Modified Projection Algorithms for Solving the Split Equality Problems

The split equality problem (SEP) has extraordinary utility and broad applicability in many areas of applied mathematics. Recently, Byrne and Moudafi (2013) proposed a CQ algorithm for solving it. In this paper, we propose a modification for the CQ algorithm, which computes the stepsize adaptively and performs an additional projection step onto two half-spaces in each iteration. We further propose a relaxation scheme for the self-adaptive projection algorithm by using projections onto half-spaces instead of those onto the original convex sets, which is much more practical. Weak convergence results for both algorithms are analyzed.


Introduction
The split equality problem (SEP) was introduced by Moudafi [1] and its interest covers many situations, for instance, in domain decomposition for PDE's, game theory, and intensitymodulated radiation therapy (IMRT) (see [2][3][4][5][6][7] for more details). Let 1 , 2 , and 3 be real Hilbert spaces; let ⊂ 1 and ⊂ 2 be two nonempty closed convex sets; let : 1 → 3 and : 2 → 3 be two bounded linear operators. The SEP can mathematically be formulated as the problem of finding and with the property ∈ , ∈ , such that = , which allows asymmetric and partial relations between the variables and . If 2 = 3 and = , then the split equality problem (1) reduces to the split feasibility problem (originally introduced in Censor and Elfving [8]) which is to find ∈ with ∈ . For solving the SEP (1), Moudafi [1] introduced the following alternating algorithm: where ∈ ( , min(1/ , 1/ ) − ) and and are the spectral radii of * and * , respectively. By studying the projected Landweber algorithm of the SEP (1) in a product space, Byrne and Moudafi [7] obtained the following algorithm: where , the stepsize at the iteration , is chosen in the interval ( , (2/( + )) − ). It is easy to see that the alternating algorithm (2) is sequential but the algorithm (3) is simultaneous.
Observe that in the algorithms (2) and (3), the determination of the stepsize depends on the operator (matrix) norms ‖ ‖ and ‖ ‖ (or the largest eigenvalues of * and * ). This means that, in order to implement the alternating algorithm (2), one has first to compute (or, at least, estimate) operator norms of and , which is in general not an easy work in practice. Considering this, Dong and He [9] proposed algorithms without prior knowledge of operator norms.
In this paper, we first propose a modification for algorithm (3), inspired by Tseng [10] (also see [11]). Our modified projection method computes the stepsize adaptively and performs an additional projection step onto two halfspaces, ⊂ 1 and ⊂ 2 , in each iteration. Then we 2 The Scientific World Journal give a relaxation scheme for this modification by replacing the orthogonal projections onto the sets and by projections onto the two half-spaces and , respectively. Since projections onto half-spaces can be directly calculated, the relaxed scheme will be more practical and easily implemented.
The rest of this paper is organized as follows. In the next section, some useful facts and tools are given. The weak theorem of the proposed self-adaptive projection algorithm is obtained in Section 3. In Section 4, we consider a relaxed self-adaptive projection algorithm, where the sets and are level sets of convex functions.

Preliminaries
In this section, we review some definitions and lemmas which will be used in this paper.
Let be a Hilbert space and let be the identity operator on . If : → R is a differentiable functional, then denote by ∇ the gradient of . If : → R is a subdifferentiable functional, then denote by the subdifferential of . Given a sequence ( , ) in 1 × 2 , ( , ) stands for the set of cluster points in the weak topology. " → " (resp., " ⇀ ") means the strong (resp., weak) convergence of ( ) to .
Definition 2. The graph of an operator is called to be weaklystrongly closed if ∈ ( ) with strongly converging to and weakly converging to ; then ∈ ( ). The next lemma is well known (see [10,12]) and shows that the maximal monotone operators are weakly-strongly closed.

Lemma 3. Let be a Hilbert space and let
: be a maximal monotone mapping. If ( ) is a sequence in bounded in norm and converging weakly to some and ( ) is a sequence in converging strongly to some and ∈ ( ) for all , then ∈ ( ).
The projection is an important tool for our work in this paper. Let Ω be a closed convex subset of real Hilbert space . Recall that the (nearest point or metric) projection from onto Ω, denoted by Ω , is defined in such a way that, for each ∈ , Ω is the unique point in Ω such that The following two lemmas are useful characterizations of projections.

Lemma 4. Given ∈ and ∈ Ω, then = Ω if and only if
Lemma 5. For any , ∈ and ∈ Ω, it holds Throughout this paper, assume that the split equality problem (1) is consistent and denote by Γ the solution of (1); that is, Then Γ is closed, convex, and nonempty. The split equality problem (1) can be written as the following minimization problem: where ( ) is an indicator function of the set defined by By writing down the optimality conditions, we obtain which implies, for > 0 and > 0, which in turn leads to the fixed point formulation The following proposition shows that solutions of the fixed point equations (17) are exactly the solutions of the SEP (1).

A Self-Adaptive Projection Algorithm
Based on Proposition 6, we construct a self-adaptive projection algorithm for the fixed point equations (13) and prove the weak convergence of the proposed algorithm.
The Scientific World Journal 3 Define the function : 1 × 2 → 1 by and the function : The self-adaptive projection algorithm is defined as follows.
where is chosen to be the largest ∈ { , , 2 , . . .} satisfying Construct the half-spaces and , the bounding hyperplanes of which support and at and V , respectively, If then set = 0 ; otherwise, set = .
In this algorithm, (19) involves projection onto halfspaces (resp., ) rather than onto the set (resp., ) and it is obvious that projections on (resp., ) are very simple. It is easy to show ⊂ and ⊂ . The last step is used to reduce the inner iterations for searching the stepsize .
The Scientific World Journal 5 To show the uniqueness of the weak cluster points, we will use the same strick as in the celebrated Opial Lemma. Indeed, let ( , ) be other weak cluster point of ( , ). By passing to the limit in the relation we obtain Reversing the role of (̂,̂) and ( , ), we also have By adding the two last equalities, we obtain Hence (̂,̂) = ( , ); this implies that the whole sequence ( , ) weakly converges to a solution of the SEP (1), which completes the proof.

A Relaxed Self-Adaptive Projection Algorithm
In Algorithm 7, we must calculate the orthogonal projections, and , many times even in one iteration step, so they should be assumed to be easily calculated; however, sometimes it is difficult or even impossible to compute them. In this case, we always turn to relaxed methods [13,14], which were introduced by Fukushima [15] and are more easily implemented. For solving the SEP (1), Moudafi [16] followed the ideas of Fukushima [15] and introduced a relaxed alternating algorithm which depends on the norms ‖ ‖ and ‖ ‖. In this section, we propose a relaxed scheme for the self-adaptive Algorithm 7.
Assume that the convex sets and are given by where : 1 → R and : 2 → R are convex functions which are subdifferentiable on and , respectively, and we assume that their subdifferentials are bounded on bounded sets.
In the th iteration, let ( ) and ( ) be two sequences of closed convex sets defined by where ∈ ( ) and where ∈ ( ). It is easy to see that ⊃ and ⊃ for every ≥ 0.
Following the proof of Lemma 8, we easily obtain the following.
Theorem 12. Let ( , ) be the sequence generated by Algorithm 10 and let and be nonempty closed convex sets in 1 and 2 with simple structures, respectively. If ( × ) ∩ Γ is nonempty, then ( , ) converges weakly to a solution of the SEP (1).