Abstract
Let and be Banach spaces and . Let be a single valued function which is nonsmooth. Suppose that is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation and analyze its semilocal and local convergence under the conditions that is Lipschitz-like and admits a certain type of approximation which generalizes the concept of point-based approximation so-called -point-based approximation. Applications of -point-based approximation are provided for smooth functions in the cases and as well as for normal maps. In particular, when and the derivative of , denoted , is -Hölder continuous, we have shown that admits -point-based approximation for while admits -point-based approximation for , when and the second derivative of , denoted , is -Hölder. Moreover, we have constructed an -point-based approximation for the normal maps when has an -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.
1. Introduction
Robinson [1, 2] introduced the generalized equation as a general tool for describing, analyzing and solving different problems in a unified manner and it has been studied extensively. Typical examples are systems of inequalities, variational inequalities, linear and nonlinear complementary problems, system of nonlinear equations, equilibrium problems, first-order necessary conditions for nonlinear programming etc. They also have plenty of applications in engineering and economics. For more details on these applications and many other ones that we did not mention here, one can refer to [1–3].
In this study, let and be Banach spaces, be a set-valued mapping with closed graph and be a nonsmooth single-valued function that admits -point-based approximation on with a constant . We are concerned with the problem of approximating the point (which is called the solution of (1)) of the following nonsmooth generalized equation:
The classical Newton method is very well known and extensively used to find solutions of (1) when , where has Lipschitz continuous Fréchet derivatives. A survey of local and semilocal convergence results for Newton method’s are found and mentioned in [4–7]. When is nonsmooth, such a classical linearization is no longer available and we need to seek a replacement. In other words, if doesn’t possess Fréchet derivatives, it is not so clear how a Newton algorithm should be designed. There are many investigators have worked on this question and the applicants have presented different methods for a few things that are important in certain cases and have proved their justification; see for example [4, 8–14]. Several papers have worked on the Newton-type methods for nonsmooth equations and variational inequalities; see for example [8, 9, 15] for inspiration and advanced works on these topics.
In the case when and is a nonsmooth function, Robinson [9, Theorem 3.2] considered point-based approximation with Lipschitzian property to show the convergence of Newtons method under the Newton-Kantorovich-type hypothesis. Argyros [10] presented a semilocal convergence analysis of Newtons method based on a suitable point-based approximation. More explicitly, he has taken weaker assumptions in point-based approximation by considering Hölderian property instead of Lipschitzian property in order to cover a wider range of problems than those discussed in [9] and hence showed the convergence result for Newton’s method.
In addition, Kummer [16] presented a necessary and adequate condition for superlinear convergence of the Newton method, which was originally designed for derivative-type approximations of a nonsmooth function around an isolated zero. Relevant results, for solving the nonsmooth generalized (1) are given in [8, 17, 18].
To solve the nonsmooth generalized (1), Geoffroy and Piétrus in [19] considered the following methodwhere is an approximation of , and presented a local convergence result by using the assumptions that admits an -point-based approximation and the set-valued map is -pseudo-Lipschitz around . For the first time, Dontchev [11] introduced the iterative procedure (2) for solving (1) and presented the nonsmooth analogue of the Kantorovich-type theorem for this procedure by assuming the Aubin continuity of the map at (or, equivalently, is Aubin continuous at ), where is the first iterate of (2).
Let . The subset of , denoted by , is defined by
Although the method (2) guarantees the existence of a convergent sequence for solving (1), the constructed points are not unique and therefore, for a starting point near to a solution, the sequences generated by the method (2) are not uniquely defined. For example, the convergence result established in [19, Theorem 3.3], guarantees the existence of a convergent sequence. Hence, in view of numerical computation, this kind of Newton-type methods are not convenient in practical application. Based on these ideas, Rashid [8] introduced and studied the following algorithm and presented semilocal and local convergence results under the assumptions that has a point-based approximation and is Lipsctiz-like mapping:
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It is noted that, in the case when is replaced by the classical linearization of , the Algorithm 1 is reduced to the Gauss-Newton-type method introduced by Rashid . [20].
Moreover, when the single-valued function involved in (1) is smooth, there has been increased amount of interest on semilocal and local analysis (see, for example, [8, 20–23] and the references therein).
Our approach is somewhat different. In this study, we give a more general approach, namely -point-based approximation, which is an extension of the concept of point-based approximation introduced by Robinson [9] and it can apply to a wide range of particular problems. Because of the presence of Step 3 in Algorithm 1, we have shown in the main proof (Theorem 2) that each of the constructed points has limit. Therefore, in numerical computational view point, Algorithm 1 gives the more accurate result than the result given by the method (2).
In the present paper, we present semilocal and local convergence of Algorithm 1 under some mild conditions for the function and the set-valued mapping . In fact, the main motivation of this research is to analyze the semilocal and local convergence of the sequence generated by Algorithm 1 for solving the nonsmooth generalized (1) using the notion of -point-based approximation introduced by Geoffroy and Piétrus [19] and Lipschitz-like property. Based on the information around the initial point, the main result is the convergence criterion, developed in the section 3, which provides some sufficient conditions, for a starting point near to the solution, ensuring the convergence to the solution of any sequence generated by Algorithm 1. As a result, local convergence result for the extended Newton-type method is obtained.
This paper is organized as follows: In section 2, we recall some definitions, notations and preliminarily results that will need afterwards. In Section 3, we show the existence of the sequence generated by Algorithm 1 and then establish the convergence of the extended Newton-type method by using the concept of -point-based approximation as well as Lipchitz-like property. In Section 4, we have given some applications of -point-based approximation for smooth functions in the case when , and and for normal maps which is reformulated by Rashid [8]. In the last section, a numerical experiment is provided to justify the theoretical result of this study.
2. Preliminaries
Throughout this paper, we assume that and are two real or complex Banach spaces and is the set of all Natural numbers and . Suppose that is a Fréchet differentiable function and is a set-valued mapping with closed graph. Let and . The closed ball centered at with radius is denoted by .
All the norms are denoted by . The domain and the inverse are respectively defined by
Let . The distance from a point to a set is defined bywhile the excess from the set to the set is defined by
Definition 1. Consider the set-valued mapping Then the graph of is defined by
Definition 2. A set-valued function is said to be a closed graph if the set is a closed subset of in the product topology i.e. for all sequences and such that and and for all , we have .
The notions of pseudo-Lipschitz and Lipchitz-like set-valued mappings were introduced by Aubin in [24] and have been studied extensively; see for example [25, 26]. We recall these notions from [20].
Definition 3. Let be a set-valued mapping and let . Let and . is said to be(a)Lipchitz-like on relative to with constant if the following inequality holds:(b)pseudo-Lipschitz around if there exist constants , and such that is Lipchitz-like on relative to with constant .
Remark 1. The pseudo-Lipschitz property of a set-valued mapping is equivalent to the openness with linear rate of (the covering property) and to the metric regularity of (a basic well-posedness property in optimization) (see [23, 24, 27, 28] for more details).
Remark 2. Equivalently for the property (a) we can say that is Lipschitz-like at on with constant if for every and for every , there exists such thatThe following lemma is useful and it was proven by Rashid . in [20, Lemma 2.1].
Lemma 1. Let be a set-valued mapping and let . Assume that is Lipschitz-like on relative to with constant . Thenholds for every and satisfying .
The following concept of -point-based approximation is extracted from Geoffroy and Piétrus [19].
Definition 4. Let be a function and . Then a function is said to be a -point-based approximation (-PBA in brief) on for with modulus if there exists a scalar such that, for each , both of the following assertions hold:(a), where(b)The function is Lipschitz continuous on with modulus .It is clear that when and , Definition 4 agrees with Robinson’s definition of point-based approximation introduced in [9].
Recall the following definition of strict differentiability, which has been taken from [11].
Definition 5. A function is said to be strictly differentiable at with strict derivative if for every there exists such thatThe following result is a version of [11, Lemma 2]. This result establishes the connection between the strict differentiability of and -PBA of a function .
Lemma 2. Let be a -point-based approximation of a function in with a constant and let . Then the function is strictly differentiable at the point and its strict derivative at is zero.
The following lemma is taken from [25, Corollary 2].
Lemma 3. Let be a set-valued mapping with closed graph and let be two continuous functions. Let , let and let the difference be strictly differentiable at the point with . Let be a positive constant. Then the following are equivalent:(i)The map is Lipschitz-like at with modulus ;(ii)The map is Lipschitz-like at with modulus .
Remark 3. Combining Lemma 2 and Lemma 3 we conclude that if is a -PBA of a function in an open neighborhood of some , then is Lipschitz-like at if and only if the map possesses the same property.
The following theorem on the convergence of the nonsmooth function using -point-based approximation is due to Geoffroy and Piétrus; see [19, Theorem 3.3]:
Theorem 1. Let be a solution of (1). Fix . Suppose that has closed graph and admits a -point-based approximation with modulus , denoted by , on some open neighborhood of ; The set-valued map is M-pseudo-Lipschitz around . Then for every , one can find such that for every starting point , there exists a sequence generated by (2), which satisfiesDontchev and Hager [25] proved Banach fixed point theorem, which has been employing the standard iterative concept for contracting mapping. To prove the existence of the sequence generated by Algorithm 1, the following lemma will be played an important rule in this study.
Lemma 4. Let be a set-valued mapping. Let , and be such thatandThen has a fixed point in , that is, there exists such that . Furthermore, if is single-valued, then there exists a fixed point such that .
3. Convergence Analysis
Throughout the whole study we assume that and are real or complex Banach spaces. Let and be a set-valued mapping with closed graph. Suppose that is a nonsmooth function that admits -point-based approximation on with a constant , where is an open neighborhood of a point . Let and define the mapping by
Then
Furthermore, the following equivalence is clear:
In particular,
Let and let . Furthermore, throughout in this section we assume that . Suppose that is defined in Definition 4.
Define
Then
Let us recall that (1) is an abstract model for various problems. From now on, we make the following assumptions.(i) has closed graph;(ii) admits an -point-based approximation with modulus , denoted by , on some open neighborhood of (iii)The set valued map is Lipschitz-like on relative to with constant .
The following lemma plays an important role to the convergence analysis of the extended Newton-type method which is defined by Algorithm 1. The proof is a refinement of the one for [11, Lemma 1].
Lemma 5. Suppose the assumptions (i)-(iii) hold and let be defined in (10), so that (11) is satisfied. Let . Then is Lipschitz-like on relative to with constant , that is,
Proof. Since has a -point-based approximation on an open neighbourhood of with a constant and the map is Lipschitz-like around with a constant , then by Remark 3 we have that is Lipschitz-like around with a constant , that is, there exist constants , and such thatNote, by (20 and 21), that . Now letIt is sufficent to show that there exist such thatTo this end, we shall verify that there exists a sequence such thatandhold for each . We proceed by mathematical induction. DenoteNote by (24) thatIt follows, from (13) and the relation by (20) thatThis implies that for each . Letting . Then by (13) and it follows from (18) thatwhich can be rewritten asThis, by the definition of , means that . Hence by (18). This together with (24) implies thatAccording to the concept of Lipschitz-like property of and noting that , it follows from (23) that there exists such thatMoreover, by the definition of and noting , we havewhich together with (18) implies thatThis shows that 26 and 27 are true with constructed points and .
Suppose that the points have constructed so that 26 and 27 are true for . We need to construct such that (26 and 27) are also true for . To do this, settingThen, by the inductional assumption together with the concept of -point-based approximation of A, we obtain thatWe have and from (24) and using (27) we getBy (20), we have and then (39) becomesConsequently,Furthermore, using 13 and 20, we get that, for each It follows that for each . Since assumption (14) holds for , we havewhich can be written asThen by definition of , it follows that . This, together with 18 and 40, yields thatUsing (23) again, inasmuch as , there exists an element such thatwhere the last inequality holds by (38). By the definition of , we havewhich together with (18) impliesThis together with (46) completes the induction step and the existence of sequence satisfying (14) and (15).
Since , we see from (27) that is a Cauchy sequence. Define . Note that has closed graph. Then, taking limit in (26), we get and so . Moreover,This completes the proof of the Lemma 5.
Before going to state the main theorem in this study, for our convenience, we define the map , for each , byand the set-valued map byThen we have thatThe main result of this study read as follows, which provides some sufficient conditions ensuring the convergence of the extended Newton-type method for nonsmooth generalized (1) from starting point .
Theorem 2. Suppose that . Let , be an open and convex subset of containing and let be a function which has -point-based approximation on with a constant . Suppose that the map has closed graph and the map is Lipschitz-like on relative to with constant . Let be defined by (10) so that (11) holds. Let be such that(a),(b),(c).Suppose thatThen there exists some such that any sequence generated by Algorithm 1 with starting point converges to a solution of nonsmooth generalized (1), that is, satisfies .
Proof.
By assumption (b), it can be easily written thatSetIt follows from (54) thatSince by assumption (c) and (26) holds, there exists be such thatLet . We will proceed by mathematical induction. We will show that Algorithm 1 generates at least one sequence and any sequence generated by Algorithm 1 for (1) satisfies the following assertions:andfor each . For this purpose we defineOwing to the fact in assumption (a) and , by assumption (b) we can write as followsThe above inequality gives eitherBy the facts from condition (c)and (34), the inequality (33) reduces to, for each It is trivial that (58) is true for . To show, (32) holds for , firstly we need to verify that exists, that is, we need to show that . To do this, we consider the mapping defined by (24) and apply Lemma 4 to the map with . Let us check that both assumptions (5) and (6) of Lemma 4, with and hold. Noting that by (3) and according to the definition of the excess and the map , we obtainFrom the notion of -point-based approximation of with , we obtain thatNote that because of assumption (a), by assumption (c) and . It follows from (65), for each , thatThis implies thatIn particular, letting in (65). Then we have thatand henceHence, by the assumed Lipschitz-like property of and (68), we have from (64) thatthat is, the assumption (5) of Lemma 4 is satisfied.
Below, we will show that the assumption (6) of Lemma 4 holds. To do this, let . Then from assumption (a) and (35), we have that and by (39). This, together with the assumed Lipschitz-like property of , implies thatApplying (52), we get thatWith the help of first relation in (62) and combining the above two inequalities we get,This means that the assumption (6) of Lemma 4 is also satisfied. Since both assumptions (5) and (6) of Lemma 4 are satisfied, we can say that Lemma 4 is applicable and therefore, we conclude that there exists such that , that is, and so . This fact reflects that .
Since and , we can choose such thatBy Algorithm 1, is defined. Hence is generated for (1).
Furthermore, by the definition of , we can writesoNow we are ready to show that (59) is hold for . Note that by assumption (a). Then (21) is satisfied by (20). Lemma 5 states us that the mapping is Lipschitz-like on relative to with constant for each when is Lipschitz-like on relative to . Particularly, is Lipschitz-like on relative to with constant as by assumption (a) and the choice of .
Furthermore, assumptions (a), (c) and the 2nd relation of the inequality (62) imply thatNow (57) becomesNoting that as mentioned earlier and by (78)) we have that .
Thus, by applying Lemma 1, we obtain thatAccording to Algorithm 1 and using (77 and 80) we haveFrom (56 and 81) we get,This shows that (59) is hold for .
Suppose that the points have obtained by Algorithm 1 satisfying (2) such that (31 and 32) are hold for . We show that assertions (31) and (32) are also hold for . Since (31) and (32) are true for each , we have the following inequalityand so . This shows that (58) holds for .
Next we show that the assertion (59) is also hold for . Let . If we apply Lemma 4 to the map with , and , then by the analogue argument as we did for the case one can find that . Because of , Lemma 5 permit us to say that is Lipschitz-like on relative to with constant .
Moreover, inasmuch as , using the idea of -point-based approximation of , the inequality from assumption (a), we obtain thatIt is noted earlier that . Moreover, (78) implies that . This, together with (84), implies that Lemma 1 is applicable for the map and hence we have thatSince , Algorithm 1 ensure us the existence of a point which satisfy the following inequalityThis shows that (59) holds for . Thus, we can see from (59) that is a Cauchy sequence and hence convergent to some . Since the graph of is closed, we can pass to the limit in obtaining that is a solution of (1). Therefore, the proof is completed.
In particular, in the case when is a solution of (1), that is, , Theorem 2 is reduced to the following corollary, which gives the local convergent result of the extended Newton-type method for solving nonsmooth generalized (1).
Corollary 1. Suppose that and be a solution of (1). Let be an open and convex subset of containing and be such that is an open and convex set. Suppose that the function is continuous which has an -point-based approximation on with a constant , the map has closed graph. Assume that the map is Lipschitz-like around with constant . Suppose that
Then there exists some such that any sequence generated by Algorithm 1 starting from converges to a solution of nonsmooth generalized (1), that is, satisfies that .
Proof. By hypothesis is pseudo-Lipschitz around . Then there exists constants , and such that is Lipschitz-like on relative to with constant . Then, for each , one has thatthat is, the map is Lipschitz-like on relative to with constant .
Let and choose such thatand is a -point-based approximation of on . Then, defineandThus we can choose such thatandNow it is routine to check that all the conditions of Theorem 2 are hold. Thus, Theorem 2 is applicable to complete the proof of the corollary 1.
4. Application of -point-based approximation
This section is devoted to present applications of -point-based approximation. In particular, when the Fréchet derivative of is -Hölder, the function is an - point-based approximation for . Moreover, when is a twice Fréchet differentiable function such that is -Hölder, then the function is an -point-based approximation for . In addition, application of -point-based approximation is provided for normal maps.
4.1. Application of -PBA for smooth function
Let and be a convex subset of . Let .(1)Suppose that the Fréchet derivative of is -Hölder continuous. We show that the functionis an -point-based approximation for . In this case, by using the Algorithm 1 we can infer that there exists a sequence which converges superlinearly and this result recovers the convergence result of Geoffroy and Piétrus in [19].
In this regards, define the function by
It follows that
This yields that satisfies the first property of -point-based approximation on . To proof the second property of -point-based approximation, we assume that . Then, we have that
This shows that the second property of -PBA for also holds. Therefore, we say that when the Fréchet derivative of is -Hölder with exponent , the function is an -point-based approximation.(2)Let be such that . Suppose that is a twice Fréchet differentiable function on such that is -Hölder on and with exponent . Choose and be such that
Let and define the function
Then, Theorem 2 ensures the existence of a sequence which converges super-quadratically and the result of Theorem 2 coincides with the result of [22, 29].
To show the first property of -point-based approximation, denote . Then we have that
Since, , then (100) reduces to
Therefore, A satisfies the first property of an -point-based approximation on .
For the proof of second property, we assume that be any elements of , Then, we have that
This also can be written as
Since there exist an open subset and a positive number such that on . Let . Then, . Then, by applying the notion of -Hölder continuity property of and -Hölder continuity property of , we get
This shows that the second property of -PBA is satisfied. Thus, both of properties for -PBA hold on and . Hence, is an -PBA for on .
4.2. Application to Normal Maps
In this subsection we deal with a class of nonsmooth functions, i.e. normal maps. Normal maps have been studied by many authors to obtain solutions of variational inequalities and comprehensive accounts on this topic can be found in [9, 12, 13, 17, 30].
A detailed discussion about normal maps is given by Robinson [13]. Recall the following notion of normal maps which was introduced by Robinson [9, 13].
Definition 6. Let be a nonempty closed convex subset of a Banach space and let be the metric projector from onto . Let be an open subset of meeting and let be a function from to . The normal map is defined from the set to byMoreover, the following variational problemis completely equivalent to the normal-map equation through the transformation . Robinson has shown that how the first-order necessary optimality conditions for nonlinear optimization, as well as linear and nonlinear complementarity problems and more general variational inequalities, can all be expressed compactly and conveniently in the form of equations involving normal maps.
However, sometimes the use of normal maps enables one to gain insight into special properties of problem classes that might have remained obscure in the formalism of variational inequalities. A particular illustration of this is the characterization of the local and global homeomorphism properties of linear normal maps, given in [13] and improved in [31, 32].
In [8, Proposition 4.1], Rashid proved that for any function admitting a on a nonempty closed convex subset of a Hilbert space , the normal map associated with admits a on . In our study we will show that the same result holds when we replace the normal maps in lieu of the normal maps . Rashid [8, 14] reformulate the normal maps by simple modification of the definition of normal maps given by Robinson [13]. In [8, 14] Rashid assumed the concept of point-based approximation and -point-based approximation. Here we extend that concept to -point-based approximation which is reformulated by Rashid [8, 14], then we show that if have a -point-based approximation, then one can easily be constructed a -point-based approximation for .
The following reformulation of the normal maps is due to [14].
Definition 7. Let be a nonempty closed convex subset of a Banach space and let be the metric projector from onto . Let be an open subset of meeting and let and . The normal map is defined from the set to byWe are now able to construct a -point-based approximation for the normal map provided that a -point-based approximation exists for . The following proposition can be extracted from [14, Proposition 4.3].
Proposition 1. Let be a Banach space and be a nonempty closed convex subset of and let be the metric projector on which is nonexpansive. Let , be functions and let be a set-valued map with closed graph. If is a -point-based approximation for on with a constant , then the function defined by is a -point-based approximation for on with the same constant .
Proof. Let . We note that by the definition of normal map, and are respectively defined byandBy hypothesis we have that has the two properties for given in Definition 4 with a constant . We need to show that also has these same two properties for with the constant . Since is the -point-based approximation for on , then using the notion of first property of -point-based approximation and the non expansiveness of the metric projector we have thatThis implies that satisfies the first property of -point-based approximation. For proving the second property, we suppose that . To this end, let . We will show that is Lipschitz continuous on with lipschitz constant . Again using the concept of second property of -point-based approximation and non expansiveness of metric projector, we obtain thatThis shows that the second property of the -point-based approximation is satisfied. Since the both properties in Definition 4 are fulfilled for , we can conclude that is a -point-based approximation for on . The proof is completed.
5. Numerical Experiment
In this section, to present the numerical experiment we recall some necessary notations and notions . Let and be a Fréchet differentiable function at . Suppose that the set of all points is denoted by at which the derivative exists. The B-subdifferential of at , denoted by , is the set
Then, Clarke’s generalized Jacobian of at is the set =conv . If is differentiable near , and is continuous at , then obviously . Otherwise, is not necessarily a singleton, even if is differentiable at . In this case, holds. Now, in order to illustrate the theoretical result of the extended Newton-type method, we consider the following example.
Example 1. Let and . Let and be defined, respectively, byThen Algorithm 1 generates a sequence which converges superlinearly to and , respectively, with initial points and in the case . On the other hand, Algorithm 1 generates a superlinear convergent sequence which converges to and , respectively, with initial points and in the case .
Solution: It is manifest that is not differentiable at and hence is nonsmooth function on . But this function is differentiable on and hence . So, we getWe mark thatInitially, we study the set-valued mapping for the case and note that has a closed graph at with and . Thus, gph and , that is, is Lipschitz-like at . By taking , it is easily shown that is Lipschitz-like at for and . Therefore, the assumptions of Theorem 2 hold. From the definition of , we getAlternatively, if we takeAlso, from (86) with we consumeHereafter, for the given values of and , w get that Algorithm 1 generates a superlinearly convergent sequence with initial point in a neighborhood of . The following Tables 1 and 2, obtained by using Matlab code, indicate that the solution of the variational inclusion has the solutions and in the case and and in the case . The graphs of are plotted in Figure 1.
(a)
(b)
Remark 4. If we set in Example 1, we get the quadratic convergence of Algorithm 1.
6. Concluding Remarks
We have established semilocal and local convergence of the extended Newton-type method for solving the nonsmooth generalized (1) under the conditions is Lipschitz-like and the nonsmooth function has a -point-based approximation. Moreover, when and is -Hölder, we have presented an application of -point-based approximation for smooth function with , that is, we have shown is an -point-based approximation. In this case Theorem 2 provides the superlinear convergent result and this result extends the convergence theorem of Geoffroy and Piétrus [19]. On the other hand, for and , if is a twice Fréchet differentiable function and is -Hölder, we have given an application of -point-based approximation, that is, we have shown is an -point-based approximation. In this case Theorem 2 yields the superquadratic convergent result and this result extends the convergence result of [22, 29]. Furthermore, we have given another application of normal maps for which extends the concept of point-based-approximation reformulated by Rashid [8]. That is, we have shown that if has an -point-based approximations, it is easy to construct an -point-based approximation for the . Finally, we have presented a numerical experiment to validate the theoretical result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.