The inertial relaxed algorithm with Armijo-type line search for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem is the generalization of split feasibility problem, which has been widely used in fuzzy image reconstruction and sparse signal processing systems. In this paper, we present an inertial relaxed algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. The advantage of this algorithm is that the choice of stepsize is determined by Armijo-type line search, which avoids calculating the norms of operators. The weak convergence of the sequence obtained by our algorithm is proved under mild conditions. In addition, the numerical experiments are given to verify the convergence and validity of the algorithm.


Introduction
Let H 1 and H 2 be real Hilbert spaces, t ≥ 1 and r ≥ 1 be integers, {C i } t i=1 and {Q j } r j=1 be the nonempty, closed, and convex subsets of H 1 and H 2 .
In this paper, we study the multiple-sets split feasibility problem (MSSFP). This problem is to find a point x * such that It is well known that the split feasibility problem amounts to the following minimization problem: where P C is the metric projection on C and P Q is the metric projection on Q. It is important to note that since a projection on an ordinary closed convex set has not closed form method, it is difficult to calculate its projection. Fukushima [11] proposed a new relaxation projection formula to overcome this difficulty. Specifically, he calculated a projection on a convex functions level set by calculating a series of projections onto half-spaces containing the general level set. Yang [26] proposed a relaxed CQ algorithm for working out the split feasibility problem in the context of a finite-dimensional Hilbert space, in which closed convex subsets C and Q are the level sets of convex functions, which are proposed as follows: C = x ∈ H 1 : c(x) ≤ 0 and Q = y ∈ H 2 : q(y) ≤ 0 , (1.3) where c : H 1 → R and q : H 2 → R are convex functions which are weakly lower semicontinuous. Meanwhile, they assumed that c is subdifferentiable on H 1 and ∂c is a bounded operator in any bounded subset of H 1 . Similarly, q is subdifferentiable on H 2 and ∂q is also a bounded operator in any bounded subset of H 2 . Then two sets are defined at point x n as follows: C n = x ∈ H 1 : c(x n ) ≤ ξ n , x nx , (1.4) where ξ n ∈ ∂c(x n ), and Q n = y ∈ H 2 : q(Ax n ) ≤ ζ n , Ax ny , (1.5) where ζ n ∈ ∂q(Ax n ). We can easily see that C n and Q n are half-spaces. For all n ≥ 1, we easily know that C n ⊃ C and Q n ⊃ Q. Under this framework, the projection can be simply computed because of the particular form of the metric projection of the sets C n and Q n , for details, please see [18]. Using this framework, Yang [26] built a new relaxed CQ algorithm, which was used to solve the split feasibility problem by using the semi-spaces C n and Q n , rather than the sets C and Q. Whereafter, Shehu [19] came up with a relaxed CQ method with alternating inertial extrapolation step, which was used to solve the split feasibility problem by using the half spaces C n and Q n . At the same time, they verified their convergence in certain appropriate step size.
In this paper, we consider a class of multiple-sets split feasibility problem (1.1), where the convex sets are defined by  [6] invented the following distance function: where l i (i = 1, 2, . . . , t) and λ j (j = 1, 2, . . . , r) are positive constants such that t i=1 l i + r j=1 λ j = 1. Then we know that They proposed the following algorithm: where ⊆ R N is the auxiliary brief nonempty closed convex set satisfying ∩ S = ∅ and ρ > 0. When L was the Lipschitz constant of ∇f (x) and ρ ∈ (0, 2/L), they proved that the sequence {x n } produced by (1.9) converged to a solution of the multiple-sets split feasibility problem.
In order to improve the practicability of the method, in allusion to the split convex programming problem, Nesterov [17] proposed the next iterative process. y n = x n + θ n (x nx n-1 ), x n+1 = y nλ n ∇f (y n ), n ≥ 1, where λ n is a positive array and θ n ∈ [0, 1) is an inertial element. Besides that, there are many other correlative algorithms, for example, the inertial forward-backward splitting method, the inertial Mann method, and the moving asymptotes method, for details, please see [1-4, 10, 12, 14-16].
Under the motivation of the above study, we provide a relaxed CQ algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. In this algorithm, the stepsize is determined by line search. Hence, it avoids the calculation of the operators norms. Furthermore, we prove the weak convergence of the algorithm under some mild conditions. In addition, the inertial factor of the controlling parameters β n can be selected as far as possible to close to the one, such as [7-9, 20-23, 27].
The structure of the paper is as follows. The basic concepts, definitions, and related results are described in Sect. 2. The third section presents the algorithm and its proof, and the fourth section provides the corresponding numerical experiment, which verifies the validity and stability of the algorithm. The final summarization is offered in Sect. 5.

Preliminaries
In this section, we give some basic concepts and relevant conclusions. Suppose that H is a Hilbert space.
Look back upon that a mapping T: Equivalently, for all x, y ∈ H, Tx -Ty 2 ≤ xy, Tx -Ty . As we all know, T is firmly nonexpansive if and only if I -T is firmly nonexpansive. For a point u ∈ H and C is a nonempty, closed, and convex set of H, there is a unique point P C u ∈ C such that where P C is the metric projection of H on C. The following is a list of the significant quality of the metric projection. It is well known that P C is the firmly nonexpansive mapping on C. Meanwhile, P C possesses Moreover, the characteristic of the P C x is This representation means that Suppose that a function f : H → R, the element g ∈ H is thought to be the subgradient (2.5) Besides, ∂f (x) is the subdifferential of f at the point x which is described by The function f : H → R is thought to be weakly lower semi-continuous on a point x if {x n } converges weakly to x. It means that (2.7)

Lemma 2.3 ([18]) Suppose that the half-spaces C k and Q k are defined as (1.4) and (1.5).
Then the projections onto them from the points x and y are given as follows, respectively:
Step 2: For the iterations x n , x n-1 , calculate y n = ⎧ ⎨ ⎩ x n n = even, x n + β n (x nx n-1 ) n = odd. (3.6) Step 3: Calculate z n = P y nτ n ∇f n (y n ) , (3.7) where τ n = γ l m n and m n is the smallest nonnegative integer such that τ n ∇f n (y n ) -∇f n (z n ) ≤ μ y nz n .
In the following, we prove the convergence of Algorithm 3.1.

Lemma 3.1
Suppose that the solution set of MSSFP is nonempty, that is, S = ∅ and {x n } is any sequence generated by Algorithm 3.1. Then {x 2n } is Fejer monotone with respect to S (i.e., x 2n+2z ≤ x 2nz , ∀z ∈ S).
Thus, x * ∈ S. Now, we are going to prove that {x 2n+1 } converges to x * . Just for the sake of convenience, we are still going to use {x 2n+1 } for the proof. According to lim n→∞ x 2nx * exists and lim n→∞ x 2n kx * = 0, these mean that lim n→∞ x 2nx * = 0. Thus, x * is sole.

Numerical examples
As in Example 4.1, we will provide the results in this section. The whole codes are written in Matlab R2012a. All the numerical results are carried out on a personal Lenovo Thinkpad   Table 1 and Fig. 1. Secondly, we contrast Algorithm 3.1 in this paper and Algorithm 3.1 in [23]. From the numerical results of Example 1 in [23], it is better than the results in [13]. So our algorithm is compared to Algorithm 3.1 in [23]. These results are provided in Table 2. Lastly, we check the stability of the iteration number for Algorithm 3.1 in this paper comparing with Algorithm 3.1 in [23]. These results are provided in Figs. 2-4.  13]) Suppose that H 1 = H 2 = R 3 , r = t = 2, and l 1 = l 2 = λ 1 = λ 2 = 1 4 . We give that To find x * ∈ C 1 ∩ C 2 such that Ax * ∈ Q 1 ∩ Q 2 .
In the first place, let γ = 2, l = 0.5, μ = 0.95, and β n = 1 n+1 . Next, we study the iteration number required for the convergence of the sequence under different initial values. The condition for stopping the iteration is Ax n -P Q n j (Ax n ) 2 < 10 -4 . (4.2) We select diverse options of x 0 and x 1 as follows.  Table 1, we can see the iteration number and running time of Algorithm 3.1 in this paper for diverse options of x 0 and x 1 .

Figure 4
The iteration number of Algorithm 3.1 in this paper and Algorithm 3.1 in [23] Algorithm 3.1 in this paper is basically stable at about 50. However, Algorithm 3.1 in [23] is basically stable at about 150.

Conclusions
In this paper, we propose the inertial relaxed CQ algorithm for solving the convex multiple-sets split feasibility problem. And the global convergence conclusions are obtained. Our consequences generalize and produce some existing associated outcomes. Moreover, the preliminary numerical conclusions reveal that our presented algorithm is superior to some existing relaxed CQ algorithms in some cases about solving the convex multiple-sets split feasibility problem.