Abstract
In this article, we study the stability properties of a Gauss-type proximal point algorithm for solving the inclusion y ϵ T (x), where T is a set-valued mapping acting on a Banach space X with locally closed graph that is not necessarily monotone and y is a parameter. Consider a sequence of bounded constants {λk} which are away from zero. Under this consideration, we present the semi-local and local convergence of the sequence generated by an iterative method in the sense that it is stable under small variation in perturbation parameter y whenever the set-valued mapping T is metrically regular at a given point. As a result, the uniform convergence of the Gauss-type proximal point method will be established. A numerical experiment is given which illustrates the theoretical result.
PUBLIC INTEREST STATEMENT
A large number of problems in engineering, optimization, economics and other disciplines can be brought in the form of equations. The unknowns of engineering equations can be differential equations, integral equations, systems of linear and nonlinear algebraic equations. The computational technique gives a lot of opportunity to researchers to solve these equations. The most commonly used solution methods for these equations are iterative and such iteration methods are applied for solving optimization problems. The generalized equation is an abstract model of a wide variety of variational problems including systems of inequalities, linear and nonlinear complementarity problems, system of nonlinear equations and first-order necessary conditions for nonlinear programming, equilibrium problems, etc. They have also plenty of applications in engineering and economics. In this communication, we have studied an iterative technique, namely Gauss-type proximal point method, to show the stability of the convergence of this method for solving generalized equation.
1. Introduction
This article is about to study the stability of the Gauss-type proximal point method for solving the perturbed inclusions involving set-valued mappings and parameters. We consider the perturbed inclusion of the following form:
where is a set-valued mapping with closed graph acting on a Banach space and is a parameter.
This type of inclusion is an abstract model for a wide variety of variational problems including complementary problems, systems of nonlinear equations and variational inequalities. In particular, it may characterize optimality or equilibrium. The classical proximal point algorithm (see Algorithm 5.1 in Section 3), whose origins can be traced back to (Krasnoselskii, Citation1955), was born in the 1960s (see, e.g. (Martinet, Citation1970; Moreau, Citation1965)).
Rockfellar (Citation1976a) presented a general convergence and rate of convergence analysis of the classical proximal point algorithm (see Algorithm 5.1 in Section 3) for solving (i.e. when y = 0) in the case when is a Hilbert space and is maximal monotone operator.
Furthermore, in his subsequent paper (Rockafellar, Citation1976b), he established its connection with the augmented Lagrangian method of constrained nonlinear optimization. In particular, Rockafellar (Rockafellar, Citation1976a, Theorem 1) showed that when is an approximate solution of and is maximal monotone, then the Algorithm 5.1 generates a sequence which is weakly convergent to a solution of for any starting point .
For solving the inclusion , Aragón Artacho, Dontchev, and Geoffroy (Citation2007) presented the following general version of the proximal point algorithm by considering a set-valued mapping acting Banach spaces and for the nonmonotone case and choosing a sequence of functions with which are Lipschitz continuous
and proved the sequence obtained by (3.2) converges linearly. The uniform convergence analysis of the method (3.2) is given by Aragón Artacho and Geoffroy in (Aragón Artacho & Geoffroy, Citation2007). Many authors have been studied on proximal point algorithm and have also found applications of this method to specific variational problems. Most of the rapidly growing study on this subject has been concentrated on various versions of the algorithm for solving inclusions involving monotone mappings, and specially, on monotone variational inequalities (see (Anh, Muu, Nguyen, & Strodiot, Citation2005; Auslender & Teboulle, Citation2000; Bauschke, Burke, Deutsch, Hundal, & Vanderwerff, Citation2005; Solodov & Svaiter, Citation1999; Yang & He, Citation2005)). In recent work, Rashid et al. (Rashid, Jinhua, & Li, Citation2013) introduced the Gauss-type proximal point method and studied the semilocal and local convergence of the sequence generated by this method for solving the inclusion (3.1) when . Moreover, by comparing with the results in Rockafellar (Rockafellar, Citation1976a, Theorem 1), these authors showed that the sequence generated by Gauss-type proximal point algorithm is more precise than the sequence generated by Algorithm 5.1. To see the further developments on perturbed generalized equations dealing with metrically regular mappings, one can refer to (Alom, Rashid, & Dey, Citation2016; Dontchev & Rockafellar, Citation2013; Rashid & Yuan, Citation2017).
Inspired by the works of Dontchev (Dontchev, Citation1996b) (or Aragón Artacho and Geoffroy (Citation2007)), we propose the restricted proximal point method (see Algorithm 5.2 in Section 3) and study the convergence analysis of this method for solving (3.1), which will imply the uniform convergence of the Gauss-type proximal point method introduced in (Rashid et al., Citation2013).
In this article, our approach is to study the semilocal and local convergence of the sequence generated by Algorithm 5.2 under the assumption that is metrically regular, which means the uniform convergence of the Gauss-type proximal point method in Rashid et al. (Citation2013) will be established. Indeed, we present a kind of convergence of the sequence generated by Algorithm 5.2 which is uniform in the sense that the attraction region (i.e. the ball in which the initial guess can be taken arbitrarily) does not depend on small variations in the perturbation parameter near and for such values of this method finds a solution of (3.1) whenever T is metrically regular.
The main tools, we use in this study, are metric regularity and Lipschitz-like properties for set-valued mappings. Based on the information around the initial point, we establish convergence criteria in Section 3, which provides some sufficient conditions ensuring the convergence to a solution of any sequence generated by Algorithm 5.2. As a consequence, uniformity of the local convergence result for Gauss-type proximal point method is obtained.
The content of this article is organized as follows. In Section 2, we present some notations, notions and some preliminary results. In Section 3, we introduce the restricted proximal point method defined by Algorithm 5.2. Utilizing the concept of Lipchitz-like and metric regular property, we show the existence and the convergence of the sequence generated by Algorithm 5.2. As a result, stability properties of the Gauss-type proximal point method will be justified. In Section 4, a numerical experiment is provided to illustrate the theoretical result. In the last section, we give a summary of the major results presented in this article.
2. Notations and preliminary results
Let be a real Banach space and be a set-valued mapping on , indicated by . The domain , the inverse and the graph of are, respectively, defined by
and
Let and . The closed ball centered at with radius is denoted by . All the norms are denoted by . Let . The distance function of is defined by
while the excess from the set to the set is defined by
We begin with the definition of metric regularity and pseudo-Lipschitz mappings for a set-valued mappings. The following concept of metric regularity for a set-valued mapping is extracted from Dontchev & Rockafellar (Citation2004), whereas the notion of pseudo-Lipschitz property was introduced by Aubin (Aubin, Citation1984; Aubin & Frankowska, Citation1990). In particular, connection to linear rate of openness, pseudo-Lipschitz continuty, coderivative and metric regularity of set-valued mappings were established by Penot (Penot, Citation1989) and Mordukhovich (Mordukhovich, Citation1992). To see more details on these topics, one can refer to Dontchev & Rockafellar (Citation2004, Citation2001), Ioffe (Citation2000), Mordukhovich (Citation1993) and books (Mordukhovich, Citation2006; Rockafellar & Wets, Citation1997).
Definition 4.1. Let be a set-valued mapping, and let . Let and . Then
is said to be
metrically regular at on with constant if the following inequality holds:
(4.1) (4.1)metrically regular at for if there exist constants such that is metrically regular at on with constant .
is said to be
(i) Lipchitz-like at on with constant if the following inequality holds:
(4.2) (4.2)pseudo-Lipchitz around if there exist constants and such that is Lipschitz-like at on with constant .
Remark 4.1. The infimum of the set of values for which (4.1) holds is the modulus of metric regularity, denoted by . The absence of metric regularity at for corresponds to . The inequality (4.1) has direct use in providing an estimate for how far a point is from being a solution to the generalized equation and the expression measures the residual when .
Remark 4.2. Equivalently, for the property (b–i) we can say that is Lipschitz-like at on with constant if for every and for every , there exists such that
The following lemma plays an important role to prove our main result. This lemma establishes the connection between the metric regularity and the Lipchitz-like property. To see the proof of this lemma, one can refer to Rashid et al. (Citation2013) or monogram (Dontchev & Rockafellar, Citation2009, Theorem 3E.6).
Lemma 4.1. Let be a set-valued mapping and let . Then is metrically regular at on with constant if and only if is Lipschitz-like at on with the same constant , that is, the latter condition satisfies the following inequality:
Recall the following statement which is a refinement of the Lyusternik-Graves theorem for metrically regular mapping taken from Dontchev, Lewis, & Rockafellar (Citation2002), Theorem 3.3). Analogue developments on this result appear in Dontchev (Citation1996a), Theorem 1.4) or Section 1 in Ioffe (Citation2000). This theorem plays an important role in the theory of metric regularity. This theorem proves the stability of metric regularity of a generalized equation under perturbations. Roughly says that a generalized equation with solution can be perturbed by adding a to a single-valued mapping which is Lipschitz continuous with , by fundamental estimate so as to get a generalized equation still having solution . For its statement, we recall that a set is locally closed at if there exists such that the set is closed.
Proposition 4.1. Let be a set-valued mapping and let . Let be a metrically regular at on with constant and be closed. Consider a function which is Lipschitz continuous at with Lipschitz constant such that . Then the mapping is metrically regular at on with constant
We end this section with the following fixed point lemma for set-valued mappings, which was proved in Dontchev & Hager (Citation1994), Lemma (fixed point), is a generalization of the fixed point theorem (Ioffe & Tikhomirov, Citation1979).
Lemma 4.2. Let be a set-valued mapping. Let , and be such that
and
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. Stability of convergence analysis
Throughout, we suppose that is a Banach space and let be a set-valued mapping. Let and be such that and , the image of . Assume that is metrically regular at on with constant and is closed.
Let and . For any , we define by
Recall the classical proximal point method, introduced in Rockafellar (Citation1976a), which is defined as follows:
Algorithm 5.1. (The Proximal Point Method (PPM))
Step 1. Initialize , , , and put .
Step 2. If then stop; otherwise go to Step .
Step 3. If ,choose such that .
Step 4. Set .
Step 5. Update and go to Step .
The restricted proximal point method we propose here is given in the following:
Algorithm 5.2. (The Restricted Proximal Point Method (RPPM))
Step 1. Given , , , , ,and put .
Step 2. If then stop; otherwise go to Step .
Step 3. If ,choose such that there exists and
Step 4. Set .
Step 5. Update and go to Step .
We remarked that if and the set is singleton for each , Algorithm 5.1 and Algorithm 5.2 are coincident. However, when is not singleton, Algorithm 5.2 is a restricted version of Algorithm 5.1 since it imposes a restriction on the length of , . Moreover, if , the Algorithm 5.2 coincides with the Gauss-type proximal point algorithm introduced in Rashid et al. (Citation2013).
This section is intended to prove that whenever is metrically regular at on with constant , then, for starting point and for every element , there is a sequence generated by Algorithm 5.2 which is convergent to a solution of (3.1) for .
In order to proceed, let and . For our convenience, define a mapping by
and is an identity Lipschitz continuous function on .
Then, we obtain the following equivalence
In particular,
Note that
It is obvious that the mapping is Lipschitz continuous on . Since is metrically regular at on with constant and is closed, by applying Lyusternik-Graves theorem (see Proposition 4.1) we have that the mapping is metrically regular at on with constant . Setting
Then
To prove an important result in this section, we need the following lemma. This lemma plays an important role for convergence analysis of the restricted proximal point method. Up to some minor adjustment and simplifications of (Aragón Artacho & Geoffroy, Citation2007, Lemma 3.1), we state the modified result as follows:
Lemma 5.1. Let . Assume that the mapping is metrically regular at on with constant so that
Let . Then is Lipschitz-like at on with constant , that is,
Proof. According to our assumption on , we obtain through Lemma 4.1 that the mapping is Lipschitz-like at on with , that is, the following inequality holds:
Note, by (5.5) and (5.6), that Take
Then it is clear by (5.6) and (5.9)) that . Let
It suffices to show that there exists such that
To complete this, we will proceed by mathematical induction on and verify that there exists a sequence such that
and
hold for each . Define
By (5.10), we obtain
Now, we obtain that
Then by in (5.4) together with (5.10) and (5.14), (5.15) yields that
This means that for each . Denote . Noting that by (5.14). Then, we obtain by (5.10), that is,
Inclusion (5.17) can be written as
This, by the definition of , implies that Hence, we get . This together with (5.10) gives that
From the Lipschitz-like property of and noting that by (5.16), it follows from (5.8) that there exists such that
Moreover, for and by the definition of , we have
This implies that
Therefore, (5.18) and (5.19) are ensuring us that (5.11) and (5.12) are true with constructed points
Assume that are constructed such that (5.11) and (5.12) are true for . We have to construct such that (5.11) and (5.12) are also true for . Write
Then, we have from the inductional assumption,
Since , by (17) and by (5.10), it follows from (5.12) that
Utilizing the fact from (5.4) together with (5.9) in (5.21), we have
Moreover, taking into account that
Furthermore, using (5.22) and (5.23), one has that, for each ,
By (5.4), the fact reduces the above inequality that
Inequality (5.24) shows that for each .
By our assumption (5.11) holds for , so we have
This can be written as
Then by the definition of , we have This, together with (5.22), yields that
Now, by (5.8), there exists an element such that
Then by (5.20), we have
Since , by definition of it follows that
Therefore, the inclusion (5.28) together with (5.27) completes the induction step and ensure the existence of the sequence satisfying (5.11) and (5.12).
Since , we see from (5.12) that is a Cauchy sequence and hence there exists such that . From the previous proof, we have that for each . Taking limit to (5.11) and since is closed, we obtain that
that is, Moreover,
This completes the proof of the Lemma 5.1.
Remark 5.1. Let . Then, for every , we have that
It follows that . Therefore, is Lipschitz-like at on with constant .
Before going to demonstrate the main result in this section, we need to introduce some notation. Let and . Choose a sequence of scalars such that . Set in (5.1) for every . Then the set-valued mapping can be rewritten as follows:
Then, by Algorithm 5.2, we have that
and we obtain the following equivalence
In particular,
Also, we can rewrite (5.4) as follows:
Then
Moreover, the mapping is metrically regular at on with constant by Lyusternik-Graves theorem (see Proposition 4.1). Then by Lemma 4.1, we have is Lipschitz-like at on with constant , that is, the following inequality holds:
For our convenience, we define for each and , the mapping by
and the set-valued mapping by
Then
We are now able to prove the semilocal convergence of the sequence generated by Algorithm 5.2 for solving (3.1) when is metrically regular.
Theorem 5.1. Suppose that , and let . Let be a sequence of scalars such that . Assume that the mapping is metrically regular at on with constant so that the following inequality holds:
Let be defined in (5.33) and let and be such that
• ;
• .
Then, for every , any sequence generated by Algorithm 5.2 with initial point converges to a solution of (3.1) for .
Proof. Since and , we have from (46) that
Let . Thus, Lemma 5.1 is applicable with constants , and . Moreover, inasmuch as , we have that
It follows, for , that
Note that the metric regularity of the mapping at on with constant implies through Lemma 4.1 that is Lipschitz-like at on with constant , that is, (5.35) holds.
Let . Since , then for in assumption (b), we have that
To complete the proof, we will proceed by mathematical induction. It suffices to show that the Algorithm 5.2 generates at least one sequence and any generated sequence satisfies
and
for each To this end, define
Since , by using (5.39) and the fact in assumption (b) we have from (5.45) that
First, we will prove that
To do this, we will consider the mapping defined by (5.37) and apply Lemma 4.2 to with , and . It’s sufficient to show that assertions (4.4) and (4.5) of Lemma 4.2 hold for with , and . To proceed, we note that . Then by the definition of and excess , we have
(noting that ). For each , we have that
Then by the relations and in assumptions (b) and (a), respectively, we obtain that
that is, for each , . In particular, letting in (5.49), then we obtain that
This yields that Hence, by using (5.51) and Lipschitz-like property of in (5.48), we obtain that
This implies that assertion (4.4) of Lemma 4.2 is satisfied. Below, we will show that the assertion (4.5) of Lemma 4.2 is also hold. To show this, let . Then, by the fact in assumption (a) and (4.46), we have . Moreover, we have from (4.50) that . Then, by Lipschitz-like property of , we have
Applying (5.38) and (5.39) in (5.52), we obtain
Therefore, the assertion (4.5) of Lemma 4.2 is also satisfied. Since both assertions (4.4) and (4.5) of Lemma 4.2 are fulfilled, there exists a fixed point
which translates to , that is, . This shows that and hence (5.47) is hold. Consequently, inasmuch as , we can choose such that there exists and
By Algorithm 5.2, is defined. Hence, the point is generated by Algorithm 5.2. Furthermore, by the definition of , from (5.30) we can write
and since there exists , we have
Thus, from (5.55) we have . This implies that
Then by assumptions (a) and (b), we get that
and so . This, together with the closedness of and the fact , implies that . Then, by (5.31) we have that . Because of , by (61) it follows that (5.54) holds for .
Since (5.40) holds and is metrically regular at on with constant , it follows from Lemma 5.1 that the mapping is Lipschitz-like at on with constant for each . In particular, is Lipschitz-like at on with constant as the ball contains the point . Furthermore, the facts and in assumptions (a) and (b), respectively, imply that
and hence we have that . Applying Lemma 4.1, we have that the mapping is metrically regular at on with constant such that
Using (5.56), (5.57) and (5.42) in (5.55), we obtain that
This shows that (5.44) holds for .
We assume that the points are generated by Algorithm 5.2 such that (5.43) and (5.44) are true for . We show that there exists such that (5.43) and (5.44) hold for . Because (5.43) and (51) hold for , we have, for , that
and so . Now with almost same arguments as we used for the case when , we can show that (5.43) and (5.44) hold for . Hence, (5.43) and (5.44) hold for each . This implies that is a Cauchy sequence which is generated by Algorithm 5.2 and there exists such that . Thus, passing to the limit and since is closed, it follows that . Hence, the proof is complete.
The special case is that when is a solution of (1) for , Theorem 5.1 can be reduced to the following corollary which gives the local convergence result for restricted proximal point method defined by Algorithm 5.2.
Corollary 5.1. Suppose that , and is a solution of (1) for . Let be a sequence of scalars such that . Let be metrically regular at which have locally closed graph at . Let , where . Suppose that
Then there exist constants and such that for every there exists any sequence generated by Algorithm 5.2 with initial point , which is convergent to a solution of (1) for .
Proof. Let be such that . Since is locally closed at and is metrically regular at , there exist constants such that is metrically regular at on with constant and is closed. Since is Lipschitz continuous on , by Proposition 4.1 we have that is metrically regular at on with constant .
Choose and be such that . Since and , we have that
This yields that . Then define
It follows that
Let be such that
Let . Since (5.60) holds, we can take so that for each there exists near 0 such that , that is, . Then for such we have that so that
It follows that and and hence is closed. Thus, by the property of , we conclude that is metrically regular at on with constant . Now, it is routine to check that all assumptions in Theorem 5.1 hold. Thus, Theorem 5.1 is applicable to complete the proof of the Corollary 5.1.
4. Numerical experiment
In this section, a numerical experiment is given to validate the stability of convergence of Gauss-type proximal point method.
Example 6.1. Let . Define a set-valued mapping on by . Then Algorithm 5.2 generates a sequence for solving (3.1), which is converges to .
Solution: Let us consider . It is obvious from the statement that has a closed graph at . From the definition of , we have that
On the other hand, the nonemptyness of implies that
and we have, Theorem 5.1, that
Then by the definition of , we obtain that , and hence for given values of and , we see that . Thus, this implies that the sequence generated by Algorithm 5.2 converges linearly. Using Mat lab program, we present the solution of (3.1), which is , when the number of iterations are . Similarly, we can use the same approach for finding the solution of (3.1) when . The Table shows the numerical results and Figure gives the graphical representation of .
5. Concluding remarks
When , we have established the semilocal and local convergence of the restricted proximal point method defined by Algorithm 5.2 under the assumption that is metrically regular. Our proposed method coincides with the Gauss-type proximal point algorithm introduced by Rashid et al. in Rashid et al. (Citation2013) when . Moreover, when , and the set is singleton, the Algorithm 5.2 reduces to the classical proximal point algorithm defined by Algorithm 5.1. The convergence result established in the present article is ensuring the validity of the Gauss-type proximal point method, introduced by Rashid et al. in Rashid et al. (Citation2013), in the sense that the convergence result is uniform. Therefore, this study improves and extends the result corresponding to (Rashid et al., Citation2013). Finally, we have presented a numerical experiment that illustrated the theoretical result.
Acknowledgements
The author thanks the anonymous referees for their insightful comments and constructive suggestions, which contribute to the improvement of the initial versions of this manuscript. The author is also grateful to the associate editor for his constructive suggestions which have improved the presentation of this manuscript.
Additional information
Funding
Notes on contributors
M.H. Rashid
Mohammed Harunor Rashid is an Associate Professor in the Department of Mathematics, University of Rajshahi, Bangladesh, where he teaches calculus, geometry, real analysis, numerical analysis, vector and tensor analysis, operations research, functional analysis and other courses in both undergraduate and graduate level. Presently, he is a Postdoctoral Fellow at the Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. He has completed his PhD from Zhejiang University, China. His research interests focus on fuzzy Mathematics, nonlinear numerical functional analysis and nonlinear optimization, especially on generalized equations. In his investigation, he has shown how to solve generalized equations using an iterative method under some suitable conditions. He has published his research contributions in some internationally renowned journals whose publishers are Springer, Taylor & Francis, Yokohama and other journals.
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