Strong convergence of a relaxed CQ algorithm for the split feasibility problem

where A is a given $M\times N$ real matrix (where $A^{\ast }$ is the transpose of A), C and Q are nonempty, closed and convex subsets in ${\mathbb{R}}^N$ and ${\mathbb{R}}^M$, respectively. This problem has received much attention [2] due to its applications in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy [3–5], and many other applied fields.

where ${\tau }_n\in (0,\frac{2}{{\parallel A\parallel }^2})$, and $P_C$ and $P_Q$ are the orthogonal projections onto the sets C and Q, respectively. Compared with Censer and Elfving’ algorithm [1], the Byrne’ algorithm is easily executed since it only deal with orthogonal projections with no need to compute matrix inverses.

where the stepsize ${\tau }_n$ is chosen in the interval $(0,\frac{2}{L})$, where L is the Lipschitz constant of ∇f.

The computation of a projection onto a general closed convex subset is generally difficult. To overcome this difficulty, Fukushima [20] suggested a so-called relaxed projection method to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. In the setting of finite-dimensional Hilbert spaces, this idea was followed by Yang [13], who introduced the relaxed CQ algorithms for solving SFP (1.1) where the closed convex subsets C and Q are level sets of convex functions.

Recently, for the purpose of generality, the SFP (1.1) is studied in a more general setting. For instance, Xu [12] and López et al. [19] considered the SFP (1.1) in infinite-dimensional Hilbert spaces (i.e., the finite-dimensional Euclidean spaces ${\mathbb{R}}^N$ and ${\mathbb{R}}^M$ are replaced with general Hilbert spaces). Very recently, López et al. proposed a relaxed CQ algorithm with a new adaptive way of determining the stepsize sequence $({\tau }_n)$ for solving the SFP (1.1) where the closed convex subsets C and Q are level sets of convex functions. This algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, their algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed CQ algorithm such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results of López et al. and some others.

The rest of this paper is organized as follows. Some useful lemmas are listed in Section 2. In Section 3, the strong convergence of the new relaxed CQ algorithm of this paper is proved.

Throughout the rest of this paper, we denote by H or K a Hilbert space, A is a bounded linear operator from H to K, and by I the identity operator on H or K. If $f:H\to \mathbb{R}$ is a differentiable function, then we denote by ∇f the gradient of the function f. We will also use the notations:

This relation is called the subdifferentiable inequality.

A function $f:H\to \mathbb{R}$ is said to be subdifferentiable at x, if it has at least one subgradient at x. The set of subgradients of f at the point x is called the subdifferentiable of f at x, and it is denoted by $\partial f(x)$. A function f is called subdifferentiable, if it is subdifferentiable at all $x\in H$. If a function f is differentiable and convex, then its gradient and subgradient coincide.

The following lemma is not hard to prove (see [17, 21]).

The following are characterizations of firmly nanexpansive mappings (see [22]).

Lemma 2.3 [23]

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

where $c:H\to \mathbb{R}$ and $q:K\to \mathbb{R}$ are convex functions. We assume that both c and q are subdifferentiable on H and K, respectively, and that ∂c and ∂q are bounded operators (i.e., bounded on bounded sets). By the way, we mention that every convex function defined on a finite-dimensional Hilbert space is subdifferentiable and its subdifferential operator is a bounded operator (see [24]).

where ${\zeta }_n\in \partial q(A{x}_n)$.

Firstly, we recall the relaxed CQ algorithm of López et al. [19] for solving the SFP (1.1) where C and Q are given in (3.1) as follows.

López et al. proved that under some certain conditions the sequence $(x_n)$ generated by Algorithm 3.1 converges weakly to a solution of the SFP (1.1). Since the projections onto half-spaces $C_n$ and $Q_n$ have closed forms and ${\tau }_n$ is obtained adaptively via the formula (3.4) (no need to know a priori the norm of operator A), the above relaxed CQ algorithm 3.1 is implementable. But the weak convergence is its a weakness. To overcome this weakness, inspired by Algorithm 3.1, we will introduce a new relaxed CQ algorithm for solving the SFP (1.1) where C and Q are given in (3.1) so that the strong convergence is guaranteed.

It is well known that Halpern’s algorithm has a strong convergence for finding a fixed point of a nonexpansive mapping [25, 26]. Then we are in a position to give our algorithm. The algorithm given below is referred to as a Halpern-type algorithm [27].

where the sequence $({\alpha }_n)\subset (0,1)$ and $({\tau }_n)$ and $({\rho }_n)$ are given as in (3.4).

The convergence result of Algorithm 3.2 is stated in the next theorem.

Theorem 3.3 Assume that $({\alpha }_n)$ and $({\rho }_n)$ satisfy the assumptions:

(a1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$.

(a2) ${inf}_n{\rho }_n(4-{\rho }_n)>0$.

Then the sequence $(x_n)$ generated by Algorithm  3.2 converges in norm to $P_{S}u$.

Now, following an idea in [28], we prove $s_n\to 0$ by distinguishing two cases.

This implies that $\parallel \mathrm{\nabla }{f}_n(x_n)\parallel$ is bounded and it yields $f_n(x_n)\to 0$, namely $\parallel (I-P_{Q_n})A{x}_n\parallel \to 0$.

Applying Lemma 2.3 to (3.16), we obtain $s_n\to 0$.

which, together with (3.17), in turn implies that $s_n\to 0$, that is, $x_n\to z$. □

Remark 3.4 Since u can be chosen in H arbitrarily, one can compute the minimum-norm solution of SFP (1.1) where C and Q are given in (3.1) by taking $u=0$ in Algorithm 3.2 whether $0\in C$ or $0\notin C$.