# Extended Newton-type Method for Nonsmooth Generalized Equation undern,α-point-based Approximation

Authors: M. Z. Khaton; M. H. Rashid
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

LetX and Y be Banach spaces and Ω⊆X. Let f:Ω⟶Y be a single valued function which is nonsmooth. Suppose that F:X⇉2Y 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 0∈fx+Fx and analyze its semilocal and local convergence under the conditions that f+F−1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n,α-point-based approximation. Applications of n,α-point-based approximation are provided for smooth functions in the cases n=1 and n=2 as well as for normal maps. In particular, when 0<α<1 and the derivative of f, denoted ∇f, is ℓ,α-Hölder continuous, we have shown that f admits 1,α-point-based approximation for n=1 while f admits 2,α-point-based approximation for n=2, when 0<α<1 and the second derivative of f, denoted ∇2f, is K,α-Hölder. Moreover, we have constructed an n,α-point-based approximation for the normal maps fC+F when f has an n,α-point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.


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## Body

## 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, letX and Y be Banach spaces, F:X⇉2Y be a set-valued mapping with closed graph and f:Ω⊆X⟶Y be a nonsmooth single-valued function that admits n,α-point-based approximation A on Ω with a constant L>0. We are concerned with the problem of approximating the point x¯∈Ω (which is called the solution of (1)) of the following nonsmooth generalized equation:(1)0∈fx¯+Fx¯.The classical Newton method is very well known and extensively used to find solutions of (1) when F=0, where f 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 f is nonsmooth, such a classical linearization is no longer available and we need to seek a replacement. In other words, if f 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 whenF=0 and f 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 method(2)0∈Axk,xk+1+Fxk+1for eachk=0,1,2,…,where A:X×X⟶Y is an approximation of f, and presented a local convergence result by using the assumptions that f admits an n,α-point-based approximation A and the set-valued map Ax∗,⋅+F⋅−1 is M-pseudo-Lipschitz around 0,x∗. 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 Ax0,⋅+F⋅−1 at 0,x1 (or, equivalently, f+F−1 is Aubin continuous at fx1−Ax0,x1,x1 ), where x1 is the first iterate of (2).Letx∈Ω⊆X. The subset of Ω, denoted by ℳx, is defined by(3)ℳx:=d∈Ω:0∈Ax,x+d+Fx+d.Although the method (2) guarantees the existence of a convergent sequencexk for solving (1), the constructed points x1,x2,…,xk 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 f has a point-based approximation and f+F−1 is Lipsctiz-like mapping:Algorithm 1: (The Extended Newton-type Method)(ENM).
Step 1
Selectη∈1,∞, x0∈Ω, and put i:=0.Step 2
If0∈ℳxi, then stop; otherwise, go to Step 3.Step 3
If0∉ℳxi, choose di such that di∈ℳxi and di≤ηdist0,ℳxi.Step 4
Setxi+1:=xi+di.Step 5
Replacei by i+1 and go to Step 2.It is noted that, in the case whenA is replaced by the classical linearization of f, the Algorithm 1 is reduced to the Gauss-Newton-type method introduced by Rashid et al. [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, namelyn,α-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 x1,x2,…,xk 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 Algorithm1 under some mild conditions for the function f and the set-valued mapping f+F. 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 n,α-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 section2, 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 n,α-point-based approximation as well as Lipchitz-like property. In Section 4, we have given some applications of n,α-point-based approximation for smooth functions in the case when n=1, n=2 and 0<α<1 and for normal maps fC+F 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 thatX and Y are two real or complex Banach spaces and ℕ is the set of all Natural numbers and ℕ∗=ℕ−0. Suppose that f:X⟶Y is a Fréchet differentiable function and F:X⇉2Y is a set-valued mapping with closed graph. Let x∈X and r>0. The closed ball centered at x with radius r is denoted by Brx.All the norms are denoted by‖⋅‖. The domain domF and the inverse F−1 are respectively defined by(4)domF:=x∈X:Fx≠∅,F−1y:=x∈X:y∈Fxfor eachy∈Y.LetD⊆X. The distance from a point x to a set D is defined by(5)distx,D:=infx−a:a∈Dfor eachx∈X,while the excess from the set D to the set C⊆X is defined by(6)eC,D=supdistx,D:x∈C.Definition 1.
Consider the set-valued mappingF:X⇉2Y. Then the graph of F is defined by(7)gphF≔x,y∈X×Y:y∈Fx.Definition 2.
A set-valued functionF:X⇉2Y is said to be a closed graph if the set x,y:y∈Fx is a closed subset of X×Y in the product topology i.e. for all sequences xkk∈ℕ and ykk∈ℕ such that xk⟶x and yk⟶y and yk∈Fxk for all n, we have y∈Fx.
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.
LetΓ:y⇉2X be a set-valued mapping and let y¯,x¯∈gphΓ. Let rx¯>0,ry¯>0 and M>0. Γ is said to be(a)
Lipchitz-like onBry¯y¯ relative to Brx¯x¯ with constant M if the following inequality holds:(8)eΓy1∩Brx¯x¯,Γy2≤My1−y2for anyy1,y2∈Bry¯y¯.(b)
pseudo-Lipschitz aroundy¯,x¯ if there exist constants ry¯′>0, rx¯′>0 and M′>0 such that Γ is Lipchitz-like on Bry¯′y¯ relative to Brx¯′x¯ with constant M′.Remark 1.
The pseudo-Lipschitz property of a set-valued mappingΓ is equivalent to the openness with linear rate of Γ−1 (the covering property) and to the metric regularity of Γ−1 (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 thatΓ is Lipschitz-like at y0,x0∈gphΓ on Bry0y0×Brx0x0 with constant M if for every x1,x2∈Brx0x0 and for every x1∈Γy1∩Brx0x0, there exists x2∈Γy2 such that(9)x1−x2≤My1−y2,for everyy1,y2∈Bry0y0.
The following lemma is useful and it was proven by Rashidet al. in [20, Lemma 2.1].Lemma 1.
LetΓ:y⇉2X be a set-valued mapping and let y¯,x¯∈gphΓ. Assume that Γ is Lipschitz-like on Bry¯y¯ relative to Brx¯x¯ with constant M. Then(10)distx,Γy≤Mdisty,Γ−1x,holds for every x∈Brx¯x¯ and y∈Bry¯/3y¯ satisfying disty,Γ−1x≤ry¯/3.
The following concept ofn,α-point-based approximation is extracted from Geoffroy and Piétrus [19].Definition 4.
Letf:Ω⊆X⟶Y be a function and n∈ℕ∗,α>0. Then a function A:Ω×Ω⟶Y is said to be a n,α-point-based approximation (n,α-PBA in brief) on Ω for f with modulus κ if there exists a scalar κ such that, for each u,v∈Ω, both of the following assertions hold:(a)
fv−Au,v≤κ/πn,αu−vn+α, where(11)πn,α=∏i=1nα+i.(b)
The functionAu,⋅−Av,⋅ is Lipschitz continuous on Ω with modulus κu−vα .
It is clear that whenn=1 and α=1, 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 functionf:X⟶Y is said to be strictly differentiable at x∗ with strict derivative ∇fx∗ if for every ε>0 there exists δ>0 such that(12)fx′−fx″−∇fx∗x′−x″≤εx′−x″,for everyx′,x″∈Bδx∗.
The following result is a version of [11, Lemma2]. This result establishes the connection between the strict differentiability of f and n,α-PBA of a function f.Lemma 2.
LetA be a n,α-point-based approximation of a function f in Ω with a constant κ and let x∗∈Ω. Then the function Ax∗,⋅−f⋅ is strictly differentiable at the point x∗ and its strict derivative at x∗ is zero.The following lemma is taken from [25, Corollary 2].Lemma 3.
LetF:X⇉2Y be a set-valued mapping with closed graph and let f,g:X⟶Y be two continuous functions. Let x∗,y∗∈gphF, let fx∗=gx∗=0 and let the difference f−g be strictly differentiable at the point x∗∈X with ∇f−gx∗=0. Let L be a positive constant. Then the following are equivalent:(i)
The mapf+F−1 is Lipschitz-like at y∗,x∗ with modulus <L;(ii)
The mapg+F−1 is Lipschitz-like at y∗,x∗ with modulus <L.Remark 3.
Combining Lemma2 and Lemma 3 we conclude that if A is a n,α-PBA of a function f in an open neighborhood of some x∗∈f+F−1y∗, then f+F−1 is Lipschitz-like at y∗,x∗ if and only if the map Ax∗,⋅+F⋅−1 possesses the same property.
The following theorem on the convergence of the nonsmooth function usingn,α-point-based approximation is due to Geoffroy and Piétrus; see [19, Theorem 3.3]:Theorem 1.
Letx∗ be a solution of (1). Fix n∈ℕ∗andα>0. Suppose that F has closed graph and f admits a n,α-point-based approximation with modulus k, denoted by A, on some open neighborhood Ω of x∗; The set-valued map Ax∗,⋅+F⋅−1 is M-pseudo-Lipschitz around 0,x∗. Then for every c>Mk/πn,α, one can find δ>0 such that for every starting point x0∈Bδx∗, there exists a sequence xk generated by (2), which satisfies(13)xk+1−x∗≤cxk−x∗n+α.
Dontchev 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.
LetΦ:X⇉2X be a set-valued mapping. Let x∗∈X, 0<λ<1 and r>0 be such that(14)adistx∗,Φx∗<r1−λ,and(15)beΦx1∩Bx∗,r,Φx2≤λx1−x2for allx1,x2∈Bx∗,r.
ThenΦ has a fixed point in Bx∗,r, that is, there exists x∈Bx∗,r such that x∈Φx. Furthermore, if Φ is single-valued, then there exists a fixed point x∈Bx∗,r such that x=Φx.


## 3. Convergence Analysis

Throughout the whole study we assume thatX and Y are real or complex Banach spaces. Let n∈ℕ∗,α>0 and F:X⇉2Y be a set-valued mapping with closed graph. Suppose that f:Ω⊆X⟶Y is a nonsmooth function that admits n,α-point-based approximation A on Ω with a constant L>0, where Ω is an open neighborhood of a point x¯∈X. Let x∈X and define the mapping Rx by(16)Rx⋅≔Ax,⋅+F⋅.Then(17)ℳx=d∈X:0∈Rxx+d=d∈X:x+d∈Rx−10.Furthermore, the following equivalence is clear:(18)z∈Rx−1y⇔y∈Ax,z+Fzfor anyz∈Xandy∈Y.In particular,(19)x¯∈Rx¯−1y¯for eachx¯,y¯∈gphf+F.Letx¯,y¯∈gphf+F and let rx¯>0,ry¯>0. Furthermore, throughout in this section we assume that Brx¯x¯⊆Ω∩domF. Suppose that πn,α is defined in Definition 4.Define(20)r¯≔minry¯−Lrx¯n+α3n+α+2n+απn,α2n+α,rx¯2α−MLrx¯α4.2αM.Then(21)r¯>0⇔L<minry¯πn,α2n+αrx¯n+α3n+α+2n+α,2αMLrx¯α.Let us recall that (1) is an abstract model for various problems. From now on, we make the following assumptions.(i)
F has closed graph;(ii)
f admits an n,α-point-based approximation with modulus L, denoted by A, on some open neighborhood Ω of x¯(iii)
The set valued mapf+F−1 is Lipschitz-like on Bry¯y¯ relative to Brx¯x¯ with constant M.The following lemma plays an important role to the convergence analysis of the extended Newton-type method which is defined by Algorithm1. The proof is a refinement of the one for [11, Lemma 1].Lemma 5.
Suppose the assumptions (i)-(iii) hold and letr¯ be defined in (10), so that (11) is satisfied. Let x∈Brx¯/2x¯. Then Rx−1⋅ is Lipschitz-like on Br¯y¯ relative to Brx¯/2x¯ with constant 2αM/2α−MLrx¯α, that is,(22)eRx−1y1∩Brx¯/2x¯,Rx−1y2≤2αM2α−MLrx¯αy1−y2for anyy1,y2∈Br¯y¯.Proof.
Sincef has a n,α-point-based approximation A on an open neighbourhood of x¯∈f+F−1y¯ with a constant L and the map f+F−1 is Lipschitz-like around y¯,x¯ with a constant M, then by Remark 3 we have that Rx−1⋅ is Lipschitz-like around y¯,x¯ with a constant M<L, that is, there exist constants rx¯>0, ry¯>0 and M such that(23)eRx¯−1y1∩Brx¯x¯,Rx¯−1y2≤My1−y2for ally1,y2∈Bry¯y¯.
Note, by (20 and 21), that r¯>0. Now let(24)y1,y2∈Br¯y¯andx′∈Rx−1y1∩Brx¯/2x¯.
It is sufficent to show that there existx″∈Rx−1y2 such that(25)x′−x″≤2αM2α−MLrx¯αy1−y2.
To this end, we shall verify that there exists a sequencexk⊂Brx¯x¯ such that(26)y2∈Ax,xk−1−Ax¯,xk−1+Ax¯,xk+Fxk,and(27)xk−xk−1≤My1−y2MLrx¯α2αk−2,hold for each k=2,3,4,…. We proceed by mathematical induction. Denote(28)zi:=yi−Ax,x′+Ax¯,x′for eachi=1,2.
Note by (24) that(29)x−x′≤x−x¯+x¯−x′≤rx¯2+rx¯2≤rx¯.
It follows, from (13) and the relation r¯≤ry¯−Lrx¯n+α3n+α+2n+α/πn,α2n+α by (20) that(30)zi−y¯≤yi−y¯+Ax,x′−Ax¯,x′≤r¯+fx′−Ax,x′+fx′−Ax¯,x′≤r¯+Lπn,αx−x′n+α+x¯−x′n+α≤r¯+Lπn,αrx¯n+α+rx¯2n+α=r¯+Lrx¯n+α2n+α+1πn,α≤ry¯.
This implies thatzi∈Bry¯y¯ for each i=1,2. Letting x1:=x′. Then x1∈Rx−1y1 by (13) and it follows from (18) that(31)y1∈Ax,x1+Fx1,which can be rewritten as(32)y1−Ax,x1+Ax¯,x1∈Ax¯,x1+Fx1.
This, by the definition ofz1, means that z1∈Ax¯,x1+Fx1. Hence x1∈Rx¯−1z1 by (18). This together with (24) implies that(33)x1∈Rx¯−1z1∩Brx¯x¯.
According to the concept of Lipschitz-like property ofRx¯−1⋅ and noting that z1,z2∈Bry¯y¯, it follows from (23) that there exists x2∈Rx¯−1z2 such that(34)x2−x1≤Mz1−z2=My1−y2.
Moreover, by the definition ofz2 and noting x1=x′, we have(35)x2∈Rx¯−1z2=Rx¯−1y2−Ax,x1+Ax¯,x1,which together with (18) implies that(36)y2∈Ax,x1−Ax¯,x1+Ax¯,x2+Fx2.
This shows that26 and 27 are true with constructed points x1 and x2.
Suppose that the pointsx1,x2,…,xm have constructed so that 26 and 27 are true for k=2,3,…,m. We need to construct xm+1 such that (26 and 27) are also true for k=m+1. To do this, setting(37)zim:=y2−Ax,xm+i−1+Ax¯,xm+i−1for eachi=0,1.
Then, by the inductional assumption together with the concept ofn,α-point-based approximation of A, we obtain that(38)z0m−z1m=Ax,xm−1−Ax¯,xm−1−Ax,xm−Ax¯,xm≤Lx−x¯αxm−xm−1≤Lrx¯α2αxm−xm−1≤y1−y2MLrx¯α2αm−1.
We havex1−x¯≤rx¯/2 and y1−y2≤2r¯ from (24) and using (27) we get(39)xm−x¯≤∑k=2mxk−xk−1+x1−x¯≤2Mr¯∑k=2mMLrx¯α2αk−2+rx¯2≤2.2αMr¯2α−MLrx¯α+rx¯2.
By (20), we have 4.2αMr¯≤rx¯2α−MLrx¯α and then (39) becomes(40)xm−x¯≤rx¯.
Consequently,(41)xm−x≤xm−x¯+x¯−x≤32rx¯.
Furthermore, using13 and 20, we get that, for each i=0,1(42)zim−y¯≤y2−y¯+Ax,xm+i−1−Ax¯,xm+i−1≤r¯+fxm+i−1−Ax,xm+i−1+fxm+i−1−Ax¯,xm+i−1≤r¯+Lπn,αx−xm+i−1n+α+x¯−xm+i−1n+α≤r¯+Lπn,α32rx¯n+α+rx¯n+α=r¯+L3n+α+2n+αrx¯n+απn,α2n+α≤ry¯.
It follows thatzim∈Bry¯y¯ for each i=0,1. Since assumption (14) holds for k=m, we have(43)y2∈Ax,xm−1−Ax¯,xm−1+Ax¯,xm+Fxm,which can be written as(44)y2−Ax,xm−1+Ax¯,xm−1∈Ax¯,xm+Fxm.
Then by definition ofz0m, it follows that z0m∈Ax¯,xm+Fxm. This, together with 18 and 40, yields that(45)xm∈Rx¯−1z0m∩Brx¯x¯.
Using (23) again, inasmuch as z0m,z1m∈Bry¯y¯, there exists an element xm+1∈Rx¯−1z1m such that(46)xm+1−xm≤Mz0m−z1m≤My1−y2MLrx¯α2αm−1,where the last inequality holds by (38). By the definition of z1m, we have(47)xm+1∈Rx¯−1z1m=Rx¯−1y2−Ax,xm+Ax¯,xm,which together with (18) implies(48)y2∈Ax,xm−Ax¯,xm+Ax¯,xm+1+Fxm+1.
This together with (46) completes the induction step and the existence of sequence xk satisfying (14) and (15).
SinceMLrx¯α/2α<1, we see from (27) that xk is a Cauchy sequence. Define x″:=limk⟶∞xk. Note that F has closed graph. Then, taking limit in (26), we get y2∈Ax,x″+Fx″ and so x″∈Rx−1y2. Moreover,(49)x′−x″≤limm⟶∞sup∑k=2mxk−xk−1≤limm⟶∞sup∑k=2mMLrx¯α2αk−2My1−y2≤2αM2α−MLrx¯αy1−y2.
This completes the proof of the Lemma5.
Before going to state the main theorem in this study, for our convenience, we define the mapZx:X⟶Y, for each x∈X, by(50)Zx⋅≔Ax¯,⋅−Ax,⋅,and the set-valued map Φx:X⇉2X by(51)Φx⋅=Rx¯−1Zx⋅.
Then we have that(52)Zxx′−Zxx″=Ax¯,x′−Ax,x′−Ax¯,x″−Ax,x″≤Lx¯−xαx′−x″for anyx′,x″∈X.
The 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 x0.Theorem 2.
Suppose thatη>1. Let x¯∈X, Ω be an open and convex subset of X containing x¯ and let f be a function which has n,α-point-based approximation A on Ω with a constant L>0. Suppose that the map F has closed graph and the map Rx¯−1⋅ is Lipschitz-like on Bry¯y¯ relative to Brx¯x¯ with constant M>0. Let r¯ be defined by (10) so that (11) holds. Let δ>0 be such that(a)
δ≤minrx¯/4,r¯.πn,α/4n+α,ry¯πn,α/L3n+α+2n+α+11/n+α,1,(b)
M+1L2α+1ηδα+rx¯α≤2α,(c)
y¯<L/πn,αδn+α.
Suppose that(53)limx⟶x¯disty¯,Ax,x+Fx=0.
Then there exists someδ^>0 such that any sequence xm generated by Algorithm 1 with starting point x0∈Bδ^x¯ converges to a solution x∗ of nonsmooth generalized (1), that is, x∗ satisfies 0∈fx∗+Fx∗.Proof.
By assumption (b), it can be easily written that(54)ML2α+1ηδα+rx¯α≤M+1L2α+1ηδα+rx¯α≤2α.
Set(55)t≔2αηMLδα2α−MLrx¯α.
It follows from (54) that(56)t≤12.
Sinceπn,αy¯<Lδn+α by assumption (c) and (26) holds, there exists 0<δ^≤δ be such that(57)dist0,Ax0,x0+Fx0≤Lπn,αδn+αfor eachx0∈Bδ^x¯.
Letx0∈Bδ^x¯. We will proceed by mathematical induction. We will show that Algorithm 1 generates at least one sequence and any sequence xm generated by Algorithm 1 for (1) satisfies the following assertions:(58)xm−x¯≤2δ,and(59)xm+1−xm≤t1πn,αn+αmδ,for each m=0,1,2,…. For this purpose we define(60)rx≔32MLπn,αx−x¯n+α+My¯for eachx∈X.
Owing to the fact4δ≤rx¯ in assumption (a) and η>1, by assumption (b) we can write as follows(61)M+1L2α.3δα≤M+1L.2α2δα+δα=M+1L2α+1δα+2δα≤M+1L2α+1ηδα+4δα≤M+1L2α+1ηδα+rx¯α≤2α.
The above inequality gives either(62)MLδα≤2α2α.3=13orLδα≤2α2α.3=13.
By the factsπn,αy¯<Lδn+α from condition (c)and (34), the inequality (33) reduces to, for each x∈B2δx¯(63)rx=32MLπn,αx−x¯n+α+My¯≤32MLπn,αx−x¯n+α+MLπn,αδn+α≤32MLπn,α2δn+α+MLπn,αδn+α=32MLπn,αδα2n+α+1.δnSinceδn≤δ,we get that,≤32MLπn,αδα2n+α+1.δ≤32.13πn,α2n+α+1.δ≤12πn,α2n+α+1.δ≤2δfor eachx∈B2δx¯.
It is trivial that (58) is true for m=0. To show, (32) holds for m=0, firstly we need to verify that x1 exists, that is, we need to show that ℳx0≠∅. To do this, we consider the mapping Φx0 defined by (24) and apply Lemma 4 to the map Φx0 with η0=x¯. Let us check that both assumptions (5) and (6) of Lemma 4, with r:=rx0 and λ:=1/3 hold. Noting that x¯∈Rx¯−1y¯∩Brx0x¯ by (3) and according to the definition of the excess e and the map Φx0, we obtain(64)distx¯,Φx0x¯≤eRx¯−1y¯∩Brx0x¯,Φx0x¯≤eRx¯−1y¯∩B2δx¯,Rx¯−1Zx0x¯≤eRx¯−1y¯∩Brx¯x¯,Rx¯−1Zx0x¯≤My¯−Zx0x¯.
From the notion ofn,α-point-based approximation A of f with L, we obtain that(65)Zx0x−y¯=Ax¯,x−Ax0,x−y¯≤Ax¯,x−Ax0,x+y¯≤fx−Ax¯,x+fx−Ax0,x+y¯≤Lπn,αx¯−xn+α+Lπn,αx0−xn+α+y¯≤Lπn,αx¯−xn+α+x0−xn+α+y¯.
Note thatLδn+α2n+α+3n+α+1≤πn,αry¯ because of assumption (a), πn,αy¯<Lδn+α by assumption (c) and x0−x¯≤δ^≤δ. It follows from (65), for each x∈Brx0x¯⊆B2δx¯, that(66)Zx0x−y¯≤Lπn,αx¯−xn+α+x0−x¯+x¯−xn+α+y¯≤Lπn,α2δn+α+δ+2δn+α+y¯=Lπn,α2δn+α+3δn+α+y¯≤Lπn,αδn+α2n+α+3n+α+Lπn,αδn+α=Lπn,αδn+α2n+α+3n+α+1≤ry¯.
This implies that(67)Zx0x∈Bry¯y¯for eachx∈Brx0x¯.
In particular, lettingx=x¯ in (65). Then we have that(68)Zx0x¯−y¯≤Lπn,αx0−xn+α+y¯,(69)≤Lπn,αδn+α+Lπn,αδn+α≤2Lπn,αδn+α≤ry¯,and hence(70)Zx0x¯∈Bry¯y¯.
Hence, by the assumed Lipschitz-like property ofRx¯−1 and (68), we have from (64) that(71)distx¯,Φx0x¯≤My¯−Zx0x¯≤MLπn,αx0−xn+α+My¯=1−13rx0=1−λr,that is, the assumption (5) of Lemma 4 is satisfied.
Below, we will show that the assumption (6) of Lemma4 holds. To do this, let x′,x″∈Brx0x¯. Then from assumption (a) and (35), we have that x′,x″∈Brx0x¯⊆B2δx¯⊆Brx¯x¯ and Zx0x′,Zx0x″∈Bry¯y¯ by (39). This, together with the assumed Lipschitz-like property of Rx¯−1, implies that(72)eΦx0x′∩Brx0x¯,Φx0x″≤eΦx0x′∩Brx¯x¯,Φx0x″=eRx¯−1Zx0x′∩Brx¯x¯,Rx¯−1Zx0x″≤MZx0x′−Zx0x″.
Applying (52), we get that(73)Zx0x′−Zx0x″≤Lx¯−x0αx′−x″.
With the help of first relation in (62) and combining the above two inequalities we get,(74)eΦx0x′∩Brx0x¯,Φx0x″≤MLx¯−x0αx′−x″≤MLδαx′−x″≤13x′−x″=λx′−x″.
This means that the assumption (6) of Lemma4 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 x^1∈Brx0x¯ such that x^1∈Φx0x^1, that is, 0∈Ax0,x^1+Fx^1 and so x^1−x0∈ℳx0. This fact reflects that ℳx0≠∅.
Sinceη>1 and ℳx0≠∅, we can choose d0∈ℳx0 such that(75)d0≤ηdist0,ℳx0.
By Algorithm1, x1:=x0+d0 is defined. Hence x1 is generated for (1).
Furthermore, by the definition ofℳx0, we can write(76)ℳx0≔d0∈Ω:0∈Ax0,x0+d0+Fx0+d0=d0∈Ω:x0+d0∈Rx0−10,so(77)dist0,ℳx0=distx0,Rx0−10.
Now we are ready to show that (59) is hold for m=0. Note that r¯>0 by assumption (a). Then (21) is satisfied by (20). Lemma 5 states us that the mapping Rx−1⋅ is Lipschitz-like on Br¯y¯ relative to Brx¯/2x¯ with constant 2αM/2α−MLrx¯α for each x∈Brx¯/2x¯ when Rx¯−1⋅ is Lipschitz-like on Bry¯y¯ relative to Brx¯x¯. Particularly, Rx0−1⋅ is Lipschitz-like on Br¯y¯ relative to Brx¯/2x¯ with constant 2αM/2α−MLrx¯α as x0∈Bδ^x¯⊆Bδx¯⊆Brx¯/2x¯ by assumption (a) and the choice of δ^.
Furthermore, assumptions (a), (c) and the 2nd relation of the inequality (62) imply that(78)y¯≤Lπn,αδn+α=Lπn,αδα.δn≤Lπn,αδα.δ≤13πn,α.δ≤13πn,α.r¯πn,α4n+α≤r¯3.
Now (57) becomes(79)dist0,Rx0x0=dist0,Ax0,x0+Fx0≤Lπn,αδn+α≤r¯3.
Noting thatx0∈Brx¯/2x¯ as mentioned earlier and by (78)) we have that 0∈Br¯/3y¯ .
Thus, by applying Lemma1, we obtain that(80)distx0,Rx0−10≤2αM2α−MLrx¯αdist0,Rx0x0.
According to Algorithm1 and using (77 and 80) we have(81)x1−x0=d0≤ηdist0,ℳx0=ηdistx0,Rx0−10≤2αηM2α−MLrx¯αdist0,Rx0x0≤2αηMLπn,α2α−MLrx¯αδn+α≤2αηMLπn,α2α−MLrx¯αδn.δα≤2αηMLδαπn,α2α−MLrx¯αδ.Sinceδn≤δ.
From (56 and 81) we get,(82)x1−x0=d0≤tπn,αδ≤t1πn,αδ.
This shows that (59) is hold for m=0.
Suppose that the pointsx1,x2,…,xk have obtained by Algorithm 1 satisfying (2) such that (31 and 32) are hold for m=0,1,2,…,k−1. We show that assertions (31) and (32) are also hold for m=k. Since (31) and (32) are true for each m≤k−1, we have the following inequality(83)xk−x¯≤∑i=0k−1di+x0−x¯≤tδ∑i=0k−11πn,αn+αi+δ≤2δ,and so xk∈B2δx¯. This shows that (58) holds for m=k.
Next we show that the assertion (59) is also hold for m=k. Let xk∈Brxkx¯. If we apply Lemma 4 to the map Φxk with η=x¯, r:=rxk and λ:=1/3, then by the analogue argument as we did for the case k=0 one can find that ℳxk≠∅. Because of xk∈Brxkx¯⊆B2δx¯⊆Brx¯/2x¯, Lemma 5 permit us to say that Rxk−1⋅ is Lipschitz-like on Br¯y¯ relative to Brx¯/2x¯ with constant 2αM/2α−MLrx¯α.
Moreover, inasmuch as−Axk−1,xk∈Fxk, using the idea of n,α-point-based approximation of f, the inequality 4n+αδ≤r¯πn,α from assumption (a), we obtain that(84)dist0,Rxkxk=dist0,Axk,xk+Fxk≤Axk,xk−Axk−1,xk=Lπn,αfxk−Axk−1,xkn+α≤Lπn,αxk−xk−1n+α≤Lπn,αxk−x¯+x¯−xk−1n+α≤Lπn,α2δ+2δn+α=Lπn,α4n+αδn+α≤Lπn,αδα4n+αδ=13πn,α4n+αr¯.πn,α4n+α≤r¯3.
It is noted earlier thatxk∈Brx¯/2x¯. Moreover, (78) implies that 0∈Br¯/3y¯. This, together with (84), implies that Lemma 1 is applicable for the map Rxk−1⋅ and hence we have that(85)distxk,Rxk−10≤2αM2α−MLrx¯αdist0,Rxkxk.
Sinceℳxk≠∅, Algorithm 1 ensure us the existence of a point xk+1 which satisfy the following inequality(86)xk+1−xk=dk≤ηdist0,ℳxk=ηdistxk,Rxk−10≤2αηM2α−MLrx¯αdist0,Rxkxk=2αηM2α−MLrx¯αdist0,Axk,xk+Fxk≤2αηM2α−MLrx¯Axk,xk−Axk−1,xk=2αηM2α−MLrx¯αfxk−Axk−1,xk≤2αηLMπn,α2α−MLrx¯αxk−xk−1n+α≤tδαπn,αt1πn,αn+αk−1δn+α≤tδαπn,αt1πn,αn+αk−1n+αδn+α≤tπn,αt1πn,αn+αk−1n+αδn≤tπn,αt1πn,αn+αk−1n+αδSinceδn≤δ≤t1πn,αn+αkδ.
This shows that (59) holds for m=k. Thus, we can see from (59) that xm is a Cauchy sequence and hence convergent to some x∗. Since the graph of F is closed, we can pass to the limit in xk+1∈Rxk−10 obtaining that x∗ is a solution of (1). Therefore, the proof is completed.
In particular, in the case whenx¯ is a solution of (1), that is, y¯=0, 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 thatη>1 and x¯ be a solution of (1). Let Ω be an open and convex subset of X containing x¯ and r˜>0 be such that Br˜x¯ is an open and convex set. Suppose that the function f is continuous which has an n,α-point-based approximation A on Br˜x¯ with a constant L>0, the map F has closed graph. Assume that the map Rx¯−1⋅ is Lipschitz-like around 0,x¯ with constant M. Suppose that(87)limx⟶x¯dist0,Ax,x+Fx=0.Then there exists someδ^>0 such that any sequence xm generated by Algorithm 1 starting from x0∈Bδ^x¯ converges to a solution x∗ of nonsmooth generalized (1), that is, x∗ satisfies that 0∈fx∗+Fx∗.Proof.
By hypothesisRx¯−1⋅ is pseudo-Lipschitz around 0,x¯. Then there exists constants r0, r^x¯ and M such that Rx¯−1⋅ is Lipschitz-like on Br0y¯ relative to Br^x¯x¯ with constant M. Then, for each 0<r≤r^x¯, one has that(88)eRx¯−1y1∩Brx¯,Rx¯−1y2≤My1−y2for anyy1,y2∈Br00,that is, the map Rx¯−1⋅ is Lipschitz-like on Br00 relative to Brx¯ with constant M.
LetL∈0,1 and choose rx¯∈0,r^x¯ such that(89)rx¯2≤r˜,2n+απn,αr0−L3n+α+2n+αrx¯n+α>0,and A is a n,α-point-based approximation of f on Brx¯/2x¯. Then, define(90)r¯=minr0−Lrx¯n+α3n+α+2n+απn,α2n+α,rx¯2α−MLrx¯α4.2αM>0,and(91)minrx¯4,r¯πn,α4n+α,r0πn,αL3n+α+2n+α+11/n+α.
Thus we can choose0<δ≤1 such that(92)δ≤minrx¯4,r¯.πn,α4n+α,r0πn,αL3n+α+2n+α+11/n+α,and(93)M+1L2α+1ηδα+rx¯α≤2α.
Now it is routine to check that all the conditions of Theorem2 are hold. Thus, Theorem 2 is applicable to complete the proof of the corollary 1.


## 4. Application ofn,α-point-based approximation

This section is devoted to present applications ofn,α-point-based approximation. In particular, when the Fréchet derivative of f is ℓ,α-Hölder, the function A is an 1,α- point-based approximation for f . Moreover, when f is a twice Fréchet differentiable function such that ∇2f is K,α-Hölder, then the function A is an 2,α-point-based approximation for f. In addition, application of n,α-point-based approximation is provided for normal maps.


### 4.1. Application ofn,α-PBA for smooth function f

Let0<α<1 and Ω be a convex subset of X. Let u,v∈Ω.(1)
Suppose that the Fréchet derivative off is ℓ,α-Hölder continuous. We show that the function(94)A:u,v↦fu+∇fuv−u,is an 1,α-point-based approximation for f. In this case, by using the Algorithm 1 we can infer that there exists a sequence xk which converges superlinearly and this result recovers the convergence result of Geoffroy and Piétrus in [19].In this regards, define the functionΛu,v by(95)Λu,v=fv−Au,v.It follows that(96)Λu,v=fv−fu−∇fuv−u=∫01∇fu+tv−u−∇fuv−udt≤v−u∫01∇fu+tv−u−∇fudt≤v−u∫01ℓtv−uαdt≤v−u1+αℓ∫01tαdt≤ℓα+1v−u1+α.This yields thatA satisfies the first property of 1,α-point-based approximation on Ω. To proof the second property of 1,α-point-based approximation, we assume that y,z∈Ω. Then, we have that(97)Λ′u,v,y,z=Au,y−Av,y−Au,z+Av,z,=fu+∇fuy−u−fv−∇fvy−v−fu−∇fuz−u+fv+∇fvz−v≤∇fu−∇fvy−z≤∇fu−∇fvy−z≤ℓu−vαy−z.This shows that the second property of1,α-PBA for f also holds. Therefore, we say that when the Fréchet derivative of f is ℓ,α-Hölder with exponent α∈0,1, the function A:u,v↦fu+∇fuv−u is an 1,α-point-based approximation.(2)
Letrx¯>0 be such that Brx¯/2x¯⊆X. Suppose that f is a twice Fréchet differentiable function on Brx¯/2x¯ such that ∇2f is K,α-Hölder on Brx¯/2x¯ and with exponent α∈0,1. Choose ℓ>0 and L>0 be such that(98)L>ℓ+Krx¯+1.Letp,q∈Brx¯/2x¯ and define the function(99)Ap,q=fp+∇fpq−p+12∇2fpq−p2.Then, Theorem2 ensures the existence of a sequence xk which converges super-quadratically and the result of Theorem 2 coincides with the result of [22, 29].To show the first property of2,α-point-based approximation, denote △p,q=fq−Ap,q. Then we have that(100)△p,q=fq−fp−∇fpq−p−12∇2fpq−p2.Since,∫011−t∇2fp+tq−pq−p2dt=fq−fp−∇fpq−p, then (100) reduces to(101)△p,q=∫011−t∇2fp+tq−pq−p2dt−12∇2fpq−p2=∫011−t∇2fp+tq−p−1−t∇2fpq−p2dt≤q−p2∫011−t∇2fp+tq−p−1−t∇2fpdt≤q−p2∫011−t∇2fp+tq−p−∇2fpdt≤Kq−p2∫011−ttq−pαdt≤Kq−p2+α∫011−ttαdt≤Kα+1α+2q−p2+α≤Lα+1α+2q−p2+α.Therefore, A satisfies the first property of an2,α-point-based approximation on Ω.For the proof of second property, we assume thata,b be any elements of Brx¯/2x¯, Then, we have that(102)△′p,q,a,b=Ap,a−Aq,a−Ap,b+Aq,b=fp+∇fpa−p+12∇2fpa−p2−fq−∇fqa−q−12∇2fqa−q2−fp−∇fpb−p−12∇2fpb−p2+fq+∇fqb−q+12∇2fqb−q2=∇fp−∇fqa−b+12∇2fpa−p2−∇2fqa−q2−∇2fpb−p2+∇2fqb−q2=∇fp−∇fqa−b+12∇2fpa−q+q−p,a−p−∇2fpb−q+q−p,b−p+∇2fqb−q,b−p+p−q−∇2fqa−q,a−p+p−q=∇fp−∇fqa−b+12∇2fpa−q,a−p+∇2fpq−p,a−p−∇2fpb−q,b−p−∇2fpq−p,b−p+∇2fqb−q,b−p+∇2fqb−q,p−q−∇2fqa−q,a−p−∇2fqa−q,p−q=∇fp−∇fqa−b+12∇2fqb−q,b−p−∇2fpb−q,b−p+∇2fpa−q,a−p−∇2fqa−q,a−p+∇2fpq−p,a−p−∇2fpq−p,b−p+∇2fqb−q,p−q−∇2fqa−q,p−q=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−p+12∇2fp−∇2fqa−q,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−p+12∇2fp−∇2fqa−b+b−q,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q.This also can be written as(103)△′p,q,a,b=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−a+12∇2fp−∇2fqa−b,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q.Since there exist an open subsetBrx¯/2x¯⊆X and a positive number K such that ∇2f≤K on Brx¯/2x¯. Let a,b∈Brx¯/2x¯. Then, a−b≤rx¯. Then, by applying the notion of ℓ,α-Hölder continuity property of ∇f and K,α-Hölder continuity property of ∇2f , we get(104)△′p,q,a,b≤∇fp−∇fqa−b+12∇2fq−∇2fpb−qb−a+12∇2fp−∇2fqa−ba−p+12∇2fpq−pa−b+12∇2fqb−ap−q≤ℓp−qαa−b+K2p−qαb−qb−a+K2p−qαb−aa−p+K2q−pαa−b+K2p−qαb−a≤ℓp−qαa−b+K2rx¯p−qαa−b+Kp−qαa−b≤ℓ+Krx¯+1p−qαa−b≤Lp−qαa−b,for alla,b∈Brx¯/2x¯.This shows that the second property of2,α-PBA is satisfied. Thus, both of properties for n,α-PBA hold on Brx¯/2x¯whenn=2 and 0<α<1. Hence, A is an 2,α-PBA for f on Brx¯/2x¯.


### 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.
LetC be a nonempty closed convex subset of a Banach space X and let ∏ be the metric projector from X onto C. Let Ω be an open subset of X meeting C and let f be a function from Ω to X. The normal map fC is defined from the set ∏−1Ω to X by(105)fCx=f∏x+x−∏x.
Moreover, the following variational problem(106)findy0∈C:fy0,c−y0≥0,for allc∈C,is completely equivalent to the normal-map equation fCx0=0 through the transformation x0=y0−fy0. 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 fCx=0 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 functionf admitting a PBA on a nonempty closed convex subset C of a Hilbert space H, the normal map associated with f admits a PBA on H. In our study we will show that the same result holds when we replace the normal maps fC+F in lieu of the normal maps fC. Rashid [8, 14] reformulate the normal maps fC+F 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 p-point-based approximation. Here we extend that concept to n,α-point-based approximation which is reformulated by Rashid [8, 14], then we show that if f have a n,α-point-based approximation, then one can easily be constructed a n,α-point-based approximation for fC+F.
The following reformulation of the normal mapsfC+F is due to [14].Definition 7.
LetC be a nonempty closed convex subset of a Banach space X and let ∏ be the metric projector from X onto C. Let Ω be an open subset of X meeting C and let f:Ω⟶X and F:Ω⇉X. The normal map fC+F is defined from the set ∏−1Ω to X by(107)fC+Fx=f∏x+F∏x+x−∏x.
We are now able to construct an,α-point-based approximation for the normal map fC+F provided that a n,α-point-based approximation exists for f. The following proposition can be extracted from [14, Proposition 4.3].Proposition 1.
LetX be a Banach space and C be a nonempty closed convex subset of X and let ∏ be the metric projector on C which is nonexpansive. Let f:C⟶X , A:C×C⟶X be functions and let F:C⇉X be a set-valued map with closed graph. If A is a n,α-point-based approximation for f on C with a constant L, then the function H:X×X⟶X defined by Hy,x=A∏y,⋅C+F⋅x is a n,α-point-based approximation for fC+F on X with the same constant L.Proof.
Lety,x∈X. We note that by the definition of normal map, fC+Fx and Hy,x are respectively defined by(108)fC+Fx=f∏x+F∏x+x−∏x,and(109)Hy,x=A∏x,∏x+F∏x+x−∏x.
By hypothesis we have thatA has the two properties for f given in Definition 4 with a constant L. We need to show that H also has these same two properties for fC+F with the constant L. Since A is the n,α-point-based approximation for f on C, then using the notion of first property of n,α-point-based approximation and the non expansiveness of the metric projector we have that(110)fC+Fx−Hy,x=f∏x+F∏x+x−∏x−A∏y,∏x+F∏x+x−∏x=f∏x−A∏y,∏x≤Lπn,α∏y−∏xn+α≤Lπn,αy−xn+α.
This implies thatH satisfies the first property of n,α-point-based approximation. For proving the second property, we suppose that x,x′∈X. To this end, let y,z∈X. We will show that Hx,⋅−Hx′,⋅ is Lipschitz continuous on X with lipschitz constant Lx−x′α. Again using the concept of second property of n,α-point-based approximation and non expansiveness of metric projector, we obtain that(111)Hx,y−Hx′,y−Hx,z−Hx′,z=A∏x,∏y+F∏y+y−∏y−A∏x′,∏y−F∏y−y−∏y−A∏x,∏z+F∏z+z−∏z−A∏x′,∏z−F∏z−z−∏x=A∏x,∏y−A∏x′,∏y−A∏x,∏z−A∏x′,∏z≤L∏x−∏x′∏x−∏z≤Lx−x′αy−z.
This shows that the second property of then,α-point-based approximation is satisfied. Since the both properties in Definition 4 are fulfilled for H, we can conclude that H is a n,α-point-based approximation for fC+F on X. The proof is completed.


## 4.1. Application ofn,α-PBA for smooth function f

Let0<α<1 and Ω be a convex subset of X. Let u,v∈Ω.(1)
Suppose that the Fréchet derivative off is ℓ,α-Hölder continuous. We show that the function(94)A:u,v↦fu+∇fuv−u,is an 1,α-point-based approximation for f. In this case, by using the Algorithm 1 we can infer that there exists a sequence xk which converges superlinearly and this result recovers the convergence result of Geoffroy and Piétrus in [19].In this regards, define the functionΛu,v by(95)Λu,v=fv−Au,v.It follows that(96)Λu,v=fv−fu−∇fuv−u=∫01∇fu+tv−u−∇fuv−udt≤v−u∫01∇fu+tv−u−∇fudt≤v−u∫01ℓtv−uαdt≤v−u1+αℓ∫01tαdt≤ℓα+1v−u1+α.This yields thatA satisfies the first property of 1,α-point-based approximation on Ω. To proof the second property of 1,α-point-based approximation, we assume that y,z∈Ω. Then, we have that(97)Λ′u,v,y,z=Au,y−Av,y−Au,z+Av,z,=fu+∇fuy−u−fv−∇fvy−v−fu−∇fuz−u+fv+∇fvz−v≤∇fu−∇fvy−z≤∇fu−∇fvy−z≤ℓu−vαy−z.This shows that the second property of1,α-PBA for f also holds. Therefore, we say that when the Fréchet derivative of f is ℓ,α-Hölder with exponent α∈0,1, the function A:u,v↦fu+∇fuv−u is an 1,α-point-based approximation.(2)
Letrx¯>0 be such that Brx¯/2x¯⊆X. Suppose that f is a twice Fréchet differentiable function on Brx¯/2x¯ such that ∇2f is K,α-Hölder on Brx¯/2x¯ and with exponent α∈0,1. Choose ℓ>0 and L>0 be such that(98)L>ℓ+Krx¯+1.Letp,q∈Brx¯/2x¯ and define the function(99)Ap,q=fp+∇fpq−p+12∇2fpq−p2.Then, Theorem2 ensures the existence of a sequence xk which converges super-quadratically and the result of Theorem 2 coincides with the result of [22, 29].To show the first property of2,α-point-based approximation, denote △p,q=fq−Ap,q. Then we have that(100)△p,q=fq−fp−∇fpq−p−12∇2fpq−p2.Since,∫011−t∇2fp+tq−pq−p2dt=fq−fp−∇fpq−p, then (100) reduces to(101)△p,q=∫011−t∇2fp+tq−pq−p2dt−12∇2fpq−p2=∫011−t∇2fp+tq−p−1−t∇2fpq−p2dt≤q−p2∫011−t∇2fp+tq−p−1−t∇2fpdt≤q−p2∫011−t∇2fp+tq−p−∇2fpdt≤Kq−p2∫011−ttq−pαdt≤Kq−p2+α∫011−ttαdt≤Kα+1α+2q−p2+α≤Lα+1α+2q−p2+α.Therefore, A satisfies the first property of an2,α-point-based approximation on Ω.For the proof of second property, we assume thata,b be any elements of Brx¯/2x¯, Then, we have that(102)△′p,q,a,b=Ap,a−Aq,a−Ap,b+Aq,b=fp+∇fpa−p+12∇2fpa−p2−fq−∇fqa−q−12∇2fqa−q2−fp−∇fpb−p−12∇2fpb−p2+fq+∇fqb−q+12∇2fqb−q2=∇fp−∇fqa−b+12∇2fpa−p2−∇2fqa−q2−∇2fpb−p2+∇2fqb−q2=∇fp−∇fqa−b+12∇2fpa−q+q−p,a−p−∇2fpb−q+q−p,b−p+∇2fqb−q,b−p+p−q−∇2fqa−q,a−p+p−q=∇fp−∇fqa−b+12∇2fpa−q,a−p+∇2fpq−p,a−p−∇2fpb−q,b−p−∇2fpq−p,b−p+∇2fqb−q,b−p+∇2fqb−q,p−q−∇2fqa−q,a−p−∇2fqa−q,p−q=∇fp−∇fqa−b+12∇2fqb−q,b−p−∇2fpb−q,b−p+∇2fpa−q,a−p−∇2fqa−q,a−p+∇2fpq−p,a−p−∇2fpq−p,b−p+∇2fqb−q,p−q−∇2fqa−q,p−q=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−p+12∇2fp−∇2fqa−q,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−p+12∇2fp−∇2fqa−b+b−q,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q.This also can be written as(103)△′p,q,a,b=∇fp−∇fqa−b+12∇2fq−∇2fpb−q,b−a+12∇2fp−∇2fqa−b,a−p+12∇2fpq−p,a−b+12∇2fqb−a,p−q.Since there exist an open subsetBrx¯/2x¯⊆X and a positive number K such that ∇2f≤K on Brx¯/2x¯. Let a,b∈Brx¯/2x¯. Then, a−b≤rx¯. Then, by applying the notion of ℓ,α-Hölder continuity property of ∇f and K,α-Hölder continuity property of ∇2f , we get(104)△′p,q,a,b≤∇fp−∇fqa−b+12∇2fq−∇2fpb−qb−a+12∇2fp−∇2fqa−ba−p+12∇2fpq−pa−b+12∇2fqb−ap−q≤ℓp−qαa−b+K2p−qαb−qb−a+K2p−qαb−aa−p+K2q−pαa−b+K2p−qαb−a≤ℓp−qαa−b+K2rx¯p−qαa−b+Kp−qαa−b≤ℓ+Krx¯+1p−qαa−b≤Lp−qαa−b,for alla,b∈Brx¯/2x¯.This shows that the second property of2,α-PBA is satisfied. Thus, both of properties for n,α-PBA hold on Brx¯/2x¯whenn=2 and 0<α<1. Hence, A is an 2,α-PBA for f on Brx¯/2x¯.


## 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.
LetC be a nonempty closed convex subset of a Banach space X and let ∏ be the metric projector from X onto C. Let Ω be an open subset of X meeting C and let f be a function from Ω to X. The normal map fC is defined from the set ∏−1Ω to X by(105)fCx=f∏x+x−∏x.
Moreover, the following variational problem(106)findy0∈C:fy0,c−y0≥0,for allc∈C,is completely equivalent to the normal-map equation fCx0=0 through the transformation x0=y0−fy0. 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 fCx=0 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 functionf admitting a PBA on a nonempty closed convex subset C of a Hilbert space H, the normal map associated with f admits a PBA on H. In our study we will show that the same result holds when we replace the normal maps fC+F in lieu of the normal maps fC. Rashid [8, 14] reformulate the normal maps fC+F 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 p-point-based approximation. Here we extend that concept to n,α-point-based approximation which is reformulated by Rashid [8, 14], then we show that if f have a n,α-point-based approximation, then one can easily be constructed a n,α-point-based approximation for fC+F.
The following reformulation of the normal mapsfC+F is due to [14].Definition 7.
LetC be a nonempty closed convex subset of a Banach space X and let ∏ be the metric projector from X onto C. Let Ω be an open subset of X meeting C and let f:Ω⟶X and F:Ω⇉X. The normal map fC+F is defined from the set ∏−1Ω to X by(107)fC+Fx=f∏x+F∏x+x−∏x.
We are now able to construct an,α-point-based approximation for the normal map fC+F provided that a n,α-point-based approximation exists for f. The following proposition can be extracted from [14, Proposition 4.3].Proposition 1.
LetX be a Banach space and C be a nonempty closed convex subset of X and let ∏ be the metric projector on C which is nonexpansive. Let f:C⟶X , A:C×C⟶X be functions and let F:C⇉X be a set-valued map with closed graph. If A is a n,α-point-based approximation for f on C with a constant L, then the function H:X×X⟶X defined by Hy,x=A∏y,⋅C+F⋅x is a n,α-point-based approximation for fC+F on X with the same constant L.Proof.
Lety,x∈X. We note that by the definition of normal map, fC+Fx and Hy,x are respectively defined by(108)fC+Fx=f∏x+F∏x+x−∏x,and(109)Hy,x=A∏x,∏x+F∏x+x−∏x.
By hypothesis we have thatA has the two properties for f given in Definition 4 with a constant L. We need to show that H also has these same two properties for fC+F with the constant L. Since A is the n,α-point-based approximation for f on C, then using the notion of first property of n,α-point-based approximation and the non expansiveness of the metric projector we have that(110)fC+Fx−Hy,x=f∏x+F∏x+x−∏x−A∏y,∏x+F∏x+x−∏x=f∏x−A∏y,∏x≤Lπn,α∏y−∏xn+α≤Lπn,αy−xn+α.
This implies thatH satisfies the first property of n,α-point-based approximation. For proving the second property, we suppose that x,x′∈X. To this end, let y,z∈X. We will show that Hx,⋅−Hx′,⋅ is Lipschitz continuous on X with lipschitz constant Lx−x′α. Again using the concept of second property of n,α-point-based approximation and non expansiveness of metric projector, we obtain that(111)Hx,y−Hx′,y−Hx,z−Hx′,z=A∏x,∏y+F∏y+y−∏y−A∏x′,∏y−F∏y−y−∏y−A∏x,∏z+F∏z+z−∏z−A∏x′,∏z−F∏z−z−∏x=A∏x,∏y−A∏x′,∏y−A∏x,∏z−A∏x′,∏z≤L∏x−∏x′∏x−∏z≤Lx−x′αy−z.
This shows that the second property of then,α-point-based approximation is satisfied. Since the both properties in Definition 4 are fulfilled for H, we can conclude that H is a n,α-point-based approximation for fC+F on X. The proof is completed.


## 5. Numerical Experiment

In this section, to present the numerical experiment we recall some necessary notations and notions . Lets∈ℝn and ψ:ℝn⟶ℝm be a Fréchet differentiable function at s. Suppose that the set of all points s∈ℝn is denoted by Pψ at which the derivative ψ′s exists. The B-subdifferential of ψ at s∈ℝn, denoted by ∂Bψs, is the set(112)∂Bψs=J∈ℝm×n:J=limk⟶+∞ψ′skfor somesk⊂Pψsuch thatsk⟶s.Then, Clarke’s generalized Jacobian ofψ at s∈ℝn is the set ∂ψs =conv ∂Bψs. If ψ is differentiable near s, and ψ′ is continuous at s, then obviously ∂ψs=∂Bψs=ψ′s. Otherwise, ∂Bψs is not necessarily a singleton, even if ψ is differentiable at s. In this case, ψ′s∈∂Bψs holds. Now, in order to illustrate the theoretical result of the extended Newton-type method, we consider the following example.Example 1.
LetX=Y=ℝ,s0=−1.7,η=5,L=0.5,rs¯=3,n=1,α=0.9 and M=1. Let ζ:ℝ⟶ℝ and ξ:ℝ⇉ℝ be defined, respectively, by(113)ζs=s7+s2,ifs<0,10s27−2s,ifs≥0,ξs=s14−17,s+17.
Then Algorithm1 generates a sequence which converges superlinearly to s∗=−0.5000 and s∗=−1.0000 , respectively, with initial points s0=−1.7 and s0=−1.5 in the case s<0. On the other hand, Algorithm 1 generates a superlinear convergent sequence which converges to s∗=01.4204 and s∗=0.5000 , respectively, with initial points s0=1.5 and s0=1.7 in the case s≥0.
Solution: It is manifest thatζ is not differentiable at s=0 and hence ζ is nonsmooth function on ℝ. But this function is differentiable on ℝ−0 and hence ∂Bζs=ζ′s. So, we get(114)∂Bζs=ζ′s=17+2s,ifs<0,20s7−2,ifs≥0.
We mark that(115)Γs:=ζ+ξss2+3s14−17,s2+8s7+17,ifs<0,10s27−27s14−17,10s27−s+17,ifs≥0.
Initially, we study the set-valued mappingΓs=s2+3s/14−1/7 for the case x<0 and note that Γ has a closed graph at s¯,t¯ with s¯=−1 and t¯=0.64. Thus, −1,0.64∈ gph Γ and ζ+ξ−1, that is, Γ−1 is Lipschitz-like at 0.64,−1. By taking As,⋅=ζs+∂Bζs⋅−s, it is easily shown that Rs¯⋅=ζs¯+∂Bζs¯⋅−s¯+ξ⋅−1 is Lipschitz-like at t¯,s¯ for t¯=0.64 and s¯=−1. Therefore, the assumptions of Theorem 2 hold. From the definition of ℳsk, we get(116)ℳsk=dk∈ℝ:0∈ζsk+∂Bζskdk+ξsk+dk=dk∈ℝ:dk=2−3sk−14sk23+28sk.
Alternatively, ifℳsk≠∅ we take(117)0∈ζsk+∂Bζsksk+1−sk+ξsk+1⇒sk+1=2+14sk23+28sk.
Also, from (86) with 0≤α≤1 we consume(118)dk≤2αηLMπn,α2α−MLrs¯αdk−1n+α.
Hereafter, for the given values ofL,M,η,rs¯,n and α, w get that Algorithm 1 generates a superlinearly convergent sequence with initial point s0=−1.7 in a neighborhood of s¯=−1.9. The following Tables 1 and 2, obtained by using Matlab code, indicate that the solution of the variational inclusion Γs∋0 has the solutions s∗=−1.0000 and s∗=−0.5000 in the case s<0 and s∗=0.5000 and s∗=1.4202 in the case s≥0. The graphs of Γ are plotted in Figure 1.Table 1
Numerical results for Example1 for the case s<0.
iteration no.skΓ=s2+3s/14−1/7skΓ=s2+8s/7+1/71-1.70002.3829-1.50000.67862-0.95200.5595-1.13460.13353-0.62090.1096-1.01610.01404-0.51420.0114-1.00030.00025-0.50020.0002-1.00000.00006-0.50000.0000-1.00000.00007-0.50000.0000-1.00000.0000Table 2
Numerical results for Example1 for the case s≥0.
iteration no.skΓ=10s2/7−27s/14−1/7skΓ=10s2/7−s+1/711.50000.17861.70002.571421.42420.00821.03330.634931.42040.00000.70810.151141.42040.00000.56050.031151.42040.00000.50870.003861.4204-0.00000.50020.000171.42040.00000.50000.000081.42040.00000.50000.0000Figure 1
Superlinear rate of convergence of Algorithm1 at -1.0000 (-0.5000) and 0.5000 (1.4204).
(a)(b)Remark 4.
If we setα=1 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 η>1,f+F−1 is Lipschitz-like and the nonsmooth function f has a n,α-point-based approximation. Moreover, when 0<α<1 and ∇f is ℓ,α-Hölder, we have presented an application of n,α-point-based approximation for smooth function with n=1, that is, we have shown A is an 1,α-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 n=2 and 0<α<1, if f is a twice Fréchet differentiable function and ∇2f is K,α-Hölder, we have given an application of n,α-point-based approximation, that is, we have shown A is an 2,α-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 fC+F which extends the concept of point-based-approximation reformulated by Rashid [8]. That is, we have shown that if f has an n,α-point-based approximations, it is easy to construct an n,α-point-based approximation for the fCC+F. Finally, we have presented a numerical experiment to validate the theoretical result.



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