Iterative methods for constrained convex minimization problem in Hilbert spaces

Abstract

In this paper, based on Yamada’s hybrid steepest descent method, a general iterative method is proposed for solving constrained convex minimization problem. It is proved that the sequences generated by proposed implicit and explicit schemes converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality.

MSC:58E35, 47H09, 65J15.

1 Introduction

Let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and induced norm $\parallel \cdot \parallel$. Let C be a nonempty, closed and convex subset of H. We need some nonlinear operators which are introduced below.

Let $T,A:H\to H$ be nonlinear operators.

• T is nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in H$.

• T is Lipschitz continuous if there exists a constant $L>0$ such that $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$, for all $x,y\in H$.

• $A:H\to H$ is monotone if $〈x-y,Ax-Ay〉\ge 0$, for all $x,y\in H$.

• Given is a number $\eta >0$, $A:H\to H$ is η-strongly monotone if $〈x-y,Ax-Ay〉\ge \eta {\parallel x-y\parallel }^2$, for all $x,y\in H$.

• Given is a number $\upsilon >0$. $A:H\to H$ is υ-inverse strongly monotone (υ-ism) if $〈x-y,Ax-Ay〉\ge \upsilon {\parallel Ax-Ay\parallel }^2$, for all $x,y\in H$.

It is known that inverse strongly monotone operators have been studied widely (see [1–3]), and applied to solve practical problems in various fields; for instance, in traffic assignment problems (see [4, 5]).

• $T:H\to H$ is said to be an averaged mapping if $T=(1-\alpha )I+\alpha S$, where α is a number in $(0,1)$ and $S:H\to H$ is nonexpansive. In particular, projections are ($1/2$)-averaged mappings.

Averaged mappings have received many investigations, see [6–10].

Consider the following constrained convex minimization problem:

$$\underset{x\in C}{min}f(x),$$

(1.1)

where $f:C\to R$ is a real valued convex function. Assume that the minimization problem (1.1) is consistent, and let S denote its solution set. It is known that the gradient-projection algorithm is one of the powerful methods for solving the minimization problem (1.1) (see [11–18]), and sometimes the minimization problem (1.1) has more than one solution. So, regularization is needed. We can use the idea of regularization to design an iterative algorithm for finding the minimum-norm solution of (1.1).

We consider the regularized minimization problem:

$$\underset{x\in C}{min}{f}_{\alpha }(x)=f(x)+\frac{\alpha }{2}{\parallel x\parallel }^2.$$

(1.2)

Here, $\alpha >0$ is the regularization parameter, f is convex function with L-Lipschitz continuous gradient ∇f. Let $x_{\min }$ be minimum-norm solution of (1.1), namely, $x_{\min }$ satisfies the property:

$$\parallel x_{\min }\parallel =min\{\parallel x\parallel :x\in S\}.$$

$x_{\min }$ can be obtained by two steps. First, observing that the gradient $\mathrm{\nabla }{f}_{\alpha }=\mathrm{\nabla }f+\alpha I$ is $(L+\alpha )$-Lipschitzian and α-strongly monotone, the mapping ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{\alpha })$ is a contraction with coefficient $\sqrt{1-\gamma (2\alpha -\gamma {(L+\alpha )}^2)}\le 1-\frac{1}{2}\alpha \gamma$, where $0<\gamma \le \frac{\alpha }{{(L+\alpha )}^2}$. So, the regularized problem (1.2) has a unique solution, which is denoted as $x_{\alpha }\in C$ and which can be obtained via the Banach contraction principle. Secondly, letting $\alpha \to 0$ yields $x_{\alpha }\to x_{\min }$ in norm. The following result shows that for suitable choices of γ and α, the minimum-norm solution $x_{\min }$ can be obtained by a single step.

Theorem 1.1 [9]

Assume that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is L-Lipschitz continuous. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be generated by the following iterative algorithm:

$$x_{n+1}={Proj}_C(I-{\gamma }_n\mathrm{\nabla }{f}_{{\alpha }_n})x_n={Proj}_C(I-{\gamma }_n(\mathrm{\nabla }f+{\alpha }_{n}I))x_n,n\ge 0.$$

(1.3)

Let $\{{\gamma }_n\}$ and $\{{\alpha }_n\}$ satisfy the following conditions:

• (i) $0<{\gamma }_n\le {\alpha }_n/{(L+{\alpha }_n)}^2$ for all n;

• (ii) ${\alpha }_n\to 0$ (and ${\gamma }_n\to 0$) as $n\to \mathrm{\infty }$;

• (iii) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n{\gamma }_n=\mathrm{\infty }$;

• (iv) $(|{\gamma }_n-{\gamma }_{n-1}|+|{\alpha }_n{\gamma }_n-{\alpha }_{n-1}{\gamma }_{n-1}|)/{({\alpha }_n{\gamma }_n)}^2\to 0$ as $n\to \mathrm{\infty }$.

Then $x_n\to x_{\min }$ as $n\to \mathrm{\infty }$.

In the assumptions of Theorem 1.1, the sequence $\{{\gamma }_n\}$ is forced to tend to zero. If we keep it as a constant, then we have weak convergence as follows.

Theorem 1.2 [19]

Assume that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is L-Lipschitz continuous. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be generated by the following iterative algorithm:

$$x_{n+1}={Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\alpha }_n})x_n={Proj}_C(I-\gamma (\mathrm{\nabla }f+{\alpha }_{n}I))x_n,n\ge 0.$$

(1.4)

Assume that $0<\gamma <2/L$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n<\mathrm{\infty }$. Then ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ converges weakly to a solution of the minimization problem (1.1).

In 2001, Yamada [10] introduced the following hybrid steepest descent method:

$$x_{n+1}=(I-s_n\mu F)T{x}_n,$$

(1.5)

where $F:H\to H$ is k-Lipschitzian and η-strongly monotone, and $0<\mu <2\eta /k^2$. It is proved that the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ generated by (1.5) converges strongly to $x^{\ast }\in Fix(T)$, which solves the variational inequality:

$$〈F\left(x^{\ast }\right),x^{\ast }-z〉\le 0,\mathrm{\forall }z\in Fix(T).$$

In this paper, we introduce a modification of algorithm (1.4) which is based on Yamada’s method. It is proved that the sequence generated by our proposed algorithm converges strongly to a minimizer of (1.1), which is also a solution of a certain variational inequality.

2 Preliminaries

In this section, we introduce some useful properties and lemmas which will be used in the proofs for the main results in the next section.

Proposition 2.1 [7, 8]

Let the operators $S,T,V:H\to H$ be given:

• (i) If $T=(1-\alpha )S+\alpha V$, for some $\alpha \in (0,1)$ and if S is averaged and V is nonexpansive, then T is averaged.

• (ii) The composition of finitely many averaged mappings is averaged. That is, if each of the mappings ${\{T_i\}}_{i=1}^N$ is averaged, then so is the composite $T_1\cdots T_N$. In particular, if $T_1$ is ${\alpha }_1$-averaged and $T_2$ is ${\alpha }_2$-averaged, where ${\alpha }_1,{\alpha }_2\in (0,1)$, then the composite $T_1{T}_2$ is α-averaged, where $\alpha ={\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2$.

• (iii) If the mappings ${\{T_i\}}_{i=1}^N$ are averaged and have a common fixed point, then

  $$\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1\cdots T_N).$$

Here, the notations $Fix(T)$ denotes the set of fixed point of the mapping T; that is, $Fix(T):=\{x\in H:Tx=x\}$.

Proposition 2.2 [7, 20]

Let $T:H\to H$ be given. We have:

• (i) T is nonexpansive, if and only if the complement $I-T$ is ($1/2$)-ism;

• (ii) If T is υ-ism, then for $\gamma >0$, γT is ($\upsilon /\gamma$)-ism;

• (iii) T is averaged, if and only if the complement $I-T$ is υ-ism for some $\upsilon >1/2$; indeed, for $\alpha \in (0,1)$, T is α-averaged, if and only if $I-T$ is ($1/2\alpha$)-ism.

The so-called demiclosed principle for nonexpansive mappings will often be used.

Lemma 2.3 (Demiclosed Principle [21])

Let C be a closed and convex subset of a Hilbert space H and let $T:C\to C$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing }$. If ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ is a sequence in C weakly converging to x and if ${\{(I-T)x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to y, then $(I-T)x=y$. In particular, if $y=0$, then $x\in Fix(T)$.

Recall the metric (nearest point) projection ${Proj}_C$ from a real Hilbert space H to a closed convex subset C of H is defined as follows: given $x\in H$, ${Proj}_{C}x$ is the unique point in C with the property

$$\parallel x-{Proj}_{C}x\parallel =inf\{\parallel x-y\parallel :y\in C\}.$$

${Proj}_C$ is characterized as follows.

Lemma 2.4 Let C be a closed and convex subset of a real Hilbert space H. Given $x\in H$ and $y\in C$, then $y={Proj}_{C}x$ if and only if there holds the inequality

$$〈x-y,y-z〉\ge 0,\mathrm{\forall }z\in C.$$

Lemma 2.5 Assume that ${\{a_n\}}_{n=0}^{\mathrm{\infty }}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\gamma }_n{\delta }_n+{\beta }_n,n\ge 0,$$

where ${\{{\gamma }_n\}}_{n=0}^{\mathrm{\infty }}$ and ${\{{\beta }_n\}}_{n=0}^{\mathrm{\infty }}$ are sequences in $(0,1)$ and ${\{{\delta }_n\}}_{n=0}^{\mathrm{\infty }}$ is a sequence in ℝ such that

• (i) ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$;

• (ii) either ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$ or ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_n|{\delta }_n|<\mathrm{\infty }$;

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

We adopt the following notation:

• $x_n\to x$ means that $x_n\to x$ strongly;

• $x_n\rightharpoonup x$ means that $x_n\to x$ weakly.

3 Main results

Recall that throughout this paper, we use S to denote the solution set of constrained convex minimization problem (1.1).

Let H be a real Hilbert space and C be a nonempty closed convex subset of Hilbert space H. Let $F:C\to H$ be a k-Lipschitzian and η-strongly monotone operator with constant $k>0$, $\eta >0$ such that $0<\mu <2\eta /k^2$. Suppose that ∇f is L-Lipschitz continuous. We now consider a mapping $Q_s$ on C defined by:

$$Q_s(x)={Proj}_C(I-s\mu F)T_{{\lambda }_s}(x),\mathrm{\forall }x\in C,$$

where $s\in (0,1)$, and $T_{{\lambda }_s}$ is nonexpansive. Let $T_{{\lambda }_s}$ and ${\lambda }_s$ satisfy the following conditions:

• (i) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})=(1-{\theta }_s)I+{\theta }_s{T}_{{\lambda }_s}$ and $\gamma \in (0,2/L)$;

• (ii) ${\theta }_s=\frac{2+\gamma (L+{\lambda }_s)}{4}$;

• (iii) ${\lambda }_s$ is continuous with respect to s and ${\lambda }_s=o(s)$.

It is easy to see that $Q_s$ is a contraction. Indeed, we have for each $x,y\in C$,

$$\begin{array}{rcl}\parallel Q_s(x)-Q_s(y)\parallel & =& \parallel {Proj}_C(I-s\mu F)T_{{\lambda }_s}(x)-{Proj}_C(I-s\mu F)T_{{\lambda }_s}(y)\parallel \\ \le & \parallel (I-s\mu F)T_{{\lambda }_s}(x)-(I-s\mu F)T_{{\lambda }_s}(y)\parallel \\ \le & (1-s\tau )\parallel x-y\parallel ,\end{array}$$

where $\tau =\frac{1}{2}\mu (2\eta -\mu k^2)$. Hence, $Q_s$ has a unique fixed point in C, denoted by $x_s$ which uniquely solves the fixed-point equation

$$x_s={Proj}_C(I-s\mu F)T_{{\lambda }_s}(x_s).$$

(3.1)

The following proposition summarizes the properties of the net $\{x_s\}$.

Proposition 3.1 Let $x_s$ be defined by (3.1). Then the following properties for the net $\{x_s\}$ hold:

• (a) $\{x_s\}$ is bounded for $s\in (0,1)$;

• (b) ${lim}_{s\to 0}\parallel x_s-T_{{\lambda }_s}{x}_s\parallel =0$;

• (c) $x_s$ defines a continuous curve from $(0,1)$ into C.

Proof It is well known that: $\tilde{x}\in C$ solves the minimization problem (1.1) if and only if $\tilde{x}$ solves the fixed-point equation

$$\tilde{x}={Proj}_C(I-\gamma \mathrm{\nabla }f)\tilde{x}=\frac{2-\gamma L}{4}\tilde{x}+\frac{2+\gamma L}{4}T\tilde{x},$$

where $0<\gamma <2/L$ is a constant. It is clear that $\tilde{x}=T\tilde{x}$, i.e., $\tilde{x}\in S=Fix(T)$.

(a) Take a fixed $p\in S$, we obtain that

It follows that

$$\parallel x_s-p\parallel \le \frac{(1+s\mu k)\parallel T_{{\lambda }_s}(p)-T(p)\parallel }{s\tau }+\frac{\mu }{\tau }\parallel F(p)\parallel .$$

(3.2)

For $x\in C$, note that

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})x=(1-{\theta }_s)x+{\theta }_s{T}_{{\lambda }_s}x$$

and

$${Proj}_C(I-\gamma \mathrm{\nabla }f)x=(1-\theta )x+\theta Tx,$$

where ${\theta }_s=\frac{2+\gamma (L+{\lambda }_s)}{4}$ and $\theta =\frac{2+\gamma L}{4}$.

Then we get

$$\parallel (\theta -{\theta }_s)x+{\theta }_s{T}_{{\lambda }_s}x-\theta Tx\parallel =\parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})x-{Proj}_C(I-\gamma \mathrm{\nabla }f)x\parallel \le \gamma {\lambda }_s\parallel x\parallel .$$

Since ${\theta }_s=\frac{2+\gamma (L+{\lambda }_s)}{4}$ and $\theta =\frac{2+\gamma L}{4}$, there exists a real positive number $M>0$ such that

$$\parallel T_{{\lambda }_s}x-Tx\parallel \le \frac{{\lambda }_s\gamma (5\parallel x\parallel +\parallel Tx\parallel )}{2+\gamma (L+{\lambda }_s)}\le {\lambda }_{s}M\parallel x\parallel .$$

(3.3)

It follows from (3.2) and (3.3) that

$$\parallel x_s-p\parallel \le \frac{1+s\mu k}{\tau }\cdot \frac{{\lambda }_s}{s}\cdot M\parallel p\parallel +\frac{\mu }{\tau }\parallel F(p)\parallel .$$

Since ${\lambda }_s=o(s)$, there exists a real positive number $M^{\mathrm{\prime }}>0$ such that $\frac{{\lambda }_s}{s}\le M^{\mathrm{\prime }}$, and

Hence, $\{x_s\}$ is bounded.

(b) Note that the boundedness of $\{x_s\}$ implies that $\{F{T}_{{\lambda }_s}(x_s)\}$ is also bounded. Hence, by the definition of $\{x_s\}$, we have

(c) For $\gamma \in (0,2/L)$, there exists

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})=(1-{\theta }_s)I+{\theta }_s{T}_{{\lambda }_s}$$

and

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_{s_0}})=(1-{\theta }_{s_0})I+{\theta }_{s_0}{T}_{{\lambda }_{s_0}},$$

where ${\theta }_s=\frac{2+\gamma (L+{\lambda }_s)}{4}$ and ${\theta }_{s_0}=\frac{2+\gamma (L+{\lambda }_{s_0})}{4}$.

So for $x_s\in C$, we get

for some appropriate constant $N>0$ such that

$$N\ge \gamma \parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})x_s\parallel +5\gamma \parallel x_s\parallel .$$

Now take $s,s_0\in (0,1)$ and calculate

It follows that

$$\parallel x_s-x_{s_0}\parallel \le \frac{\mu \parallel F{T}_{{\lambda }_s}(x_s)\parallel }{s_0\tau }|s-s_0|+\frac{(1+s_0\mu k)N}{s_0\tau }|{\lambda }_s-{\lambda }_{s_0}|.$$

Since $\{F{T}_{{\lambda }_s}(x_s)\}$ is bounded, and ${\lambda }_s$ is continuous with respect to s, $x_s$ defines a continuous curve from $(0,1)$ into C. □

The following theorem shows that the net $\{x_s\}$ converges strongly as $s\to 0$ to a minimizer of (1.1), which solves some variational inequality.

Theorem 3.2 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of Hilbert space H. Let $F:C\to H$ be a k-Lipschitzian and η-strongly monotone operator with constant $k>0$, $\eta >0$ such that $0<\mu <2\eta /k^2$. Suppose that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is Lipschitzian with constant $L>0$. Let $x_s$ be defined by (3.1), where the parameter $s\in (0,1)$ and $T_{{\lambda }_s}$ is nonexpansive. Let $T_{{\lambda }_s}$ and ${\lambda }_s$ satisfy the following conditions:

• (i) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_s})=(1-{\theta }_s)I+{\theta }_s{T}_{{\lambda }_s}$ and $\gamma \in (0,2/L)$;

• (ii) ${\theta }_s=\frac{2+\gamma (L+{\lambda }_s)}{4}$;

• (iii) ${\lambda }_s$ is continuous with respect to s and ${\lambda }_s=o(s)$.

Then the net $\{x_s\}$ converges strongly as $s\to 0$ to a minimizer $x^{\ast }$ of (1.1), which solves the variational inequality

$$〈F{x}^{\ast },x^{\ast }-z〉\le 0,\mathrm{\forall }z\in S.$$

(3.4)

Equivalently, we have ${Proj}_S(I-\mu F)x^{\ast }=x^{\ast }$.

Proof It is easy to see the uniqueness of a solution of the variational inequality (3.4). Indeed, suppose both $\tilde{x}\in S$ and $\hat{x}\in S$ are solutions to (3.4), then

$$〈F\tilde{x},\tilde{x}-\hat{x}〉\le 0$$

(3.5)

and

$$〈F\hat{x},\hat{x}-\tilde{x}〉\le 0.$$

(3.6)

Adding up (3.5) and (3.6) gets

$$〈F\tilde{x}-F\hat{x},\tilde{x}-\hat{x}〉\le 0.$$

The strong monotonicity of F implies that $\tilde{x}=\hat{x}$ and the uniqueness is proved. Below we use $x^{\ast }\in S$ to denote the unique solution of the variational inequality (3.4).

Let us show that $x_s\to x^{\ast }$ as $s\to 0$. Set

$$y_s=(I-s\mu F)T_{{\lambda }_s}(x_s).$$

Then we have $x_s={Proj}_C{y}_s$. For any given $z\in S$, we get

(3.7)

Since ${Proj}_C$ is the metric projection from H onto C, we have

$$〈y_s-x_s,z-x_s〉\le 0.$$

Note that ${Proj}_C(I-\gamma \mathrm{\nabla }f)z=z$ and ${Proj}_C(I-\gamma \mathrm{\nabla }f)=\frac{2-\gamma L}{4}I+\frac{2+\gamma L}{4}T$, so we get $z=Tz$, i.e., $z\in S=Fix(T)$.

It follows from (3.7) that

By (3.3), we obtain that

(3.8)

Since $\{x_s\}$ is bounded, it is obvious that if $\{s_n\}$ is a sequence in $(0,1)$ such that $s_n\to 0$, and $x_{s_n}\rightharpoonup \overline{x}$.

By Proposition 3.1(b) and (3.3), we have

So, by Lemma 2.3, we get $\overline{x}\in Fix(T)=S$.

Since ${\lambda }_s=o(s)$, we obtain from (3.8) that $x_{s_n}\to \overline{x}\in S$.

Next, we show that $\overline{x}$ solves the variational inequality (3.4). Observe that

$$x_s={Proj}_C{y}_s={Proj}_C{y}_s-y_s+(I-s\mu F)T_{{\lambda }_s}(x_s).$$

Hence, we conclude that

$$\mu F(x_s)=\frac{1}{s}({Proj}_C{y}_s-y_s)+\frac{1}{s}[(I-s\mu F)T_{{\lambda }_s}(x_s)-(I-s\mu F)(x_s)].$$

Since $T_{{\lambda }_s}$ is nonexpansive, $I-T_{{\lambda }_s}$ is monotone. Note that, for any given $z\in S$, $z=Tz$ and $〈{Proj}_C{y}_s-y_s,{Proj}_C{y}_s-z〉\le 0$.

By (3.3), it follows that

(3.9)

Since ${\lambda }_s=o(s)$, by Proposition 3.1(b), we obtain from (3.9) that

$$〈\mu F(\overline{x}),\overline{x}-z〉\le 0.$$

So $\overline{x}\in S$ is a solution of the variational inequality (3.4). We get $\overline{x}=x^{\ast }$ by uniqueness. Therefore, $x_s\to x^{\ast }$ as $s\to 0$.

The variational inequality (3.4) can be rewritten as

$$〈(I-\mu F)x^{\ast }-x^{\ast },x^{\ast }-z〉\ge 0,\mathrm{\forall }z\in S.$$

So in terms of Lemma 2.4, it is equivalent to the following fixed point equation:

$${Proj}_S(I-\mu F)x^{\ast }=x^{\ast }.$$

Next, we study the following iterative method. For a given arbitrary initial guess $x_0\in C$, we propose the following explicit scheme that generates a sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ in an explicit way:

$$x_{n+1}={Proj}_C(I-s_n\mu F)T_{{\lambda }_n}(x_n),$$

(3.10)

where the parameters $\{s_n\}\subset (0,1)$. Let $T_{{\lambda }_n}$ and ${\lambda }_n$ satisfy the following conditions:

• (i) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=(1-{\theta }_n)I+{\theta }_n{T}_{{\lambda }_n}$ and $0<\gamma <2/L$;

• (ii) ${\theta }_n=\frac{2+\gamma (L+{\lambda }_n)}{4}$;

• (iii) ${\lambda }_n=o(s_n)$.

It is proved that the sequence ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ converges strongly to a minimizer $x^{\ast }\in S$ of (1.1), which also solves the variational inequality (3.4). □

Theorem 3.3 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of Hilbert space H. Let $F:C\to H$ be a k-Lipschitzian and η-strongly monotone operator with constant $k>0$, $\eta >0$ such that $0<\mu <2\eta /k^2$. Suppose that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is Lipschitzian with constant $L>0$. Let ${\{x_n\}}_{n=0}^{\mathrm{\infty }}$ be generated by algorithm (3.10) and the parameters $\{s_n\}\subset (0,1)$. Let $T_{{\lambda }_n}$, ${\lambda }_n$ and $s_n$ satisfy the following conditions:

• (C1) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=(1-{\theta }_n)I+{\theta }_n{T}_{{\lambda }_n}$ and $\gamma \in (0,2/L)$;

• (C2) ${\theta }_n=\frac{2+\gamma (L+{\lambda }_n)}{4}$ for all n;

• (C3) ${lim}_{n\to \mathrm{\infty }}{s}_n=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{s}_n=\mathrm{\infty }$;

• (C4) ${\sum }_{n=0}^{\mathrm{\infty }}|s_{n+1}-s_n|<\mathrm{\infty }$;

• (C5) ${\lambda }_n=o(s_n)$ and ${\sum }_{n=0}^{\mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|<\mathrm{\infty }$.

Then the sequence $\{x_n\}$ generated by the explicit scheme (3.10) converges strongly to a minimizer $x^{\ast }$ of (1.1), which is also a solution of the variational inequality (3.4).

Proof It is well known that:

• (a) $\tilde{x}\in C$ solves the minimization problem (1.1) if and only if $\tilde{x}$ solves the fixed-point equation

  $$\tilde{x}={Proj}_C(I-\gamma \mathrm{\nabla }f)\tilde{x}=\frac{2-\gamma L}{4}\tilde{x}+\frac{2+\gamma L}{4}T\tilde{x},$$

where $0<\gamma <2/L$ is a constant. It is clear that $\tilde{x}=T\tilde{x}$, i.e., $\tilde{x}\in S=Fix(T)$.

• (b) the gradient ∇f is $1/L$-ism.

• (c) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})$ is $\frac{2+\gamma (L+{\lambda }_n)}{4}$ averaged for $\gamma \in (0,2/L)$, in particular, the following relation holds:

  $${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=\frac{2-\gamma (L+{\lambda }_n)}{4}I+\frac{2+\gamma (L+{\lambda }_n)}{4}{T}_{{\lambda }_n}=(1-{\theta }_n)I+{\theta }_n{T}_{{\lambda }_n}.$$

We observe that $\{x_n\}$ is bounded. Indeed, take a fixed $p\in S$, we get

It follows that

$$\parallel x_{n+1}-p\parallel \le (1-s_n\tau )\parallel x_n-p\parallel +(1+s_n\mu k)\parallel T_{{\lambda }_n}(p)-T(p)\parallel +s_n\parallel \mu F(p)\parallel .$$

Note that, by using the same argument as in the proof of (3.3), there exists a real positive number $M>0$ such that

$$\parallel T_{{\lambda }_n}p-Tp\parallel \le \frac{{\lambda }_n\gamma (5\parallel p\parallel +\parallel Tp\parallel )}{2+\gamma (L+{\lambda }_n)}\le {\lambda }_{n}M\parallel p\parallel .$$

(3.11)

Since ${\lambda }_n=o(s_n)$, there exists a real positive number $M^{\mathrm{\prime }}>0$ such that $\frac{{\lambda }_n}{s_n}\le M^{\mathrm{\prime }}$ and by (3.11) we get

It follows from induction that

$$\parallel x_n-p\parallel \le max\{\parallel x_0-p\parallel ,\frac{\parallel \mu F(p)\parallel +(1+\mu k)M^{\mathrm{\prime }}M\parallel p\parallel }{\tau }\},n\ge 0.$$

(3.12)

Consequently, $\{x_n\}$ is bounded. It implies that $\{T_{{\lambda }_n}(x_n)\}$ is also bounded.

We claim that

$$\parallel x_{n+1}-x_n\parallel \to 0.$$

(3.13)

Indeed, since

$${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=\frac{2-\gamma (L+{\lambda }_n)}{4}I+\frac{2+\gamma (L+{\lambda }_n)}{4}{T}_{{\lambda }_n},$$

we obtain that

$$T_{{\lambda }_n}=\frac{4{Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})-[2-\gamma (L+{\lambda }_n)]I}{2+\gamma (L+{\lambda }_n)}.$$

By using the same argument as in the proof of Proposition 3.1(c), we obtain that

for some appropriate constant $K>0$ such that

$$K\ge \gamma \parallel {Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})(x_{n-1})\parallel +5\gamma \parallel x_{n-1}\parallel ,n\ge 0.$$

Thus, we get

for some appropriate constant $E>0$ such that

$$E\ge \parallel F{T}_{{\lambda }_n}(x_{n-1})\parallel ,n\ge 0.$$

Consequently, we get

$$\parallel x_{n+1}-x_n\parallel \le (1-s_n\tau )\parallel x_n-x_{n-1}\parallel +\mu E|s_n-s_{n-1}|+|{\lambda }_n-{\lambda }_{n-1}|(K+\mu k\cdot K).$$

By Lemma 2.5, we obtain $\parallel x_{n+1}-x_n\parallel \to 0$.

Next, we show that

$$\parallel x_n-T_{{\lambda }_n}{x}_n\parallel \to 0.$$

(3.14)

Indeed, it follows from (3.13) that

Now we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-x^{\ast },-\mu F\left(x^{\ast }\right)〉\le 0,$$

(3.15)

where $x^{\ast }\in S$ is a solution of the variational inequality (3.4).

Indeed, take a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-x^{\ast },-\mu F\left(x^{\ast }\right)〉=\underset{k\to \mathrm{\infty }}{lim}〈x_{n_k}-x^{\ast },-\mu F\left(x^{\ast }\right)〉.$$

(3.16)

Without loss of generality, we may assume that $x_{n_k}\rightharpoonup \tilde{x}$.

We observe that

$$\parallel x_n-T{x}_n\parallel \le \parallel x_n-T_{{\lambda }_n}(x_n)\parallel +\parallel T_{{\lambda }_n}(x_n)-T{x}_n\parallel .$$

It follows from (3.11) that

$$\parallel x_n-T{x}_n\parallel \le \parallel x_n-T_{{\lambda }_n}(x_n)\parallel +{\lambda }_{n}M\parallel x_n\parallel .$$

By (3.14), we get $\parallel x_n-T{x}_n\parallel \to 0$.

In terms of Lemma 2.3, we get $\tilde{x}\in Fix(T)=S$.

Consequently, from (3.16) and the variational inequality (3.4), it follows that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈x_n-x^{\ast },-\mu F\left(x^{\ast }\right)〉=〈\tilde{x}-x^{\ast },-\mu F\left(x^{\ast }\right)〉\le 0.$$

Finally, we show that $x_n\to x^{\ast }$.

As a matter of fact, set

$$y_n=(I-s_n\mu F)T_{{\lambda }_n}(x_n),n\ge 0.$$

Then, $x_{n+1}={Proj}_C{y}_n-y_n+y_n$.

In terms of Lemma 2.4 and (3.11), we obtain

It follows that

since $\{x_n\}$ is bounded, we can take a constant $L^{\mathrm{\prime }}>0$ such that

$$L^{\mathrm{\prime }}\ge (M+\mu kM)\parallel x^{\ast }\parallel \parallel x_{n+1}-x^{\ast }\parallel ,n\ge 0.$$

It then follows that

$${\parallel x_{n+1}-x^{\ast }\parallel }^2\le (1-s_n\tau ){\parallel x_n-x^{\ast }\parallel }^2+s_n{\delta }_n,$$

(3.17)

where ${\delta }_n=\frac{2}{1+s_n\tau }〈-\mu F(x^{\ast }),x_{n+1}-x^{\ast }〉+\frac{2{\lambda }_n}{s_n}{L}^{\mathrm{\prime }}$.

By (3.15) and ${\lambda }_n=o(s_n)$, we get ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$. Now applying Lemma 2.5 to (3.17) concludes that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$. □

4 Application

In this section, we give an application of Theorem 3.3 to the split feasibility problem (say SFP, for short), which was introduced by Censor and Elfving [22]. Since its inception in 1994, the split feasibility problem (SFP) has received much attention (see [7, 23, 24]) due to its applications in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy.

The SFP can mathematically be formulated as the problem of finding a point x with the property

$$x\in C\text{and}Bx\in Q,$$

(4.1)

where C and Q are nonempty, closed and convex subset of Hilbert space $H_1$ and $H_2$, respectively. $B:H_1\to H_2$ is a bounded linear operator.

It is clear that $x^{\ast }$ is a solution to the split feasibility problem (4.1) if and only if $x^{\ast }\in C$ and $B{x}^{\ast }-{Proj}_{Q}B{x}^{\ast }=0$. We define the proximity function f by

$$f(x)=\frac{1}{2}{\parallel Bx-{Proj}_{Q}Bx\parallel }^2,$$

and consider the constrained convex minimization problem

$$\underset{x\in C}{min}f(x)=\underset{x\in C}{min}\frac{1}{2}{\parallel Bx-{Proj}_{Q}Bx\parallel }^2.$$

(4.2)

Then $x^{\ast }$ solves the split feasibility problem (4.1) if and only if $x^{\ast }$ solves the minimization problem (4.2) with the minimize equal to 0. Byrne [7] introduced the so-called CQ algorithm to solve the (SFP).

$$x_{n+1}={Proj}_C(I-\gamma B^{\ast }(I-{Proj}_Q)B)x_n,n\ge 0,$$

(4.3)

where $0<\gamma <2/{\parallel B\parallel }^2$. He obtained that the sequence $\{x_n\}$ generated by (4.3) converges weakly to a solution of the (SFP).

In order to obtain strong convergence iterative sequence to solve the (SFP), we propose the following algorithm:

$$x_{n+1}={Proj}_C(I-s_n\mu F)T_{{\lambda }_n}(x_n),$$

(4.4)

where the parameters $\{s_n\}\subset (0,1)$ and $\{T_{{\lambda }_n}\}$ satisfy the following conditions:

• (C1) ${Proj}_C(I-\gamma (B^{\ast }(I-{Proj}_Q)B+{\lambda }_{n}I))=(1-{\theta }_n)I+{\theta }_n{T}_{{\lambda }_n}$ and $\gamma \in (0,2/L)$;

• (C2) ${\theta }_n=\frac{2+\gamma (L+{\lambda }_n)}{4}$ for all n,

where $F:C\to H$ is k-Lipschitzian and η-strongly monotone operator with constant $k>0$, $\eta >0$ such that $0<\mu <2\eta /k^2$. We can show that the sequence $\{x_n\}$ generated by (4.4) converges strongly to a solution of the (SFP) (4.1) if the sequence $\{s_n\}\subset (0,1)$ and the sequence $\{{\lambda }_n\}$ of parameters satisfy appropriate conditions.

Applying Theorem 3.3, we obtain the following result.

Theorem 4.1 Assume that the split feasibility problem (4.1) is consistent. Let the sequence $\{x_n\}$ be generated by (4.4). Where the sequence $\{s_n\}\subset (0,1)$ and the sequence $\{{\lambda }_n\}$ satisfy the conditions (C3)-(C5). Then the sequence $\{x_n\}$ converges strongly to a solution of the split feasibility problem (4.1).

Proof By the definition of the proximity function f, we have

$$\mathrm{\nabla }f(x)=B^{\ast }(I-{Proj}_Q)Bx,$$

and ∇f is Lipschitz continuous, i.e.,

$$\parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel \le L\parallel x-y\parallel ,$$

where $L={\parallel B\parallel }^2$.

Set $f_{{\lambda }_n}(x)=f(x)+\frac{{\lambda }_n}{2}{\parallel x\parallel }^2$, consequently

$$\begin{array}{rcl}\mathrm{\nabla }{f}_{{\lambda }_n}(x)& =& \mathrm{\nabla }f(x)+{\lambda }_{n}I(x)\\ =& B^{\ast }(I-{Proj}_Q)Bx+{\lambda }_{n}x.\end{array}$$

Then the iterative scheme (4.4) is equivalent to

$$x_{n+1}={Proj}_C(I-s_n\mu F)T_{{\lambda }_n}(x_n),$$

where the parameters $\{s_n\}\subset (0,1)$. $\{T_{{\lambda }_n}\}$ satisfy the following conditions:

• (C1) ${Proj}_C(I-\gamma \mathrm{\nabla }{f}_{{\lambda }_n})=(1-{\theta }_n)I+{\theta }_n{T}_{{\lambda }_n}$ and $\gamma \in (0,2/L)$;

• (C2) ${\theta }_n=\frac{2+\gamma (L+{\lambda }_n)}{4}$ for all n.

Due to Theorem 3.3, we have the conclusion immediately. □