Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces

Proximal splitting algorithms for monotone inclusions (and convex optimization problems) in Hilbert spaces share the common feature to guarantee for the generated sequences in general weak convergence to a solution. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity (respectively, strong convexity for optimization problems). In this paper, we propose a modified Krasnosel'skiĭ–Mann algorithm in connection with the determination of a fixed point of a nonexpansive mapping and show strong convergence of the iteratively generated sequence to the minimal norm solution of the problem. Relying on this, we derive a forward–backward and a Douglas–Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. Furthermore, we formulate strong convergent primal–dual algorithms of forward–backward and Douglas–Rachford-type for highly structured monotone inclusion problems involving parallel-sums and compositions with linear operators. The resulting iterative schemes are particularized to the solving of convex minimization problems. The theoretical results are illustrated by numerical experiments on the split feasibility problem in infinite dimensional spaces.


Introduction and preliminaries
Let H be a real Hilbert space endowed with inner product ·, · and associated norm · = √ ·, · . Let T : H → H be a nonexpansive mapping, that is Tx − Ty ≤ x − y for all x, y ∈ H.
One of the most popular iterative methods for finding a fixed point of the operator T is the Krasnosel'skiȋ-Mann algorithm where x 0 ∈ H is arbitrary and (λ n ) n≥0 is a sequence of nonnegative real numbers. Provided FixT = {x ∈ H : Tx = x} = ∅, one can show under mild conditions imposed on (λ n ) n≥0 , that the sequence (x n ) n≥0 converges weakly to an element in FixT (see for instance [6]).
The applications and the impact of this fundamental result go beyond the usual fixed point theory, representing in fact the starting point for the derivation of algorithms and related convergence statements in connection with the solving of monotone inclusions. In this context, we mention the classical forward-backward algorithm for determining a zero of the sum of a set-valued maximally monotone operator and a single-valued and cocoercive one and the Douglas-Rachford algorithm for determining a zero of the sum of two set-valued maximally monotone operators. The paradigms behind these classical methods can be transferred to the solving of convex optimization problems, too (see [6]).
The iterative algorithms mentioned above share the common property that the generated sequences converge weakly to a solution of the problem under investigation. However, for applications where infinite dimensional functional spaces are involved, weak convergence is not satisfactory. In order to achieve strong convergence, one usually needs to impose more restrictive properties for the involved operators, like strong monotonicity when considering monotone inclusions and strong convexity when solving optimization problems. Since there evidently are applications for which these stronger properties are not fulfilled, the interest of the applied mathematics community in developing algorithms which generate iterates that strongly convergence is justified.
We mention in this sense the Halpern algorithm and its numerous variants designed for finding a fixed point of a nonexpansive mapping (see for instance [13]). In the context of solving monotone inclusions, we mention the proximal-Tikhonov algorithm where A : H ⇒ H is a maximally monotone operator, Id is the identity operator on H and (λ n ) n≥0 and (μ n ) n≥0 are sequences of nonnegative real numbers. Under mild conditions imposed on (λ n ) n≥0 and (μ n ) n≥0 , one can prove strong convergence of (x n ) n≥0 to the minimal norm solution of the set of zeros of A (see [19,27]). It is important to emphasize the Tikhonov regularization terms (μ n Id) n≥0 in the above scheme, which actually enforces the strong convergence property. In the absence of the regularization term, the above numerical scheme becomes the classical proximal algorithm for determining a zero of the operator A, for which in general only weak convergence can be proved (see [23]). For more theoretical results concerning Tikhonov regularization and more motivational arguments for using such techniques, especially for optimization problems, we refer the reader to Attouch's paper [2]. For other techniques and tools in order to achieve strong convergence we mention also the works of Haugazeau [18] and [5].
In this article, we will first introduce and investigate a modified Krasnosel'skiȋ-Mann algorithm with relaxation parameters, having the outstanding property that it generates a sequence of iterates which converges strongly to the minimal norm solution of the fixed points set of a nonexpansive mapping. In contrast to [13,20,27] (see also the references therein), where the techniques and tools used have their roots in fixed point theory results for contractions, our convergence statements follow more directly. Relying on this, we derive a forward-backward and a Douglas-Rachford algorithm, both endowed with Tikhonov regularization terms, which generate iterates that strongly converge to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators. The resulting iterative schemes are particularized to the minimization of the sum of two convex functions.
Furthermore, we deal with complexly structured monotone inclusions where parallel-sums and compositions with linear operators are involved. By making use of modern primal-dual techniques (see [10,12,15,17,28] and also [8,9]), we derive strongly convergent numerical schemes of forward-backward and Douglas-Rachford-type, both involving Tikhonov regularization terms and having the remarkable property that all the operators are evaluated separately. Moreover, the designed algorithms solve both the structured monotone inclusion problem and its dual monotone inclusion problem in the sense of Attouch-Théra (see [3]). When particularized to convex optimization problems, this means the concomitantly solving of a primal problem and its Fenchel dual one. For other types of primal-dual algorithms with strong convergence properties and their applications we refer the reader to [1,4]. Finally, in the last section we carry out numerical experiments on the split feasibility problem in infinite dimensional Hilbert spaces which illustrate the potential of the algorithm endowed with Tikhonov regularization terms.
In the remaining of this section, we recall some results which will play a decisive role in the convergence analysis of the proposed algorithms. The following result is related to the convergence of a sequence satisfying a sharp quasi-Fejér monotonicity property and follows as a direct consequence of [27,Lemma 2.5].
Then the sequence (a n ) n≥0 converges to 0.
We close this section with a result that is a consequence of the demiclosedness principle (see [6,Corollary 4.18]) and it will be used in the proof of Theorem 3, which is the main result of this paper.
Lemma 2 Let T : H → H be a nonexpansive operator and let (x n ) n≥0 be a sequence in H and x ∈ H be such that w − lim n→+∞ x n = x and (Tx n − x n ) n≥0 converges strongly to 0 as n → +∞. Then x ∈ FixT.

A strongly convergent Krasnosel'skiȋ-Mann algorithm
In order to induce strong convergence into the asymptotic behaviour of the Krasnosel'skiȋ-Mann algorithm for determining a fixed point of a nonexpansive mapping T : H → H, we propose the following modified version of it: where x 0 ∈ H is the starting point and (λ n ) n≥0 and (β n ) n≥0 suitably chosen sequences of positive numbers. In the proof of the theorem below, we denote by proj C : H → C, proj C (x) = argmin c∈C x − c , the projection operator onto the nonempty closed convex set C ⊆ H. We notice that for a nonexpansive mapping T : H → H, its set of fixed points FixT is closed and convex (see [6,Corollary 4.15]).
Consider the iterative scheme (2) with and arbitrary starting point x 0 ∈ H and a nonexpansive mapping T : H → H fulfilling FixT = ∅. Then (x n ) n≥0 converges strongly to proj FixT (0).
Proof For the beginning, we prove that (x n ) n≥0 is bounded. Let x ∈ FixT. Due to the nonexpansiveness of T, we have for any n ≥ 0: A simple induction leads to the inequality We claim that Indeed, by taking into account that T is nonexpansive and that (x n ) n≥0 is bounded, we obtain for any n ≥ 1 the following estimates: where C 1 > 0. Further, we derive for any n ≥ 1: (3) is a consequence of Lemma 1, for a n := x n − x n−1 , b n := 0, ε n := |β n − β n−1 |C 2 + |λ n − λ n−1 |C 1 and θ n := 1 − β n , n ≥ 1.
In the following we prove that For any n ≥ 0 we have the following inequalities: From here we deduce that for any n ≥ 0: Taking into account that (x n ) n≥0 is bounded, (3) and the properties of the sequences involved, we derive from the last inequality that (4) holds.
In what follows we show that (x n ) n≥0 actually converges strongly to proj FixT (0) :=x. Since T is nonexpansive, we have for any n ≥ 0: Hence, Next we show that lim sup Assuming the contrary, there would exist a positive real number l and a subsequence ( Due to the boundedness of the sequence (x n ) n≥0 , we can assume without losing the generality that (x k j ) j≥0 weakly converges to an element y ∈ H. According to Lemma 2, by taking into consideration (4), it follows that y ∈ FixT. From this and the variational characterization of the projection we easily derive which leads to a contradiction. This shows that (6) holds. Thus A direct application of Lemma 1 to (5), for a n := x n −x 2 , b n := 2β n −x, x n −x + (1 − β n ) x 2 , ε n := 0 and θ n := 1 − β n , n ≥ 0, delivers the desired conclusion.
An immediate consequence of Theorem 3 is the following corollary, which proposes an iterative scheme that finds a minimal norm solution of the set of fixed points of an averaged operator. Let α ∈ (0, 1) be fixed. We say that R : H → H is an α-averaged operator if there exists a nonexpansive operator T : H → H such that R = (1 − α)Id + αT. It is obvious that α-averaged operators are also nonexpansive. The 1 2 -averaged operators are nothing else than the firmly nonexpansive ones and form the most important representatives of this class. For properties and insights into these families of operators we refer the reader to [6]. The following result will play in the next section a determinant role in the converge analysis of the forward-backward method endowed with Tikhonov regularization term.
Proof Since R is α-averaged, there exists a nonexpansive operator T : H → H such that R = (1 − α)Id + αT. The conclusion follows from Theorem 3, by taking into account that (7) is equivalent to x n+1 = β n x n + αλ n (T(β n x n ) − β n x n ) ∀n ≥ 0 and that FixR = FixT.

A forward-backward algorithm with Tikhonov regularization term
This section is dedicated to the formulation and convergence analysis of a forward-backward algorithm with Tikhonov regularization terms, which generates a sequence of iterates that converges strongly to the minimal norm solution of the set of zeros of the sum of two maximally monotone operators, one of them being single-valued.
For readers' convenience, we recall some standard notions and results in monotone operator theory which will be used in the following (see also [6,7,26]). For an arbitrary set-valued operator A : H ⇒ H we denote by GrA = {(x, u) ∈ H × H : u ∈ Ax} its graph. Then A −1 : H ⇒ H, which is the operator with GrA −1 = {(x, u) ∈ H × H : x ∈ Au}, denotes the inverse operator of A. We use also the notation zerA = {x ∈ H : 0 ∈ Ax} for the set of zeros of A. We say that A is Let γ > 0. We say that B : The following technical result (see [21,Theorem 3(b)] and [16, Proposition 2.4]) gives an expression for the averaged parameter of the composition of two averaged operators. We refer also to [6,Proposition 4.32] for other results of this type.

Theorem 7 Let A : H ⇒ H be a maximally monotone operator and B
with x 0 ∈ H as starting point and (λ n ) n≥0 and (β n ) n≥0 real sequences satisfying the conditions: Then (x n ) n≥0 converges strongly to proj zer(A+B) (0).
Proof It is immediate that the iterative scheme (8) can be written in the form We consider two cases. The first one is when γ ∈ (0, 2β). According to [6, Corollary 23.8 and Remark 4.24(iii)], J γ A is 1 2 -cocoercive. Moreover, by [6, Proposition 4.33], Id − γ B is γ /2β-averaged. Combining this with Proposition 6, we derive that T is 2β/(4β − γ )-averaged. The statement follows now from Corollary 5, by noticing that The second case is when γ = 2β. The cocoercivity of B implies that Id − γ B is nonexpansive, hence the operator T = J γ A • (Id − γ B) is nonexpansive, too, the conclusion follows in this situation from Theorem 3.

Remark 8
The choice λ n = 1 for any n ≥ 0 in the previous theorem leads to the iterative scheme This last relation can be equivalently written as where ε n Id (with ε n := 1/β n − 1 > 0 and lim n→+∞ ε n = 0) represents the Tikhonov regularization term, which enforces the strong convergence of the sequence (x n ) n≥0 to the minimal norm solution. For other types of Tikhonov-like methods for monotone inclusion problems we refer the reader to [19,25,27] and the references therein.
In the remaining of this section we turn our attention to the solving of optimization problems of the form min where f : H → R ∪ {+∞} is a proper, convex and lower semicontinuous function and g : H → R is a convex and Fréchet differentiable function with 1/β-Lipschitz continuous gradient, for β > 0. For a proper, convex and lower semicontinuous function f : for x ∈ H with f (x) = +∞ and as ∂f (x) = ∅, otherwise. When seen as a set-valued mapping, the convex subdifferential is a maximally monotone operator (see [22]) and its resolvent is given by J ∂f = prox f (see [6] denotes the proximal operator of f. with x 0 ∈ H as starting point and (λ n ) n≥0 and (β n ) n≥0 real sequences satisfying the conditions: Then (x n ) n≥0 converges strongly to the minimal norm solution of (9).
Proof The statement is a direct consequence of Theorem 7, by choosing A := ∂f and B := ∇g and by taking into account that and the fact that ∇g is β-cocoercive due to the Baillon-Haddad Theorem (see [6,Corollary 18.16]).

A Douglas-Rachford algorithm with Tikhonov regularization term
In this section we derive from the Krasnosel'skiȋ-Mann algorithm formulated in Section 2 an iterative scheme of Douglas-Rachford-type, which generates sequences that strongly converge to a zero of of the sum of two set-valued maximally monotone operators.
In what follows, we denote by R A = 2J A − Id the reflected resolvent of a maximally monotone operator A : H ⇒ H.
Theorem 10 Let A, B : H ⇒ H be two maximally monotone operators such that zer(A + B) = ∅ and γ > 0. Consider the following iterative scheme: with x 0 ∈ H as starting point and (λ n ) n≥0 and (β n ) n≥0 real sequences satisfying the conditions: Then the following statements are true: Proof Taking into account the iteration rules and the definition of the reflected resolvent, the iterative scheme in the enunciation of the theorem can be equivalently written as where By applying Theorem 3, we obtain that (x n ) n≥0 converges strongly tox := proj FixT (0) as n → +∞, hence (a) holds. Further, by taking into account the definition of the sequence (y n ) n≥0 and the continuity of the resolvent operator, we obtain that (y n ) n≥0 converges strongly to J γ Bx ∈ zer(A + B) as n → +∞. Finally, by taking the limit in the recursive formula of the sequence (x n ) n≥0 , we obtain that z n − y n converges strongly to 0 as n → +∞, thus (b) holds, too.

Remark 11
The classical Douglas-Rachford method, which reads produces sequences for which in general only weak convergence to a zero of the A + B can be proved (see e.g. [6,Theorem 25.6]). In order to ensure strong convergence, one usually needs to impose restrictive conditions on the monotone operators involved, like uniform monotonicity (which is a generalization of strong monotonicity). This is not the case for the iterative scheme stated in Theorem 10, where we are able to guarantee strong convergence in the very general situation of maximally monotone operators.
Further, we look at optimization problems of the form where f , g : H → R ∪ {+∞} are proper, convex and lower semicontinuous functions. We denote by the effective domain of the function f. In order to proceed, we need the following notion. For S ⊆ H a convex set, we denote by is a closed linear subspace of H} its strong quasi-relative interior. Notice that we always have intS ⊆ sqriS (in general this inclusion may be strict). If H is finite-dimensional, then sqriS coincides with riS, the relative interior of S, which is the interior of S with respect to its affine hull. The notion of strong quasi-relative interior belongs to the class of generalized interiority notions and plays an important role in the formulation of regularity conditions, which are needed in convex optimization in order to guarantee duality results and also subdifferential sum formulas. The one considered in the next result is the so-called Attouch-Brézis regularity condition. We refer to [6,7,26,29] for more interiority notions and their impact on the duality theory.
Then the following statements are true: Proof The result is a direct consequence of Theorem 10 for A = ∂f and B = ∂g and by noticing that the regularity condition 0 ∈ sqri(dom f − dom g) ensures the relation (see [6,Proposition 7

Strongly convergent primal-dual algorithms
The aim of this section is to induce strong convergence in the nowadays so popular primal-dual algorithms designed for solving highly structured monotone inclusions involving parallel-sums and compositions with linear operators.

A primal-dual algorithm of forward-backward-type with Tikhonov regularization terms
In this subsection, the following monotone inclusion problem will be in the focus of our investigations.

Remark 15 (i) Since
(ii) The resolvent of the inverse operator of a maximally monotone operator M : H ⇒ H can be computed as follows (see [6]): Proof The idea is to apply Theorem 7 in an appropriate product space under the use of appropriate renorming techniques (see [28]). We consider the Hilbert space K = H × G 1 × · · · × G m endowed with inner product and associated norm defined, for respectively. Furthermore, we consider the set-valued operator which is maximally monotone, since A and B i , i = 1, . . . , m, are maximally monotone (see [6,Propositions 20.22 and 20.23]), and the linear continuous operator which is skew-symmetric (i.e. S * = −S) and hence maximally monotone (see [6, Example 20.30]). We also consider the single-valued operator We also introduce the linear continuous operator which is self-adjoint and ρ-strongly positive (see [28]), namely, the following inequality holds Therefore, its inverse operator V −1 exists and it fulfils V −1 ≤ 1/ρ. The algorithmic scheme (18) in the statement of the theorem can be written by using this notations as By introducing the sequences v 1,n , . . . , v m,n ), and y n = (p n , q 1,n , . . . , q m,n ) ∀n ≥ 0, the scheme (22) can equivalently be written in the form Furthermore, we have for any n ≥ 0 Let K V be the Hilbert space with inner product and norm defined, for x, y ∈ K, by x, y K V = x, Vy K and respectively. As the set-valued operators M + S and Q are maximally monotone on K, the operators A and B are maximally monotone on K V (see also [28]). Furthermore, B is βρ-cocoercive on K V . Moreover, since V is self-adjoint and ρ-strongly positive, weak and strong convergence in K V are equivalent with weak and strong convergence in K, respectively. Taking this into account, it follows that (23) becomes which is the algorithm presented in Theorem 7 for determining the zeros of A + B in case γ = 1. However, we have According to Theorem 7, x n converges strongly to proj zer(A+B) (0, 0, . . . , 0) in the space K V as n → +∞ and the conclusion follows from (21).
In the remaining of this subsection we investigate the convergence property of the algorithm (18) in the context of simultaneously solving complexly structured convex optimization problems and their Fenchel duals. The problem under investigation is the following one.
Problem 16 Let f ∈ (H) and h : H → R be a convex and differentiable function with a μ −1 -Lipschitz continuous gradient, for some μ > 0. Let m be a strictly positive integer and for i = 1, . . . , m, let G i be a real Hilbert space, g i , l i ∈ (G i ) such that l i is ν i -strongly convex, for some ν i > 0 and L i : H → G i a nonzero linear continuous operator. Consider the convex optimization problem and its Fenchel-type dual problem Corollary 17 In Problem 16, suppose that Let τ and σ i , i = 1, . . . , m, be strictly positive numbers such that (x 0 , v 1,0 , . . . , v m,0 ) ∈ H × G 1 × · · · × G m and set:

Remark 18
The proximal-point operator of the conjugate function can be computed via the Moreau's decomposition formula which is valid for γ > 0 and f ∈ (H) (see [6]).
Proof Consider the maximal monotone operators The Baillon-Haddad Theorem (see [6,Corollary 18.16]) ensures that C is μ-cocoercive.  (14) reads while the dual monotone inclusion problem (15) reads (32) thenx is an optimal solution of the problem (26), (v 1 , . . . ,v m ) is an optimal solution of (27) and the optimal objective values of the two problems coincide. Notice that (33) is nothing else than the system of optimality conditions for the primal-dual pair of convex optimization problems (26)- (27). The conclusion follows now from Theorem 14.
(ii) Further, let us discuss some conditions ensuring the existence of a primal optimal solution. Suppose that the primal problem (26) is feasible, which means that its optimal objective value is not identical +∞. The existence of optimal solutions for (26) is guaranteed if, for instance, f + h is coercive (that is lim x →∞ (f + h)(x) = +∞) and for all i = 1, . . . , m, g i is bounded from below. Indeed, under these circumstances, the objective function of (26) is coercive (use also [6, Corollary 11.16 and Proposition 12.14] to show that for all i = 1, . . . , m, g i l i is bounded from below and g i l i ∈ (G i )) and the statement follows via [6,Corollary 11.15]. On the other hand, if f + h is strongly convex, then the objective function of (26) is strongly convex, too, thus (26) has a unique optimal solution (see [6,Corollary 11.16]).
(iii) We discuss at this point a suitable regularity condition as mentioned at item (i) above. Since dom(g i l i ) = domg i + doml i , i = 1, . . . , m, one can use to this end the regularity condition of interiority-type (see also [15]) This is fulfilled provided that one of the following conditions is verified (see [15, Proposition 4.3]):

A primal-dual algorithm of Douglas-Rachford-type with Tikhonov regularization terms
The problem that we investigate in this section reads as follows.
together with the dual inclusion Different to Problem 13, the operators D i , i = 1, . . . , m are general maximally monotone operators, thus they will have to be addressed through their resolvents. This is why in this context a primal-dual algorithm relying on the Douglas-Rachford paradigm is more appropriate.
Proof For the proof we use Theorem 10 (see also [8]) in the same setting as in the proof of Theorem 14, namely, by considering K = H × G 1 × · · · × G m endowed with inner product and associated norm defined in (20). Furthermore, we consider again the maximally monotone operator We introduce the linear continuous operator which is self-adjoint and ρ-strongly positive (see also [8]), for namely, the following inequality holds Therefore, its inverse operator V −1 exists and it fulfils V −1 ≤ 1/ρ. The algorithmic scheme (38) is equivalent to By considering for any n ≥ 0 the notations x n = (x n , v 1,n , . . . , v m,n ), y n = (p 1,n , p 2,1,n , . . . , p 2,m,n ) and z n = (z 1,n , z 2,1,n , . . . , z 2,m,n ), the iterative scheme (40) can be written as which is further equivalent to Let K V be the Hilbert space with inner product and norm defined, for x, y ∈ K, via x, y K V = x, Vy K and are maximally monotone on K V . Furthermore, since V is self-adjoint and ρ-strongly positive, strong convergence in K V is equivalent with strong convergence in K.

Remark 24
The hypothesis (48) in Theorem 23 is fulfilled, if the primal problem (46) has an optimal solution, the regularity condition (34) holds and According to [6,Proposition 15.7], the latter also guarantees that g i l i ∈ (G i ), i = 1, . . . , m.

Numerical experiments: applications to the split feasibility problem
Let H and G be real Hilbert spaces and L : H → G a bounded linear operator. Let C and Q be nonempty, closed and convex subsets of H and G, respectively. The split feasibility problem (SFP) searches a point x ∈ H with the property The (SFP) was originally introduced by Censor and Elfving [11] for solving inverse problems in the context of phase retrieval, medical image reconstruction and intensity modulated radiation therapy. We show that the strongly convergent primal-dual algorithms which we have investigated in Section 5 are excellently suited to solve the (SFP), especially in the case when infinite dimensional Hilbert spaces are involved. For this purpose, we note that problem (51) can be written equivalently in the form where δ S (x) := 0, if x ∈ S +∞, else denotes the indicator function of a subset S of a Hilbert space. Another alternative way to write problem (51) is as the minimization problem where d C (x) = inf u∈C x − u denotes the distance of the point x ∈ H to the set C. Both optimization problems are special instances of Problem 16, consequently, we will solve them by making use of the algorithm stated in Corollary 17.
The optimization problem (52) can be stated in the framework of Problem 16 by taking f = δ C , m = 1, g 1 = δ Q , l 1 = δ {0} , L 1 = L and h = 0. The iterative scheme stated in Corollary 17 reads: The optimization problem (52) can be stated in the framework of Problem 16 by taking f = 0, m = 1, g 1 = δ Q , l 1 = δ {0} , L 1 = L and h = 1 2 d 2 C . Noticing that ∇( 1 2 d 2 C ) = Id −P C , the iterative scheme stated in Corollary 17 reads: p n = β n x n − τ (β n L * v n + β n x n − P C (β n x n )) x n+1 = β n x n + λ n (p n − β n x n ) q n = β n v n + σ L(2p n − β n x n ) − σ P Q (σ −1 β n v n + L(2p n − β n x n )) v n+1 = β n v n + λ n (q n − β n v n ). Furthermore, by using the Cauchy-Schwarz inequality yields The projection onto the set C can be computed as (see [6,Example 28.16]) x, e l s e .
On the other hand P Q is given by [  x, e l s e .
We implemented the algorithms (54) and (55) in MATLAB used symbolic computation for generating the sequences of iterates. One can easily notice that, in this particular setting, the split feasibility problem and, consequently, both addressed optimization problems are solvable. The numerical experiments confirmed that the primal-dual algorithms involving Tikhonov regularization terms outperform the ones without Tikhonov regularization terms, which correspond to the case when β n = 1 for every n ≥ 0 and for which is known that they weakly convergence to a primal-dual solution of the corresponding KKT system of optimality conditions (Tables 1  and 2).
In the two tables above we present the numbers of iterations needed by the two algorithms to approach a solution of the split feasibility problem (SFP). We consider as stopping criterion E(x n ) := 1 2 P C (x n ) − x n 2 + 1 2 P Q (Lx n ) − Lx n 2 ≤ 10 −3 , while taking as relaxation variables λ n = 0.4 for every n ≥ 0 and as Tikhonov regularization parameters β n := 1 − 1/(n + 1) for every n ≥ 0, respectively, β n = 1 for every n ≥ 0 for the variants without Tikhonov regularization terms.

Further work
We point out some directions of research related to proximal methods with Tikhonov regularization terms, which merit to be addressed starting from the investigations made in this paper: (1) To consider in the numerical algorithms of type forward-backward and Douglas-Rachford proposed in this paper dynamic step sizes, which are known to increase the flexibility of the algorithms. This can be for instance done by formulating first a Krasnosel'skiȋ-Manntype algorithm for determining an element in the intersection of the sets of fixed points of a family of nonexpansive operators (T k ) k≥0 . Suitable choices of the operators in this family, in the spirit of the investigations we made in the Sections 3 and 4, can lead to iterative algorithms with dynamic step sizes (see [14]).
(2) To employ in the iterative schemes discussed in this article inertial and memory effects, which are known to contribute to the acceleration of the convergence behaviour of the algorithms. (3) To translate the proposed numerical methods to a framework that goes beyond the Hilbert space setting considered in this article, in order to allow applications where functional spaces are involved. We refer the reader to [24,25,27] and the references therein for tools and techniques which allow to prove convergence statements in Banach spaces with some particular underlying geometric structures.