INEXACT GENERALIZED PROXIMAL POINT ALGORITHM WITH ALTERNATING INERTIAL STEPS FOR MONOTONE INCLUSION PROBLEM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we propose and study an inexact generalized proximal point algorithm with alternated inertial steps for solving monotone inclusion problem and obtain weak convergence results under some mild conditions. In the case when the operator T is such that T−1 is Lipschitz continuous at 0, we prove that the sequence of the iterates is linearly convergent. Fejér monotonicity of even subsequences of the iterates is also obtained. Finally, we give some priori and posteriori error estimates of our generated sequences.


INTRODUCTION
Let H be a real Hilbert space with inner product , and induced norm · . Given a maximal monotone set-valued operator, T : H → 2 H , we consider the following inclusion problem (1) find x ∈ H such that 0 ∈ T (x).
its sequence {z k } by the scheme which is equivalent to where c, called proximal parameter, is a positive real number. The inexact version of the PPA is defined as: In (64), the tolerance of accuracy is zero, and so (4) contains (64). However, (64) is of interest in its own right, since it requires estimating the resolvent accurately. Under different settings, both the exact and inexact versions of the PPA have been investigated in the literature. In [5], the convergence of both the exact and inexact versions of PPA was comprehensively studied.
It turns out that the PPA is a very powerful algorithmic tool and contains many well known algorithms as special cases, including the classical augmented Lagrangian method [6,7], the Douglas-Rachford splitting method [8] and the alternating direction method of multipliers [9,10]. For more facts about the PPA and generalizations of the PPA, one can consult the references [4,11,12]. The equivalent representation of the PPA (3), can be written as This can be viewed as an implicit discretization of the evolution differential inclusion problem It has been shown that the solution trajectory of (6) converges to a solution of (1) provided that T satisfies certain conditions (see e.g., [13]). To speed up convergence, the following second order evolution differential inclusion problem was introduced in the literature: where c > 0 is a friction parameter. If T = ∇ f , where f : R 2 → R is a differentiable convex function with attainable minimum, the system (7) characterizes roughly the motion of a heavy ball which rolls under its own inertia over the graph of f until friction stops it at a stationary point of f . In this case, the three terms in (7) denote, respectively, inertial force, friction force and gravity force. Consequently, the system (7) is usually referred to as the heavy-ball with friction (HBF) system. In theory, the convergence of the solution trajectories of the HBF system to a solution of (1) can be faster than those of the first-order system (6), while in practice the second order inertial term d 2 x dt 2 can be exploited to design faster algorithms (see e.g., [14,15]) . As a result of the properties of (7), an implicit discretization method was proposed in [16,17] as follows, given z k−1 and z k , the next point z k+1 is determined via which results to an iterative algorithm of the form where c = h 2 1+ch and α = 1 1+ch . Observe that (9) is the proximal point step applied to the extrapolated point z k + α(z k − z k−1 ), rather than x k itself as in the classical PPA. Hence the iterative algorithm (9) is a two-step method generally called inertial PPA (iPPA). Convergence properties of (9) were studied in [16,17] under some assumptions on the parameters α and c.
A major drawback of the iPPA is that Fejér monotonicity of z k − z * , z * ∈ zer(T ) is lost in many cases and hence, makes the sequence {z k } generated by the methods to swing back and forth around zer(T ). This situation makes these methods sometimes not converge faster than their counterpart non-inertial methods (see, e.g., [27,28,32]). Furthermore, no linear convergence rate of iPPA has been obtained in the literature (see, for example, [31]).
A search to overcome this drawback led to introduction of the so-called alternated iPPA in the literature. It has been shown that with the alternated iPPA, some sort of Fejér monotonicity of z k − z * , z * ∈ zer(T ) is recovered and that the method out-performs their non-inertial counterparts. Please, refer to [32,34] for more details. In [12], the authors studied the generalized PPA for maximal monotone set-valued operator T such that T −1 is Lipschitz continuous at 0 in real Hilbert space and obtained both weak and linear convergence results.
Motivated by the results mentioned above, Shehu and Ezeora [29] proposed and studied an alternated inertial exact generalized PPA for solving monotone inclusion problem in real Hilbert space. They obtained the following results: • Fejér monotonicity of z k − z * , z * ∈ zer(T ) to some extent, which is not obtained for iPPA in [16,17] and other related works; • weak convergence of the generated sequence {z k } to a point in zer(T ) and thereby generalizes the results obtained in [34]; • linear convergence of the generated sequence {z k } to a unique point z * under the condition that T is maximal monotone and T −1 is Lipschitz continuous at 0, which has not been obtained for iPPA in [16,17,34] and other related works • priori and posteriori error estimates of the generated sequence {z k }.
As we mentioned earlier, the inexact PPA is more general than the exact PPA.
Question of Interest: Can an inexact generalized PPA with alternated inertial terms be developed for solving problem (1) such that some or all the results of [29] are recovered?
It is our aim in this article to answer the question in the positive.

PRELIMINARIES
In this section, we present some definitions and known results needed for further discussions.

Some Definitions. Let
(ii) firmly nonexpansive if (iii) θ − strongly monotone if there exists θ > 0 such that , and for some Definition 2.3. Suppose a sequence {x n } in H converges in norm to x * ∈ H. We say that {x n } It is well known that Q− linear convergence implies R− linear convergence, but the reverse implication is not true.
• From the definitions, the following conclusions hold. If T is nonexpansive, then it is Lipschitz continuous. Furthermore, the problem (1) has a unique solution point when T −1 is Lipschitz continuous at 0.
• Example of a set-valued map T such that T −1 is Lipschitz continuous at O is given in [12]. It is also shown in [12] that Lipschitz continuity at O is weaker than strong monotonicity as assumed in the results of many authors (see e.g., [33]).

Some Known Results.
Lemma 2.1. We have the following facts.
(ii) An operator T is firmly nonexpansive if and only if 2T − I is nonexpansive.
(iii) An operator is firmly nonexpansive if and only if it is of the form 1 2 (K + I), where K is nonexpansive. iv) An operator T is firmly nonexpansive if and only if I − T is firmly nonexpansive. Lemma 2.2. (see [12]) Let T : H → 2 H be set-valued and maximal monotone, define J λ T = (ii) ||x + y|| 2 ≤ ||x|| 2 + y, x + y ∀x, y ∈ H.
Lemma 2.4. Let C be a nonempty set of H and {z k } be a sequence in H such that the following two conditions hold: (i). for any x ∈ C, lim n→∞ ||z k − x|| exists; (ii) every sequential weak cluster point of {z k } is in C. Then {z k } converges weakly to a point in C.

CONVERGENCE OF INEXACT GPPA WITH ALTERNATED INERTIA
In this section, we introduce and study the following inexact GPPA with alternated inertal term.
Observe that Hence, On the other hand, from (16), Therefore, we have using (24) that and so by (28), Applying Lemma 3.3 and condition (ii) to (31), we conclude that lim k→∞ ||z 2k − z * || exists. Hence, From (33) and (34), we have that That is This completes the proof.
Proof. Since the sequence {z 2k } is bounded,there exists a subsequence {z 2k j } of {z 2k } which converges weakly to some point z * .
Claim: z * is unique.
Assume for contradiction that there exists ξ ∈ H such that z 2k j converges weakly to ξ . Then Hence, z * is unique. Definition 2.2 of weak convergence gives From (35) and (42), we have that for all µ ∈ H, Hence {z k } converges weakly to z * ∈ T −1 (0).

linear Convergence.
Lemma 3.4. Let T : H → 2 H be maximal monotone and z * be a solution point of (1), let c > 0.
If T −1 is Lipschitz continuous at 0 with modulus a > 0, then there exists a positive number τ such that Proof. Applying Property (ii) in Lemma 2.2 withz = z * and noticing that c > 0, we get Now, using definition of J cT we get c −1 (I − J cT )(z) ∈ T (J cT (z)). Since T −1 is Lipschitz continuous at 0 with modulus a > 0, it follows from definition2.1 that there exists a positive parameter τ such that That is, Substituting this inequality (47) into (45), we obtain From (48), we get (44). This completes the proof.
Lemma 3.5. Let {z k } be the sequence generated by the algorithm (37), with γ ∈ (0, 2), and z * be a solution point of (1). If T −1 is Lipschitz continuous at 0 with modulus a > 0, and the proximal parameter c is positive, then there exists an integerk such that Proof. Observe that Lemma 3.4 above holds for all z ∈ H. Setw k = J cT (w k ), thenw k ∈ H ∀ k.
Also, w k is in H for all k. Therefore, the proof follows from Lemma 3.4 above .
assumptions hold: (iv) T −1 is Lipschitz continuous at 0 with modulus a > 0, and the proximal parameter c is positive ( c > 0 ) Given z 0 , z 1 ∈ H, let the sequence {z k } be generated by then there exists a positive integer k 1 such that and z * is the unique solution of (1) Proof.

CONCLUSION
In this paper, we introduce an alternated inertia inexact generalized PPA for solving monotone inclusion problem in infinite dimensional real Hilbert spaces. Weak and linear convergence results are obtained. Fejér monotonicity of z k − z * , z * ∈ zer(T ) to some extent, which is not obtained for iPPA in [16,17] and other related works is aslo recovered. It has been recently stated in [31] that one cannot obtain linear convergence result with inertial PPA; here in our results, we are able to show that linear convergence result for alternated inertial PPA is possible even for the more general inexact version of alternated inertial PPA.