
We divide the proof into 4 steps.
Step 1. We will show that {xn} is bounded. Let p∈U. By the strictly convexity of g, we have that U is a singleton set. Noticing from 1/L-ism of ∇g that ∇g is L-Lipschitzian. So, by (10), we have p=PC(I-λ∇g)p. Therefore, by (30), we have(34)xn+1-p=αnxn+1-αnPCI-λ∇g+βnIxn-p=αnxn+1-αnPCI-λ∇g+βnIxn-αn+1-αnp=αnxn-p+1-αnPCI-λ∇g+βnIxn-p=αnxn-p+1-αnPCI-λ∇g+βnIxn-PCI-λ∇gp≤αnxn-p+1-αnPCI-λ∇g+βnIxn-PCI-λ∇gp≤αnxn-p+1-αnPCI-λ∇g+βnIxn-PCI-λ∇g+βnIp+PCI-λ∇g+βnIp-PCI-λ∇gp≤αnxn-p+1-αn1-λβnxn-p+I-λ∇g+βnIp-I-λ∇gp=αnxn-p+1-αn1-λβnxn-p+λβnp=αnxn-p+1-αn1-λβnxn-p+1-αnλβnp=αn+1-αn1-λβnxn-p+1-αnλβnp=1-1-αnλβnxn-p+1-αnλβnp≤maxxn-p,p⋮≤maxx0-p,p. It follows that {xn} is bounded, and so are {PC(I-λ(∇g+βnI))xn} and {PC(I-λ∇g)xn}.
Step 2. We will show that limn→∞xn-PC(I-λ∇g)xn=0. Since,(35)
x
n
+
1
-
P
C
I
-
λ
∇
g
x
n
=
α
n
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
x
n
=
α
n
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
α
n
+
1
-
α
n
P
C
I
-
λ
∇
g
x
n
=
α
n
x
n
-
P
C
I
-
λ
∇
g
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
x
n
≤
α
n
x
n
-
P
C
I
-
λ
∇
g
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
x
n
≤
α
n
x
n
-
P
C
I
-
λ
∇
g
x
n
+
1
-
α
n
I
-
λ
∇
g
+
β
n
I
x
n
-
I
-
λ
∇
g
x
n
=
α
n
x
n
-
P
C
I
-
λ
∇
g
x
n
+
1
-
α
n
λ
β
n
x
n
. Therefore, by conditions (i) and (ii), we have(36)limn→∞xn+1-PCI-λ∇gxn=0.Since, ∇g is 1/L-ism, then we have(37)PCI-λ∇gx-PCI-λ∇gy2≤I-λ∇gx-I-λ∇gy2=x-y-λ∇gx-∇gy2=x-y2-2λx-y,∇gx-∇gy+λ2∇gx-∇gy2≤x-y2-2λL∇gx-∇gy2+λ2∇gx-∇gy2=x-y2-λ2L-λ∇gx-∇gy2≤x-y2. Hence, (38)PCI-λ∇gx-PCI-λ∇gy≤x-y, that is, PC(I-λ∇g) is a nonexpansive. Therefore, by (36) and NST-condition (II) in Lemma 8, we have(39)limn→∞I-PCI-λ∇gxn=limn→∞xn-PCI-λ∇gxn=0.
Step 3. Let q∈U. Since, U is a singleton set, we have(40)-q,q-y≥0,∀y∈U.Therefore, by Lemma 3, we have q=PU(0). We will show that limsupn→∞〈-q,xn+1-q〉≤0. From (10) we have (41)q=PCI-λ∇gq. Since {xn} is bounded, we consider a subsequence {xnij} of {xni}; there exists a subsequence {xnij} of {xni} which converges weakly to z. It follows by the demiclosedness to the zero in Lemma 9 and (39) that z=PC(I-λ∇g)z. So, by (10) we have z∈U (indeed, z=q). Therefore, by (40), we have(42)limsupn→∞⁡-q,xn+1-q=limsupj→∞⁡-q,xnij+1-q=max-q,z-q:forallxnij⇀zasj⟶∞≤0.
Step 4. We will show that {xn} converges strongly to q. By (30), Lemma 5, condition (i), and the linearity orthogonal projection of PC, we have(43)
x
n
+
1
-
q
2
=
α
n
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
2
=
α
n
x
n
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
α
n
+
1
-
α
n
q
2
=
α
n
x
n
-
q
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
2
≤
α
n
x
n
-
q
2
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
2
=
α
n
x
n
-
q
2
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
,
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
=
α
n
x
n
-
q
2
+
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
,
x
n
+
1
-
α
n
x
n
1
-
α
n
-
q
=
α
n
x
n
-
q
2
+
1
-
α
n
1
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
,
x
n
+
1
-
q
+
α
n
1
-
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
,
q
-
x
n
=
α
n
x
n
-
q
2
+
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
q
,
x
n
+
1
-
q
+
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
,
q
-
x
n
≤
α
n
x
n
-
q
2
+
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
+
β
n
I
q
,
x
n
+
1
-
q
+
P
C
I
-
λ
∇
g
+
β
n
I
q
-
P
C
I
-
λ
∇
g
q
,
x
n
+
1
-
q
+
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
·
q
-
x
n
≤
α
n
x
n
-
q
2
+
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
P
C
I
-
λ
∇
g
+
β
n
I
q
·
x
n
+
1
-
q
+
λ
β
n
-
q
,
x
n
+
1
-
q
+
α
n
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
·
q
-
x
n
≤
β
n
m
x
n
-
q
2
+
1
-
λ
β
n
x
n
-
q
·
x
n
+
1
-
q
+
λ
β
n
-
q
,
x
n
+
1
-
q
+
β
n
m
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
·
q
-
x
n
=
β
n
m
x
n
-
q
2
+
1
-
λ
β
n
2
x
n
-
q
2
+
x
n
+
1
-
q
2
-
x
n
-
q
-
x
n
+
1
-
q
2
+
λ
β
n
-
q
,
x
n
+
1
-
q
+
β
n
m
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
·
q
-
x
n
≤
β
n
m
x
n
-
q
2
+
1
-
λ
β
n
2
x
n
-
q
2
+
1
2
x
n
+
1
-
q
2
+
λ
β
n
-
q
,
x
n
+
1
-
q
+
β
n
m
P
C
I
-
λ
∇
g
+
β
n
I
x
n
-
q
·
q
-
x
n
,and, therefore,(44)xn+1-q2≤2βnmxn-q2+1-λβnxn-q2+2λβn-q,xn+1-q+2βnmPCI-λ∇g+βnIxn-q·q-xn=1-λβnxn-q2+λβn2βnm-1λxn-q2+2-q,xn+1-q+2βnm-1λPCI-λ∇g+βnIxn-q·q-xn=1-γnxn-q2+γnδn,where γn=λβn and (45)δn=2βnm-1λxn-q2+2-q,xn+1-q+2βnm-1λPCI-λ∇g+βnIxn-q·q-xn. It is easy to see that ∑n=0∞‍γn=∞ and limsupn→∞δn≤0. Therefore, by Lemma 10, we obtain {xn} converges strongly to q. This completes the proof.Notice that whenαn=0 for all n=0,1,2,… then the result of Theorem 11 can be reduced into the result of Tian and Zhang [26] without the control condition ∑n=0∞‍|βn+1-βn|<∞ as follows.Corollary 12 (see [26]).
LetC be a nonempty closed convex subset of a real Hilbert space H and g:C→R be a strictly real-valued convex function such that the gradient ∇g is 1/L-ism with L>0. Assume that U≠∅ and let {xn}⊂C be a sequence generated by (46)x0∈C,xn+1=PCI-λ∇g+βnIxn,∀n=0,1,2,…, where λ∈0,2/2+L and {βn}⊂(0,1) satisfies the following conditions: limn→∞βn=0 and ∑n=0∞‍βn=∞, then the sequence {xn} converges strongly to q∈U, which is the unique minimizer of the CCMP (1).


## 4. Applications

LetC and Q be closed convex subsets of real Hilbert spaces H1 and H2, respectively, and A:H1→H2 be a bounded linear operator. We now consider the split feasibility problem (SFP) which introduced in 1994 by Censor and Elfving [13], where this problem is to find an element q∈C such that Aq∈Q. Define the convex function g:C→R as follows: (47)gx=12Ax-PQAx2. It follows by Lemma 7 that the gradient of g as ∇g=A∗(I-PQ)A where A∗ is the adjoint operator of A, and ∇g is 1/A2-ism. We have the consequence results as follows.Theorem 13.
LetC and Q be closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let A:H1→H2 be a bounded linear operator. Suppose that the SFP has a nonempty solution. Let {xn}⊂C be a sequence generated by (48)x0∈C,xn+1=αnxn+1-αnPCI-λA∗I-PQA+βnIxn,∀n=0,1,2,…, where λ∈0,2/2+A2 and {αn}⊂[0,1),{βn}⊂(0,1) satisfy the following conditions: (i)
αn≤βnm such that m>1 for all n=0,1,2,…,(ii)
limn→∞βn=0 and ∑n=0∞‍βn=∞,
then the sequence{xn} converges strongly to q, which is the unique minimizer of the minimum-norm solution of the SFP.Corollary 14.
LetC and Q be closed convex subsets of real Hilbert spaces H1 and H2, respectively, and let A:H1→H2 be a bounded linear operator. Suppose that the SFP has a nonempty solution. Let {xn}⊂C be a sequence generated by (49)x0∈C,xn+1=PCI-λA∗I-PQA+βnIxn,∀n=0,1,2,…, where λ∈0,2/2+A2 and {βn}⊂(0,1) satisfies the following conditions: limn→∞βn=0 and ∑n=0∞‍βn=∞, then the sequence {xn} converges strongly to q, which is the unique minimizer of the minimum-norm solution of the SFP.


## 5. Data Availability

In order of the feasible solution, all algorithms of the iterations have to compute many inner iterations to find the appropriate result, and stack overflow often occurs in which a computer program makes too many subroutine calls and its call stack runs out of space when the parameters of iterations have using many stack arrays to compute the feasible solution.To avoid the stack overflow, we introduce how to do the mathematical programming without using the stack arrays of its parameters for solving the SFP of the algorithm in Corollary14. Indeed, the situation of the stack overflow may have occurred from calculating the floating point numbers or the significant decimal digits; to avoid it we ought to be careful of that by always using digit precision command such as the command N[expr_] in Mathematica, and the command digits(num_) in Matlab, and also define all matrix in the regular command type without using the matrix palette to avoid it.Some mathematical software has a command to give the total number of seconds of CPU time used and the total number of seconds since the beginning of computation in the session such as the commandsTimeUsed[] and AbsoluteTime[] in Mathematica, respectively, and the commands cputime and tic/toc in Matlab, respectively.We now give the formulation of orthogonal projectionPC where C is a simply closed convex sets as follows, and in the case that C is not a simply closed convex sets, for instance, C is a halfspace, we can found more the formulation in [33].Proposition 15.
Forx∈RN we have (i)
ifC=RN then PCx=x,(ii)
ifC={b} such that b∈RM then PCx=b,(iii)
ifC={x∈RN:x2≤ρ,ρ>0} then (50)PCx=ρxx2,x∉C,x,x∈C.Proof.
Obviously, the results (i) and (ii) hold by the definition of orthogonal projection ofPC, and the result (iii) also holds by the normal vector of the boundary points set of C.We are ready to introduce how to do the mathematical programming without using the stack arrays of its parameters for solving the SFP of the algorithm in Corollary14 as follows. Suppose that the SFP has the unique consistent. Taking H1=(RN,·2) and H2=(RM,·2) into Corollary 14. Let the sets C and Q, the operator A, the sequence {xn}, and the parameters {βn},λ satisfy the conditions in Corollary 14. We have that {xn} is a convergent sequence, and so it is a Cauchy sequence. Hence, we can choose the stopping criteria ϵ>0 which satisfies xn+1-xn2<ϵ for stopping the program, and also the approximate solution refers to the last iteration. Steps of the mathematical programming of the algorithm in Corollary 14 are shown as follows:Mathematical programming for the split feasibility problemFinding the solution of an augmented matrix equationAM×NXN×1=BM×1.Step 1. Declare of all parameters AM×N,BM×1, the starting point xStartN×1 and ϵ.We setM=3,N=3 and b=BM×1.The example of the commands in Mathematica is shown as follows.A = { { 1,2 , - 1 } , { 1,1 , - 1 } , { 1 , - 2,1 } } ; b = { { 1 } , { 0 } , { 3 } };x S t a r t = { { 0 } , { 0 } , { 0 } } ; ϵ = 1 0 - 6;Step 2. Define the formulations of the orthogonal projections of PC and PQ where (51)C=x∈RN:x2≤ρ,ρ>0,Q=x∈RM:x2≤σ,σ>0. If we choose C=RN and Q={b} such that b∈RM then the orthogonal projections of PC and PQ are easy to calculated, and, hence, we do not need to define its formulations in this step, and we can put directly its formulations to process.The example of the commands in Mathematica is shown as follows.P c [ u _ ] ≔ I f [ N o r m [ u ] > ρ , ρ u / ( N o r m [ u ] / / N ) , u ] ; P q [ u _ ] ≔ I f [ N o r m [ u ] > σ , σ u / ( N o r m [ u ] / / N ) , u ];Step 3. Set the starting index n=0 and fix parameter λ∈0,2/2+A2. If the parameter λ is not a fix number such that it is a sequence, then we must lie it in the while loop of step 4.We setλ=2/(2+A2)+102.The example of the commands in Mathematica is shown as follows.n = 0 ;   λ = 2 / ( 2 + ( N o r m [ A ] / / N ) 2 ) + 1 0 2;Step 4. Start to calculate the iterations of the sequence {xn} such that (52)xn+1=PCI-λA∗I-PQA+βnIxn using the while loop. Set the parameter {βn}⊂(0,1) for all n=0,1,2,… into the while loop such that it satisfies the following conditions: limn→∞βn=0 and ∑n=0∞‍βn=∞. If xn+1-xn2<ϵ then we break the while loop for approximate feasible solution, which is referred to in the last iteration.It well known that, in the case of finite-dimensional real space,A∗=AT where AT stands for matrix transposition of A, and, hence, the algorithm in Corollary 14 can be reduced to (53)xn+1=PCxn-λATAxn+λATPQAxn-λβnxnfortheclosedballsCandQ,PCxn-λATAxn+λATb-λβnxnfortheclosedballCandQ=b,xn-λATAxn+λATb-λβnxnforC=RNandQ=b where b∈RM. We set βn=1/n+2 for all n=0,1,2,… and instead of xn and xn+1 with the variables xOld and xNew, respectively, and also instead of βn with β in the while loop for avoidance using stack arrays of the parameters.The example of the commands in Mathematica is shown as follows.xOld=xStart;While[True,β=1/n+2;
(∗WesetC=RNandQ={b}whereb∈RM.∗)xNew=xOld-λ((Transpose[A].A).xOld)+λ(Transpose[A].b)-λβxOld;Print[′′x[′′,n+1,′′]=′′, Transpose[xNew]//N//MatrixForm,′′, Error=′′, ScientificForm[Norm[xNew-xOld]]];If[Norm[xNew-xOld]<ϵ,
Print[′′---Conclusion---′′];Print[′′Startingpointx[0]=′′,Transpose[xStart]//MatrixForm];Print[′′ApproximateSolutionx[′′,n+1,′′]=′′, Transpose[xNew]//N//MatrixForm];Print[′′TimeUsed=′′,TimeUsed[],′′second′′];Print[′′---End---′′];Break[],(∗GotoNextLoop∗)xOld=xNew;n++;];];Step 5. Clear memory of the system.The example of the command in Mathematica is shown as follows.Q u i t [ ];


## 6. Numerical Results

In this section, we give some insight into the behavior of the algorithm presented in Corollary14. We implemented them in Mathematica to solve and run on a computer Intel(R) Core (TM) i3 processor 2.00 GHz. We use xn+1-xn2<ϵ as the stopping criteria.Throughout the computational experiments, the parameters used in those algorithms were sets asϵ=10-6,βn=1/n+2 and λ=2/(2+A2)+102 for all n=0,1,2,…, where A is a bounded linear operator. In the results report below, all CPU times reported are in seconds. The approximate solution is referred to the last iteration.Example 1.
Find the solution of linear equation system as follows:(54)x+2y-z=1,x+y-z=0,x-2y+z=3, where x,y,z∈R.
LetH1=H2=(R3,·2). Take A=12-111-11-21, C=R3 and Q={b:b=(1,0,3)T} into Corollary 14. We have chosen x0∈C arbitrarily and(55)xn+1=PCI-λATI-PQA+βnIxn=xn-λATAxn+λATPQAxn-λβnxn=xn-λATAxn+λATb-λβnxnwhere b=(1,0,3)T for all n=0,1,2,…. As n→∞, we have xn→q such that q is the our solution. The numerical results are listed in Table 1.Table 1
Results for Example1 using algorithm in Corollary 14.
Starting points x 0 Number of iterations CPU(s) Approximate solution q (0,0,0)T 6015 2.393 ( 1.999190,0.997529,2.995430 ) T (1,1,1)T 5986 2.293 ( 1.999180,0.997518,2.995410 ) T (2,2,2)T 5954 2.333 ( 1.999180,0.997506,2.995380 ) T (2,3,1)T 5963 2.543 ( 1.999180,0.997509,2.995390 ) T (3,2,1)T 5970 2.533 ( 1.999180,0.997512,2.995400 ) T


## 7. Conclusion

In this paper, we obtain an iterative scheme using the gradient projection method based on Mann’s approximation method for solving the constrained convex minimization problem (CCMP) and also solving the split feasibility problem (SFP) such that another strong convergence theorems for the CCMP and the SFP are obtained.



---

*Source: 1580837-2018-10-01.xml*