Regularized gradient-projection methods for finding the minimum-norm solution of the constrained convex minimization problem

Abstract

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that g is a real-valued convex function and the gradient ∇g is $\frac{1}{L}$-ism with $L>0$. Let $0<\lambda <\frac{2}{L+2}$, $0<\beta_{n}<1$. We prove that the sequence $\{x_{n}\}$ generated by the iterative algorithm $x_{n+1}=P_{C}(I-\lambda(\nabla g+\beta_{n}I))x_{n}$, $\forall n\geq0$ converges strongly to $q\in U$, where $q=P_{U}(0)$ is the minimum-norm solution of the constrained convex minimization problem, which also solves the variational inequality $\langle-q, p-q\rangle\leq0$, $\forall p\in U$. Under suitable conditions, we obtain some strong convergence theorems. As an application, we apply our algorithm to solving the split feasibility problem in Hilbert spaces.

1 Introduction

Let H be a real Hilbert space with inner product $\langle\cdot,\cdot \rangle$ and norm $\Vert \cdot \Vert$. Let C be a nonempty closed convex subset of H. Let $\mathbb{N}$ and $\mathbb{R}$ denote the sets of positive integers and real numbers. Suppose that f is a contraction on H with coefficient $0<\alpha<1$. A nonlinear operator $T:H\rightarrow H$ is nonexpansive if $\Vert Tx-Ty\Vert \leq \Vert x-y\Vert$ for all $x,y\in H$. We use $\operatorname {Fix}(T)$ to denote the fixed point of T.

Firstly, consider the constrained convex minimization problem:

$$ \min_{x\in C}g(x), $$

(1.1)

where $g:C\rightarrow\mathbb{R}$ is a real-valued convex function. Assume that the constrained convex minimization problem (1.1) is solvable, let U denote its solution set. The gradient-projection algorithm (GPA) is an effective method for solving the constrained convex minimization problem (1.1). A sequence $\{x_{n}\}$ generated by the following recursive formula:

$$ x_{n+1}=P_{C}(I-\lambda\nabla g)x_{n}, \quad \forall n \geq0, $$

(1.2)

where the parameter λ is real positive number. In general, if the gradient ∇g is L-Lipschitz continuous and η-strongly monotone, $0<\lambda<\frac{2\eta}{L^{2}}$, the sequence $\{x_{n}\}$ generated by (1.2) converges strongly to a minimizer of (1.1). However, if the gradient ∇g is only to be $\frac{1}{L}$-ism with $L>0$, $0<\lambda<\frac{2}{L}$, the sequence $\{x_{n}\}$ generated by (1.2) converges weakly to a minimizer of (1.1).

Recently, many authors combined the constrained convex minimization problem with a fixed point problem [1–3] and proposed composited iterative algorithms to find a solution of the constrained convex minimization problem [4–7].

In 2000, Moudafi [8] introduced the viscosity approximation method for nonexpansive mappings.

$$ x_{n+1}=\alpha_{n}f(x_{n})+(1- \alpha_{n})Tx_{n}, \quad \forall n\geq0. $$

(1.3)

In 2001, Yamada [9] introduced the so-called hybrid steepest-descent algorithm:

$$ x_{n+1}=Tx_{n}-\mu\lambda_{n}FTx_{n},\quad \forall n\geq0, $$

(1.4)

where F is Lipschitzian and strongly monotone operator. In 2006, Marino and Xu [10] considered a generative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I- \alpha_{n}A)Tx_{n}, \quad \forall n\geq0, $$

(1.5)

where A is a strongly positive operator. In 2010, Tian [11] combined the iterative algorithm of (1.4), (1.5), and proposed a new iterative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\mu \alpha_{n}F)Tx_{n}, \quad \forall n\geq0. $$

(1.6)

In 2010, Tian [12] generalized (1.6), obtained the following iterative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma Vx_{n}+(I-\mu \alpha_{n}F)Tx_{n}, \quad \forall n\geq0, $$

(1.7)

where V is Lipschitzian operator. Based on these iterative algorithms, some authors combined GPA with averaged operator to solve the constrained convex minimization problem [13, 14].

In 2011, Ceng et al. [1] proposed a sequence $\{x_{n}\}$ generated by the following iterative algorithm:

$$ x_{n+1}=P_{C} \bigl[\theta_{n}rh(x_{n})+(I- \theta_{n}\mu F)T_{n}(x_{n}) \bigr],\quad \forall n \geq0, $$

(1.8)

where $h:C\rightarrow H$ is an l-Lipschitzian mapping with a constant $l>0$, and $F:C\rightarrow H$ is a k-Lipschitzian and η-strongly monotone operator with constants $k, \eta>0$. $\theta_{n}=\frac{2-\lambda_{n}L}{4}$, $P_{C}(I-\lambda_{n}\nabla g)=\theta_{n}I+(1-\theta_{n})T_{n}$, $\forall n\geq0$. Then a sequence $\{ x_{n}\}$ generated by (1.8) converges strongly to a minimizer of (1.1).

On the other hand, Xu [15] proposed that regularization can be used to find the minimum-norm solution of the minimization problem.

Consider the following regularized minimization problem:

$$ \min_{x\in C}g_{\beta}(x):=g(x)+\frac{\beta}{2}\Vert x \Vert ^{2}, $$

where the regularization parameter $\beta>0$. g is a convex function and the gradient ∇g is $\frac{1}{L}$-ism with $L>0$. Then the sequence $\{x_{n}\}$ generated by the following formula:

$$ x_{n+1}=P_{C}(I-\lambda\nabla g_{\beta_{n}})x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad \forall n \geq0, $$

(1.9)

where the regularization parameters $0<\beta_{n}<1$, $0<\lambda<\frac {2}{L}$ converges weakly. But, if a sequence $\{x_{n}\}$ defined by

$$ x_{n+1}=P_{C}(I-\lambda_{n} \nabla g_{\beta_{n}})x_{n}=P_{C} \bigl(I-\lambda_{n}( \nabla g+\beta_{n}I) \bigr)x_{n},\quad \forall n\geq0, $$

(1.10)

where the initial guess $x_{0}\in C$, $\{\lambda_{n}\}$, $\{\beta_{n}\}$ satisfy the following conditions:

• (i) $0<\lambda_{n}\leq\frac{\beta_{n}}{(L+\beta_{n})^{2}}$, $\forall n\geq 0$,

• (ii) $\beta_{n}\rightarrow0$ (and $\lambda_{n}\rightarrow 0$) as $n\rightarrow\infty$,

• (iii) $\sum_{n=1}^{\infty}\lambda_{n}\beta_{n} = \infty$,

• (iv) $\frac{(\vert \lambda_{n}-\lambda_{n-1}\vert +\vert \lambda_{n}\beta_{n}-\lambda_{n-1}\beta_{n-1}\vert )}{(\lambda _{n}\beta_{n})^{2}}\rightarrow0$ as $n\rightarrow\infty$.

Then the sequence $\{x_{n}\}$ generated by (1.10) converges strongly to $x^{*}$, which is the minimum-norm solution of (1.1) [15].

Secondly, Yu et al. [16] proposed a strong convergence theorem with a regularized-like method to find an element of the set of solutions for a monotone inclusion problem in a Hilbert space.

Theorem 1.1

[16]

Let H be a real Hilbert space and C be a nonempty closed and convex subset of H. Let $L>0$, F is a $\frac{1}{L}$-ism mapping of C into H. Let B be a maximal monotone mapping on H and let G be a maximal monotone mapping on H such that the domains of B and G are included in C. Let $J_{\rho}=(I+\rho B)^{-1}$ and $T_{r}=(I+rG)^{-1}$ for each $\rho>0$ and $r>0$. Suppose that $(F+B)^{-1}(0)\cap G^{-1}(0)\neq\emptyset$. Let $\{x_{n}\}\subset H$ defined by

$$ x_{n+1}=J_{\rho} \bigl(I-\rho(F+\beta_{n}I) \bigr)T_{r}x_{n}, \quad\forall n>0, $$

(1.11)

where $\rho\in(0,\infty)$, $\beta_{n}\in(0,1)$, $r\in(0,\infty)$. Assume that

• (i) $0< a\leq\rho<\frac{2}{2+L}$,

• (ii) $\lim_{n\rightarrow\infty}\beta_{n}=0$, $\sum_{n=1}^{\infty}\beta _{n}=\infty$.

Then the sequence $\{x_{n}\}$ generated by (1.11) converges strongly to x̅, where $\overline{x}= P_{(F+B)^{-1}(0)\cap G^{-1}(0)}(0)$.

From the article of Yu et al. [16], we obtain a new condition of parameter ρ, $0<\rho<\frac{2}{L+2}$, which is used widely in our article. Motivated and inspired by Lin, when $0<\lambda<\frac{2}{L+2}$, $\{\beta_{n}\}$ satisfy certain conditions, a sequence $\{x_{n}\}$ generated by the iterative algorithm (1.9):

$$ x_{n+1}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\geq0, $$

converges strongly to a point $q\in U$, where $q=P_{U}(0)$ is the minimum-norm solution of the constrained convex minimization problem.

Finally, we give concrete example and the numerical results to illustrate our algorithm is with fast convergence.

2 Preliminaries

In this part, we introduce some lemmas that will be used in the rest part. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We use ‘→’ to denote strong convergence of the sequence $\{x_{n}\}$ and use ‘⇀’ to denote weak convergence.

Recall $P_{C}$ is the metric projection from H into C, then to each point $x\in H$, the unique point $P_{C}\in C$ satisfy the property:

$$ \Vert x-P_{C}x\Vert =\inf_{y\in C}\Vert x-y \Vert =: d(x,C). $$

$P_{C}$ has the following characteristics.

Lemma 2.1

[17]

For a given $x\in H$:

• (1) $z=P_{C}x \Longleftrightarrow\langle x-z,z-y\rangle\geq0$, $\forall y\in C$;

• (2) $z=P_{C}x \Longleftrightarrow \Vert x-z\Vert ^{2}\leq \Vert x-y\Vert ^{2}-\Vert y-z\Vert ^{2}$, $\forall y\in C$;

• (3) $\langle P_{C}x-P_{C}y, x-y\rangle\geq \Vert P_{C}x-P_{C}y\Vert ^{2}$, $\forall x,y\in H$.

From (3), we can derive that $P_{C}$ is nonexpansive and monotone.

Lemma 2.2

Demiclosed principle [18]

Let $T : C\rightarrow C$ be a nonexpansive mapping with $F(T)\neq\emptyset$. If $\{x_{n}\}$ is a sequence in C weakly converging to x and if $\{ (I-T)x_{n}\}$ converges strongly to y, then $(I-T)x = y$. In particular, if $y = 0$, then $x\in F(T)$.

Lemma 2.3

[19]

Let $\{a_{n}\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1} \leq(1-\alpha_{n})a_{n} + \alpha_{n}\delta_{n}, \quad n \geq0, $$

where $\{\alpha_{n}\}_{n=0}^{\infty}$ and $\{\delta_{n}\}_{n=0}^{\infty }$ are sequences of real numbers in $(0,1)$ and such that

• (i) $\sum_{n=0}^{\infty}\alpha_{n} = \infty$;

• (ii) $\limsup_{n\rightarrow\infty}\delta_{n} \leq0$ or $\sum_{n=0}^{\infty}\alpha_{n}\vert \delta_{n}\vert < \infty$.

Then $\lim_{n\rightarrow\infty}a_{n} = 0$.

3 Main results

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that $g:C\rightarrow\mathbb{R}$ is real-valued convex function and the gradient ∇g is $\frac{1}{L}$-ism with $L>0$. Suppose that the minimization problem (1.1) is consistent and let U denote its solution set. Let $0<\lambda<\frac{2}{L+2}$, $0<\beta _{n}<1$. Consider the following mapping $G_{n}$ on C defined by

$$ G_{n}x=P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)x, \quad \forall x\in C, n\in\mathbb{N}. $$

We have

$$\begin{aligned} \Vert G_{n}x-G_{n}y\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x-P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)y \bigr\Vert ^{2} \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)y \bigr\Vert ^{2} \\ =&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}+\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ &{} -2\lambda(1-\lambda\beta_{n}) \bigl\langle x-y,\nabla g(x)- \nabla g(y) \bigr\rangle \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}+\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ &{} -\frac{2}{L}\lambda(1-\lambda\beta_{n}) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}-\lambda \biggl(\frac{2}{L}(1-\lambda)-\lambda \biggr) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}. \end{aligned}$$

That is,

$$ \Vert G_{n}x-G_{n}y\Vert \leq(1-\lambda \beta_{n})\Vert x-y\Vert . $$

Since $0<1-\lambda\beta_{n}<1$, it follows that $G_{n}$ is a contraction. Therefore, by the Banach contraction principle, $G_{n}$ has a unique fixed point $x_{n}$, such that

$$ x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}. $$

Next, we prove that the sequence $\{x_{n}\}$ converges strongly to $q\in U$, which also solves the variational inequality

$$ \langle-q, p-q\rangle\leq0, \quad \forall p\in U. $$

(3.1)

Equivalently, $q=P_{U}(0)$, that is, q is the minimum-norm solution of the constrained convex minimization problem.

Theorem 3.1

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $g:C\rightarrow\mathbb{R}$ is real-valued convex function and assume that the gradient ∇g is $\frac{1}{L}$-ism with $L>0$. Assume that $U \neq\emptyset$. Let $\{x_{n}\}$ be a sequence generated by

$$ x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\in\mathbb{N}. $$

(3.2)

Let λ, $\{\beta_{n}\}$ satisfy the following conditions:

• (i) $0<\lambda<\frac{2}{2+L}$,

• (ii) $\{\beta_{n}\}\subset(0,1)$, $\lim_{n\rightarrow\infty}\beta _{n}=0$, $\sum_{n=1}^{\infty}\beta_{n} = \infty$.

Then $\{x_{n}\}$ converges strongly to a point $q\in U$, where $q=P_{U}(0)$, which is the minimum-norm solution of the minimization problem (1.1) and also solves the variational inequality (3.1).

Proof

First, we claim that $\{x_{n}\}$ is bounded. Indeed, pick any $p\in U$, then we have

$$\begin{aligned} \Vert x_{n}-p\Vert =& \bigl\Vert P_{C} \bigl(I- \lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I- \lambda\nabla g)p \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p \bigr\Vert \\ &{}+ \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p-(I- \lambda\nabla g)p \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-p\Vert +\lambda \beta_{n}\Vert p\Vert . \end{aligned}$$

Then we derive that

$$ \Vert x_{n}-p\Vert \leq \Vert p\Vert , $$

and hence $\{x_{n}\}$ is bounded.

Next, we claim that $\Vert x_{n}-P_{C}(I-\lambda\nabla g )x_{n}\Vert \rightarrow0$. Indeed

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert =& \bigl\Vert P_{C} \bigl(I-\lambda( \nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda \nabla g)x_{n} \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\lambda\beta_{n}\Vert x_{n}\Vert . \end{aligned}$$

Since $\{x_{n}\}$ is bounded, $\beta_{n}\rightarrow0$ ($n\rightarrow \infty$), we obtain

$$ \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \rightarrow0. $$

∇g is $\frac{1}{L}$-ism. Consequently, $P_{C}(I-\lambda\nabla g)$ is a nonexpansive self-mapping on C. As a matter of fact, we have for each $x,y\in C$

$$\begin{aligned}& \bigl\Vert P_{C}(I-\lambda\nabla g)x-P_{C}(I-\lambda \nabla g)y \bigr\Vert ^{2} \\& \quad \leq \bigl\Vert (I-\lambda\nabla g)x-(I- \lambda\nabla g)y \bigr\Vert ^{2} \\& \quad = \bigl\Vert x-y-\lambda \bigl(\nabla g(x)-\nabla g(y) \bigr) \bigr\Vert ^{2} \\& \quad = \Vert x-y\Vert ^{2}-2\lambda \bigl\langle x-y, \nabla g(x)- \nabla g(y) \bigr\rangle +\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\& \quad \leq \Vert x-y\Vert ^{2}-\lambda \biggl(\frac{2}{L}-\lambda \biggr) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\& \quad \leq \Vert x-y\Vert ^{2}. \end{aligned}$$

$\{x_{n}\}$ is bounded, consider a subsequence $\{x_{n_{i}}\}$ of $\{ x_{n}\}$. Since $\{x_{n_{i}}\}$ is bounded, there exists a subsequence $\{x_{n_{i_{j}}}\}$ of $\{x_{n_{i}}\}$ which converges weakly to z. Without loss of generality, we can assume that $x_{n_{i}}\rightharpoonup z$. Then by Lemma 2.2, we obtain $z\in U$.

On the other hand

$$\begin{aligned} \Vert x_{n}-z\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)z \bigr\Vert ^{2} \\ \leq& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)z, x_{n}-z \bigr\rangle \\ =& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)z, x_{n}-z \bigr\rangle \\ &{} +\langle-\lambda\beta_{n}z, x_{n}-z\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-z\Vert ^{2}+\lambda\beta_{n}\langle-z, x_{n}-z\rangle. \end{aligned}$$

Thus

$$\Vert x_{n}-z\Vert ^{2}\leq\langle-z, x_{n}-z\rangle. $$

In particular

$$\Vert x_{n_{i}}-z\Vert ^{2}\leq\langle-z, x_{n_{i}}-z\rangle. $$

Since $x_{n_{i}}\rightharpoonup z$. Then we derive that $x_{n_{i}}\rightarrow z$ as $i\rightarrow\infty$.

Let q be the minimum-norm solution of U, that is, $q=P_{U}(0)$. Since $\{x_{n}\}$ is bounded, there exists a subsequence $\{x_{n_{i}}\}$ of $\{x_{n}\}$ such that $x_{n_{i}}\rightharpoonup z$. As the above proof, we know that $x_{n_{i}}\rightarrow z$, $z\in U$.

Then we derive that

$$\begin{aligned} \Vert x_{n}-q\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-q \bigr\Vert ^{2} \\ \leq& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)q, x_{n}-q \bigr\rangle \\ =& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)q, x_{n}-q \bigr\rangle \\ &{}+\langle-\lambda\beta_{n}q, x_{n}-q\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert ^{2}+\lambda\beta_{n}\langle-q, x_{n}-q\rangle. \end{aligned}$$

Thus

$$\Vert x_{n}-q\Vert ^{2}\leq\langle-q, x_{n}-q\rangle. $$

In particular

$$\Vert x_{n_{i}}-q\Vert ^{2}\leq\langle-q, x_{n_{i}}-q\rangle. $$

Since $x_{n_{i}}\rightarrow z$, $z\in U$,

$$\Vert z-q\Vert ^{2}\leq\langle-q, z-q\rangle\leq0. $$

So, we have $z=q$. From the arbitrariness of $z\in U$, it follows that $q\in U$ is a solution of the variational inequality (3.1). By the uniqueness of solution of the variational inequality (3.1), we conclude that $x_{n}\rightarrow q$ as $n\rightarrow\infty$, where $q=P_{U}(0)$. □

Theorem 3.2

Let C be a nonempty closed convex subset of a real Hilbert space H and $g:C\rightarrow\mathbb{R}$ is real-valued convex function and assume that the gradient ∇g is $\frac{1}{L}$-ism with $L>0$. Assume that $U\neq\emptyset$. Let $\{x_{n}\}$ be a sequence generated by $x_{1}\in C$ and

$$ x_{n+1}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\in\mathbb{N}, $$

(3.3)

where λ and $\{\beta_{n}\}$ satisfy the following conditions:

• (i) $0<\lambda<\frac{2}{L+2}$;

• (ii) $\{\beta_{n}\}\subset(0,1)$, $\lim_{n\rightarrow\infty}\beta _{n}=0$, $\sum_{n=1}^{\infty}\beta_{n} = \infty$, $\sum_{n=1}^{\infty }\vert \beta_{n+1}-\beta_{n}\vert <\infty$.

Then $\{x_{n}\}$ converges strongly to a point $q\in U$, where $q=P_{U}(0)$, which is the minimum-norm solution of the minimization problem (1.1) and also solves the variational inequality (3.1).

Proof

First, we claim that $\{x_{n}\}$ is bounded. Indeed, pick any $p\in U$, then we know that, for any $n\in\mathbb{N}$,

$$\begin{aligned} \Vert x_{n+1}-p\Vert \leq& \bigl\Vert P_{C} \bigl(I- \lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p \bigr\Vert \\ &{}+ \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)p-P_{C}(I-\lambda\nabla g)p \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-p\Vert +\lambda \beta_{n}\Vert p\Vert \\ \leq&\max \bigl\{ \Vert x_{n}-p\Vert ,\Vert p\Vert \bigr\} . \end{aligned}$$

By the introduction

$$\begin{aligned} \Vert x_{n}-p\Vert \leq \max \bigl\{ \Vert x_{1}-p \Vert ,\Vert p\Vert \bigr\} , \end{aligned}$$

and hence $\{x_{n}\}$ is bounded.

Next, we show that $\Vert x_{n+1}-x_{n}\Vert \rightarrow0$.

$$\begin{aligned} \Vert x_{n+1}-x_{n}\Vert =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n-1}I) \bigr)x_{n-1} \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n-1}I) \bigr)x_{n-1} \bigr\Vert \\ =& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n-1} \\ &{} -\lambda\beta_{n}x_{n-1}+ \lambda\beta_{n-1}x_{n-1} \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-x_{n-1} \Vert +\lambda \vert \beta_{n}-\beta_{n-1}\vert \cdot \Vert x_{n-1}\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-x_{n-1} \Vert +\lambda \vert \beta_{n}-\beta_{n-1}\vert \cdot M, \end{aligned}$$

where $M=\sup\{\Vert x_{n}\Vert :n\in\mathbb{N}\}$. Hence, by Lemma 2.3, we have

$$\Vert x_{n+1}-x_{n}\Vert \rightarrow0. $$

Then we claim that $\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n}\Vert \rightarrow0$.

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert =& \bigl\Vert x_{n}-x_{n+1}+x_{n+1}-P_{C}(I- \lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert + \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert +\lambda \beta_{n}\cdot \Vert x_{n}\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert +\lambda \beta_{n}\cdot M, \end{aligned}$$

since $\beta_{n}\rightarrow0$ and $\Vert x_{n+1}-x_{n}\Vert \rightarrow0$, we have

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \rightarrow0. \end{aligned}$$

Next, we show that

$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle-q, x_{n}-q\rangle\leq0. \end{aligned}$$

(3.4)

Let q be the minimum-norm solution of U, that is, $q=P_{U}(0)$. Since $\{x_{n}\}$ is bounded, without loss of generality, we assume that $x_{n_{j}}\rightharpoonup z$. By the same argument as in the proof of Theorem 3.1, we have $z\in U$.

$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle-q, x_{n}-q\rangle=\lim _{j\rightarrow\infty}\langle-q, x_{n_{j}}-q\rangle=\langle-q, z-q \rangle\leq0. \end{aligned}$$

Then

$$\begin{aligned} \Vert x_{n+1}-q\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)q \bigr\Vert ^{2} \\ =& \bigl\langle P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)q, x_{n+1}-q \bigr\rangle \\ &{}+ \bigl\langle P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)q-P_{C}(I-\lambda\nabla g)q, x_{n+1}-q \bigr\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert \cdot \Vert x_{n+1}-q\Vert +\lambda\beta_{n}\langle-q, x_{n+1}-q\rangle \\ \leq&\frac{1-\lambda\beta_{n}}{2}\Vert x_{n}-q\Vert ^{2}+ \frac{1}{2}\Vert x_{n+1}-q\Vert ^{2}+\lambda \beta_{n}\langle-q, x_{n+1}-q\rangle. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert x_{n+1}-q\Vert ^{2} \leq&(1-\lambda \beta_{n})\Vert x_{n}-q\Vert ^{2}+2\lambda \beta_{n}\langle-q, x_{n+1}-q\rangle \\ =&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert ^{2}+2\lambda\beta_{n}\delta_{n}, \end{aligned}$$

where $\delta_{n}=\langle-q, x_{n+1}-q\rangle$.

It is easy to see that $\lim_{n\rightarrow\infty}\lambda\beta_{n}=0$, $\sum_{n=1}^{\infty}\lambda\beta_{n} = \infty$ and $\limsup_{n\rightarrow \infty}\delta_{n}\leq0$. Hence, by Lemma 2.3, the sequence $\{x_{n}\}$ converges strongly to q, where $q=P_{U}(0)$. This completes the proof. □

4 Application

In this part, we will illustrate the practical value of our algorithm in the split feasibility problem. In 1994, Censor and Elfving [20] came up with the split feasibility problem. The SFP is formulated as finding a point x with the property:

$$ x\in C \quad \mbox{and} \quad Ax\in Q, $$

(4.1)

where C and Q are nonempty closed and convex subset of real Hilbert spaces $H_{1}$ and $H_{2}$, $A:H_{1}\rightarrow H_{2}$ is bounded linear operator.

Next, we consider the constrained convex minimization problem:

$$ \min_{x\in C}g(x)=\min_{x\in C}\frac{1}{2} \Vert Ax-P_{Q}Ax\Vert ^{2}. $$

(4.2)

If $x^{*}$ is a solution of SFP, then $Ax^{*}\in Q$ and $Ax^{*}-P_{Q}Ax^{*}=0$, $x^{*}$ is the solution of the minimization problem (4.2). The gradient of g is ∇g, where $\nabla g=A^{*}(I-P_{Q})A$. Applying Theorem 3.2, we obtain the following theorem.

Theorem 4.1

Assume that the SFP (4.1) is consistent. Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that $A:H_{1}\rightarrow H_{2}$ is bounded linear operator, $W\neq\emptyset$, where W denotes the solution set of SFP (4.1). Let $\{x_{n}\}$ be a sequence generated by $x_{1}\in C$ and

$$ x_{n+1}=P_{C} \bigl(I-\lambda \bigl(A^{*}(I-P_{Q})A+ \beta_{n}I \bigr) \bigr)x_{n}, \quad\forall n\in\mathbb{N}. $$

(4.3)

Let λ and $\{\beta_{n}\}$ satisfy the following conditions:

• (i) $0<\lambda<\frac{2}{2+\Vert A\Vert ^{2}}$;

• (ii) $\{\beta_{n}\}\subset(0,1)$, $\lim_{n\rightarrow\infty}\beta _{n}=0$, $\sum_{n=1}^{\infty}\beta_{n} = \infty$, $\sum_{n=1}^{\infty }\vert \beta_{n+1}-\beta_{n}\vert <\infty$.

Then $\{x_{n}\}$ converges strongly to a point $q\in W$, where $q=P_{W}(0)$.

Proof

We only need to show that ∇g is $\frac{1}{\Vert A\Vert ^{2}}$-ism, then Theorem 4.1 can be obtained by Theorem 3.2.

$$\begin{aligned} \nabla g=A^{*}(I-P_{Q})A. \end{aligned}$$

Since $P_{Q}$ is firmly nonexpansive, so $P_{Q}$ is $\frac {1}{2}$-averaged mapping, then $I-P_{Q}$ is 1-ism, for any $x,y\in C$, we derive that

$$\begin{aligned} \bigl\langle \nabla g(x)-\nabla g(y), x-y \bigr\rangle =& \bigl\langle A^{*}(I-P_{Q})Ax-A^{*}(I-P_{Q})Ay, x-y \bigr\rangle \\ =& \bigl\langle (I-P_{Q})Ax-(I-P_{Q})Ay, Ax-Ay \bigr\rangle \\ \geq& \bigl\Vert (I-P_{Q})Ax-(I-P_{Q})Ay \bigr\Vert ^{2} \\ =&\frac{1}{\Vert A\Vert ^{2}}\cdot \bigl\Vert A^{*} \bigl((I-P_{Q})Ax-(I-P_{Q})Ay \bigr) \bigr\Vert ^{2} \\ =&\frac{1}{\Vert A\Vert ^{2}}\cdot \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2}. \end{aligned}$$

So, ∇g is $\frac{1}{\Vert A\Vert ^{2}}$-ism. □

5 Numerical result

In this part, we use the algorithm in Theorem 4.1 to solve a system of linear equations. Then we calculate the $4\times4$ system of linear equations.

Example 1

Let $H_{1}=H_{2}=\mathbb{R}^{4}$. Take

$$\begin{aligned}& A=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1&-1&2&-1\\ 2&-2&3&-3\\ 1&1&1&0\\ 1&-1&4&3 \end{array}\displaystyle \right ), \end{aligned}$$

(5.1)

$$\begin{aligned}& b=\left ( \textstyle\begin{array}{c} -2\\ -10\\ 6\\ 18 \end{array}\displaystyle \right ). \end{aligned}$$

(5.2)

Then the SFP can be formulated as the problem of finding a point $x^{*}$ with the property

$$x^{*}\in C \quad \mbox{and} \quad Ax^{*}\in Q, $$

where $C=\mathbb{R}^{4}$, $Q=\{b\}$. That is, $x^{*}$ is the solution of the system of linear equations $Ax=b$, and

$$ x^{*}=\left ( \textstyle\begin{array}{c} 1\\ 3\\ 2\\ 4 \end{array}\displaystyle \right ).$$

(5.3)

Take $P_{C}=I$, where I denotes the $4\times4$ identity matrix. Given the parameters $\beta_{n}=\frac{1}{(n+2)^{2}}$ for $n\geq0$, $\lambda =\frac{3}{200}$. Then by Theorem 4.1, the sequence $\{x_{n}\}$ is generated by

$$x_{n+1} =x_{n}-\frac{3}{200}A^{*}Ax_{n}+ \frac{3}{200}A^{*}b-\frac{3}{200(n+2)^{2}}x_{n}. $$

As $n\rightarrow\infty$, we have $\{x_{n}\}\rightarrow x^{*}=(1,3,2,4)^{T}$.

From Table 1, we can easily see that with iterative number increasing $x_{n}$ approaches to the exact solution $x^{*}$ and the errors gradually approach zero.

Table 1 Numerical results as regards Example  1

In Tian and Jiao [21], they use another iterative algorithm to calculate the same example.

Compare Table 1 with Table 2, we find that if the parameters $\beta _{n}$ are the same, when $\lambda\rightarrow\frac{2}{L+2}$, our algorithm is with fast convergence.

Table 2 Numerical results as regards Example  1

6 Conclusion

In a real Hilbert space, there are many methods to solve the constrained convex minimization problem. However, most of them cannot find the minimum-norm solution. In this article, we use the regularized gradient-projection algorithm to find the minimum-norm solution of the constrained convex minimization problem, where $0<\lambda<\frac {2}{L+2}$. Then under some suitable conditions, new strong convergence theorems are obtained. Finally, we apply this algorithm to the split feasibility problem and use a concrete example and numerical results to illustrate that our algorithm has fast convergence.