Received 7 January 2016; accepted 17 July 2016; published 20 July 2016
We are concerned in this study with the problem of finding a point satisfying
Martinet  proposed the following algorithm for the first time for applying it to convex optimization by considering a sequence of scalars, which are different from zero:
Rockafellar  thoroughly explored the method (2) in the general framework of maximal monotone inclu- sions. In particular, Rockafellar (  , Theorem 1) shows that when is an approximate solution of (2) and T is maximal monotone, then for a sequence of positive scalars the iteration (2) generates a sequence which is weakly convergent to a solution of (1) for any starting point. In  , 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).
Let. The subset of X, denoted by, is defined by
Thus we have the following algorithms which have been presented by Aragón Artacho et al.  :
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  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 “so- called” 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:
We observe, from Algorithm 2, that
1) if and then we assume a Hilbert space, this algorithm reduces to the classical pro- ximal point algorithm defined by (2).
2) if, Algorithm 2 is equivalent to the classical Gauss-type proximal point method, which has been introduced by Rashid et al.  .
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  -  ). Spingarn  has been studied first weaker form of monotonicity and for details see in  .
There have a large study on local convergence analysis about Algorithm 1 (cf.    ), 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.   -  ). In  , 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.
2. 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. Let and. The closed ball centered at x with radius r is denoted by. The domain, the inverse and the graph of F are respectively defined by
All the norms are denoted by. Let and. The distance from x to A is defined by
while the excess from the set C to the set A is defined by
From  , we recall the following definition of metric regularity for set-valued mapping.
Definition 1 Let be a set-valued mapping and let. Let and. Then F is said to be
1) metrically regular at on with constant if
2) metrically regular at if there exist constants and such that F is metrically regular at on with constant.
The infimum of the set of values for which (4) holds is the modulus of metric regularity, denoted by. The absence of metric regularity at for corresponds to. 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 and the expression measures the residual when.
Recall the definition of Lipschitz-like continuity for set-valued mapping from  . This notion was introduced by Aubin in  and has been studied extensively.
Definition 2 Let be a set-valued mapping and let. Let and. Then is said to be Lipchitz-like at on with constant M if the following inequqlity hold:
The following result establish the equivalence relation between metric regularity of a mapping F at and the Lipschitz-like continuity of the inverse at, which is obtained from the idea in   .
Lemma 1 Let be a set valued mapping and. Let, then F is metrically regular at on if and only if its inverse is Lipschitz-like at on with constant such that
We recall the following statement of Lyusternik-Graves theorem for metrically regular mapping from  . 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 is locally closed at if there exists such that the set is closed.
Lemma 2 Consider a mapping and any at which is locally closed. Let F be metrically regular at for with constant. Consider also a function which is Lipschitz continuous at with Lipschitz constant such that. Then the mapping is metric-
ally regular at for with constant.
We finished this section with the following lemma, which is known as Banach fixed point theorem proved in  .
Lemma 3 Let be a set-valued mapping. Let, and be such that
Then has a fixed point in, that is, there exists such that. If is additionally single-valued, then the fixed point of in is unique.
3. Convergence Analysis of GG-PPA
In this section, we assume that is a set-valued mapping with locally closed graph at such that T is metrically regular at with constant. Let be a (single-valued) function such that, which is Lipschitz continuous in a neighborhood O of 0 with a Lipschitz constant Let and define a mapping by
Then we obtain the following equivalence
Let and let. Since is Lipschitz continuous on, app- lying the Lyusternik-Graves theorem (see Lemma 2) we assume that the mapping is metrically regular at
with constant, that is, by Lemma 1 we have the following inequality
The following lemma plays an important role for convergence analysis of the GG-PPA, which is due to  .
Lemma 4 Suppose that is metrically regular at on with constant such that (12) and (13) are satisfied. Let and. Then is Lipschitz-like at on with constant, that is,
For our convenience, we consider a sequence of functions with which are Lipschitz continuous in a neighbourhood O of 0, the same for all k, with Lipschitz constants satisfying
We rewrite the mapping in (8) by substituting instead of g as follows:
Since by (14), then by Lyusternik-Graves theorem (see Lemma 2) and Lemma 1 we obtain that the mapping is Lipschitz-like at on with constant satisfying (11) and hence we have
Furthermore, we define, for each, the mapping by
and the set-valued mapping by
The main result of this study given as follows, which provides some sufficient conditions ensuring the convergence of the GG-PPA with initial point.
Theorem 1 Suppose and that is metrically regular at on with constant, and let be defined in (12). Let and be such that
Then there exists some such that any sequence generated by Algorithm 2 with initial point in converges to a solution of (1), that is, satisfies that.
Then by assumption (b), (21) gives us
Assumption (c) and (20) allow us to take so that
We will proceed by mathematical induction and show that Algorithm 2 generates at least one sequence and any sequence generated by Algorithm 2 satisfies the following assertions
for each. Define
Since, by assumption (b) and (c), we have
It is trivial that (24) is true for. For showing that (25) is true for, we need to prove that exists, that is,. We will prove by applying Lemma 3 to the mapping with
. Let us check that both assertions (6) and (7) of Lemma 3 are hold with and. Noting that
by (10). Then by the mapping in (18) and the definition of excess e, we have
(noting that). Now, by the choice of, we have
Since, by the fact in assumption (a) and in assumption (c), for each, (29) implies that
that is, for each,. 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. Then by assumption (a) and (27), we have
and by (30). By assumed Lipschitz-like property of, we have
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 such that, which translates to, that is, and hence.
Now, we show that (25) is hold for.
Note that by assumption (a). Then (13) is valid for (14). Since is Lipschitz-like at on, it follows from Lemma 4 that the mapping is Lipschitz-like at on
with constant for each. In particular, is Lipschitz-like at on with constant as by assumption (a) and the choice of. Furthermore, assumptions (a) and (c) imply that
It seems that. Then by Lemma 1 we can say that the mapping is metrically regular on
relative to with constant. Thus by applying Lemma 1, we have
and (23) implies that
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.
Suppose that the points have been obtained, and (24) and (25) are true for. We will show that there exists a point such that (24) and (25) also hold for. Since (24) and (25) are true for each, we have the following inequality
This reflects that (24) holds for. Now with almost the same argument as we did for the case when, we can find that the mapping is also Lipschitz-like at on with
constant. Then by applying again Algorithm 2, we have
This shows that (25) holds for. Therefore, the proof is completed.
In the particular case, when is a solution of (1), that is, , 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 1 Suppose that and that satisfies and that is metrically regular at
with constant. Let be such that and suppose that
Then there exists some such that any sequence generated by Algorithm 2 with initial point in converges to a solution of (1), that is, satisfies that.
Proof. Since is metrically regular at, there exist constants, and such that is metrically regular at on with constant. Then, for each, one has that
Let be such that. Choose such that and Then
Thus we can choose such that
Now it is routine to check that inequalities (a)-(c) of Theorem 1 are hold. Thus Theorem 1 is applicable to complete the proof of the corollary.
4. Numerical Experiment
We will provide, in this section, a numerical example to validate the semilocal convergence results of GG-PPA.
Example 1 Let. Define a set-valued mapping T on by. Consider a sequence of Lipschitz continuous function with, which is
defined by. Then Algorithm 2 generates a sequence which is converges to.
It is obvious from the statement that T is metrically regular at and is Lipschitz continuous on the neighborhood of 0 with Lipschitz constant. Consider. Then from (3), we have that
On the other hand, if we obtain that
Thus from (40), we obtain that
For the given values of, we see that. 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.
Moreover, in the case when, we can sketch the following Figure 1:
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, a sequence of functions with which is Lipschitz continuous in a neighbourhood O of the origin
Figure 1. Graphical representation of.
Table 1. Finding a solution of generalized equation.
and T is metrically regular. Moreover, we have presented a numerical experiment to validate the semilocal convergence result for Algorithm 2. For the case where, 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, the following implication holds:
To see the detail proof of the above implication, one can refer to  .
We thank the editor and the referees for their comments. Research of this work is funded by the Ministry of Science and Technology, Bangladesh, grant No. 39.009.002.01.00.053.2014-2015/EAS-19. This support is greatly appreciated.
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