Convergence Analysis of General Version of Gauss-Type Proximal Point Method for Metrically Regular Mappings

We introduce and study in the present paper the general version of Gauss-type proximal point algorithm (in short GG-PPA) for solving the inclusion , where T is a set-valued mapping which is not necessarily monotone acting from a Banach space X to a subset of a Banach space Y with locally closed graph. The convergence of the GG-PPA is present here by choosing a sequence of functions  with , which is Lipschitz continuous in a neighbourhood O of the origin and when T is metrically regular. More precisely, semi-local and local convergence of GG-PPA are analyzed. Moreover, we present a numerical example to validate the convergence result of GG-PPA.


Introduction
We are concerned in this study with the problem of finding a point x X ∈ Ω ⊆ satisfying ( ) where : 2 Y T X  is a set-valued mapping and X and Y are Banach spaces. This type of inclusion is an abstract Md. A. Alom et al. 1249 model for a wide variety of variational problems including complementary problems, system of nonlinear equations and variational inequalities. In particular, it may characterize optimality or equilibrium problems. Choose a sequence of functions : Lipschitz continuous in a neighborhood O of the origin.
Martinet [1] proposed the following algorithm for the first time for applying it to convex optimization by considering a sequence of scalars { } k λ , which are different from zero: Rockafellar [2] thoroughly explored the method (2) in the general framework of maximal monotone inclusions. In particular, Rockafellar ([2], Theorem 1) shows that when 1 k x + is an approximate solution of (2) and T is maximal monotone, then for a sequence of positive scalars k λ the iteration (2) generates a sequence { } k x which is weakly convergent to a solution of (1) for any starting point 0 x X ∈ . In [3], Aragón Artacho et al. have been presented the general version of the proximal point algorithm (GPPA) (see Algorithm 1), for the case of nonmonotone mappings, for solving the inclusion (1).
Step 2. If , then stop; otherwise, go to Step 3.
Step 3. Put { } ( ) Step 4. Set 1 : Step 5. Replace k by 1 k + and go to Step 2. Note that, for a starting point near to a solution, the sequences generated by Algorithm 1 are not uniquely defined and not every sequence is convergent. The results obtained in [3] guarantee the existence of one sequence, which is convergent. Therefore, from the viewpoint of numerical computation, we can assume that these kinds of methods are not suitable in practical application. This drawback motivates us to introduce a method "socalled" general version of Gauss-type proximal point algorithm (GG-PPA). The difference between Algorithm 1 and our proposed Algorithm 2 is that the GG-PPA generates sequences, whose every sequence is convergent, but this does not happen for Algorithm 1. Thus we propose here the GG-PPA as follows: λ ∈ ∞ and put : 0 k = .
Step 2. If , then stop; otherwise, go to Step 3. Step Step 4. Set 1 : Step 5. Replace k by 1 k + and go to Step 2. We observe, from Algorithm 2, that 1) if ( ) k k g u u λ = and then we assume Y X = a Hilbert space, this algorithm reduces to the classical proximal point algorithm defined by (2).
2) if ( ) k k g u u λ = , Algorithm 2 is equivalent to the classical Gauss-type proximal point method, which has been introduced by Rashid et al. [4].

1250
A large number of authors have been studied on proximal point algorithm and have also found applications of this method to specific variational problems. Most of the study on this subject have been concentrated on various versions of the algorithm for solving inclusions involving monotone mappings, and specially, on monotone variational inequalities (see in [5]- [8]). Spingarn [9] has been studied first weaker form of monotonicity and for details see in [10].
There have a large study on local convergence analysis about Algorithm 1 (cf. [3] [11] [12]), but there is no semilocal analysis for Algorithm 1. A huge number of contributions have been studied on semilocal analysis for the Gauss-Newton method (cf. [4] [13]- [16]). In [4], Rashid et al. have given a semilocal convergence analysis for the classical Gauss-type proximal point method. As our best knowledge, there is no study on semilocal analysis for Algorithm 2. Therefore we conclude that the contributions presented in this study are seems new.
In the present paper, our aim is to study the semilocal convergence for the GG-PPA defined by Algorithm 2. The metric regularity property and Lipschitz-like property for set-valued mappings are mainly used in our study. The main results are convergence analysis, established in section 3, which based on the attraction region around the initial point and provide some sufficient conditions ensuring the convergence to a solution of any sequence generated by Algorithm 2. As a consequence, local convergence results for GG-PPA are obtained.
This paper is arranged as follows. In Section 2, some necessary notations, notions and preliminary results are presented. In Section 3, we consider the GG-PPA which is introduced in Section 1 and by using the concept of metric regularity property for the set valued mapping T, we will show the existence and present the convergence of the sequence generated by Algorithm 2. In Section 4, we present a numerical experiment to validate the semilocal convergence of Algorithm 2. In the last Section, we will give a summary of the major results to close our paper.

Notations and Preliminary Results
In the whole paper, we assume that X and Y are Banach spaces. Let F be a set-valued mapping from X into the subsets of Y, denoted by : 2 Y F X  . Let x X ∈ and 0 r > . The closed ball centered at x with radius r is denoted by .
All the norms are denoted by ⋅ . Let A X ⊆ and C X ⊆ . The distance from x to A is defined by while the excess from the set C to the set A is defined by , we recall the following definition of metric regularity for set-valued mapping.
2) metrically regular at ( ) with constant κ . The infimum of the set of values κ for which (4) holds is the modulus of metric regularity, denoted by | F x y . The absence of metric regularity at x for y corresponds to ( ) reg | F x y = ∞ . The inequality (4) has direct use in providing an estimate for how far a point x is from being a solution to the generalized equation Recall the definition of Lipschitz-like continuity for set-valued mapping from [17]. This notion was introduced by Aubin in [18] and has been studied extensively.
for any , .
x y r r e y x y M y y y y y The following result establish the equivalence relation between metric regularity of a mapping F at ( ) , x y and the Lipschitz-like continuity of the inverse , for all , .
x y r r e F y x F y y y y y y We recall the following statement of Lyusternik-Graves theorem for metrically regular mapping from [21]. This theorem plays an important role in the theory of metric regularity and proves the stability of metric regularity of a generalized equation under perturbations. For its statement, we use that a set C X ⊂ is locally We finished this section with the following lemma, which is known as Banach fixed point theorem proved in [22].
Then Φ has a fixed point in

Convergence Analysis of GG-PPA
In this section, we assume that : 2 Y T X  is a set-valued mapping with locally closed graph at ( ) : .
Then we obtain the following equivalence ( ) ( ) ( ) 1 for any and .
x z P y y g z x T z z X y Y − ∈ ⇔ ∈ − + ∈ ∈ In particular, The following lemma plays an important role for convergence analysis of the GG-PPA, which is due to [23].
Since 1 1 6 λκ < < by (14), then by Lyusternik-Graves theorem (see Lemma 2) and Lemma 1 we obtain that the mapping (11) and hence we have Furthermore, we define, for each x X ∈ , the mapping : and the set-valued mapping : 2 .
Then Md. A. Alom et al.
The main result of this study given as follows, which provides some sufficient conditions ensuring the convergence of the GG-PPA with initial point 0 x .
Then there exists some ˆ0 δ > such that any sequence { } k x generated by Algorithm 2 with initial point in Then by assumption (b), (21) gives us Assumption (c) and (20)   , ). Now, by the choice of λ , we have . In particular, Hence by using (31) and Lemma 1 for Lipschitz-like property in (28), we have This shows that assertion (6) of Lemma 3 is satisfied. Now, we show that the assertion (7) of Lemma 3 is satisfied. Let Applying (19) in (32), we obtain Then by (14), (33) reduces to This implies that the assertion (7) of Lemma 3 is also satisfied. Since both assertions (6) and (7) of Lemma 3 are fulfilled, we can deduce there exists a fixed point , which translates to ( ) ( ) , that is,

Md. A. Alom et al.
1255 Now, we show that (25) is hold for 0 k = . Note that 0 r > by assumption (a). Then (13) is valid for (14). Since , it follows from Lemma 4 that the mapping ( ) . In particular, and (23) Then from (16) and using (36), we obtain that From Algorithm 2 and using (21) and (37), we obtain that ( ) ( ) This implies that (25) is hold for 0 k = . Suppose that the points 1 , , n x x have been obtained, and (24) and (25) are true for 0,1, 2, , 1 k n = − . We will show that there exists a point 1 n x + such that (24) and (25) also hold for k n = . Since (24) and (25) are true for each 1 k n ≤ − , we have the following inequality This reflects that (24) holds for k n = . Now with almost the same argument as we did for the case when 0 k = , we can find that the mapping ( ) This shows that (25) holds for k n = . Therefore, the proof is completed.

Md. A. Alom et al.
1256 In the particular case, when x is a solution of (1), that is, 0 y = , Theorem 1 is reduced to the following corollary, which gives the local convergence of the sequence generated by the GG-PPA defined by Algorithm 2. Corollary . Then Algorithm 2 generates a sequence which is converges to * 0.5 x = .
It is obvious from the statement that T is metrically regular at ( ) . Thus, this implies that the sequence generated by Algorithm 2 converges linearly. Then the following Table 1, obtained by using Mat lab program, indicates that the solution of the generalized equation is 0.5 when 10 k = .

Conclusions
In this study, we have established semi-local and local convergence results for the general version of Gauss-type proximal point algorithm for solving generalized equation under the assumptions that 1 η > , a sequence of functions :  η = , the question, whether the results are true for GG-PPA, is a little bit complicated. However, from the proof of the main theorem, one sees that all the results obtained in the present paper remain true provided that, for any x X ∈ Ω ⊆ , the following implication holds: To see the detail proof of the above implication, one can refer to [17].