# Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

Authors: Messaoud Bounkhel
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The present paper is devoted to the study of the generalized projectionπK:X∗→K, where X is a uniformly convex and uniformly smooth Banach space and K is a nonempty closed (not necessarily convex) set in X. Our main result is the density of the points x∗∈X∗ having unique generalized projection over nonempty close sets in X. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.


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## Body

## 1. Introduction

In 1994, Alber [1] (see also [2]) introduced and studied an appropriate extension of the projection operator over closed convex sets from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. It is calledgeneralized projection operator. He proved various properties and extended many existing results from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces.In 2005, Li [3] extended and studied this concept from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces. This concept has been used successfully in many applications such as variational inequalities, minimization principles, and differential inclusions (see [1, 2, 4–7] and the references therein). The main result in [1–3] is the existence property of the operator πK for closed convex sets in reflexive Banach spaces (resp., in uniformly convex and uniformly smooth Banach spaces) in [3] (resp., in [1]). Our main aim is to study the existence of πK for nonempty closed sets not necessarily convex. An application of our main result to variational problems with nonconvex sets is presented at the end of the paper.


## 2. Preliminaries

LetX be a Banach space with topological dual space X∗. We denote by B and B∗ the closed unit ball in X and X∗, respectively. We recall some definitions and results on uniformly convex and uniformly smooth Banach spaces (see, e.g., [8, 9]). The moduli of convexity and smoothness of X are defined, respectively, by (1)δXϵ=inf⁡1-x+y2:x=y=1,x-y=ϵ,0≤ϵ≤2,ρXt=sup⁡12x+y+x-y-1:x=1,y=t,t>0.The space X is said to be uniformly convex whenever δX(ϵ)>0 for all 0<ϵ≤2 and is said to be uniformly smooth whenever limt↓0⁡ρX(t)=0. Let p, q>1 be real numbers. The space X is said to be p-uniformly convex (resp., q-uniformly smooth) if there is a constant c>0 such that (2)δXϵ≥cϵpresp.,ρXt≤ctq.Obviously from the definition of p-uniform convexity and q-uniform smoothness the constants p and q satisfy q∈(1,2] and p≥2. It is known (see, e.g., [8, 9]) that uniformly convex Banach spaces are reflexive strictly convex and that uniformly smooth Banach spaces are reflexive. If X is a p-uniformly convex Banach space, then X∗ is a p′-uniformly smooth Banach space, where p′=p/(p-1) is the conjugate number of p. If X is a q-uniformly smooth Banach space, then X∗ is a q′-uniformly convex Banach space, where q′=q/(q-1).The normalized duality mappingJ:X⇉X∗ is defined by (3)Jx=jx∈X∗:jx,x=x2=jx2.Many properties of the normalized duality mapping J have been studied. For the details, one may see Takahashi’s book [10] or Vainberg’s book [11]. We list some properties of J: (J1)
For anyx∈X, J(x) is nonempty.(J2)
For anyx∈X and any real number α, J(αx)=αJ(x).(J3)
IfX is reflexive, then J is a mapping of X onto X∗.(J4)
IfX∗ is strictly convex (i.e., the unit sphere in X∗ is strictly convex; i.e., the inequality x∗+y∗<2 holds for all x∗, y∗∈X∗ such that x∗=y∗=1, x∗≠y∗), then J is a single valued mapping.(J5)
J is a continuous operator in smooth Banach spaces.(J6)
IfX is strictly convex, then J is one-to-one.(J7)
IfX is a reflexive strictly convex space with strictly convex dual space X∗ and if J∗:X∗⇉X is a normalized duality mapping in X∗, then J-1=J∗.(J8)
J is the identity operator in Hilbert spaces.It is known (see [8, 9]) that a reflexive Banach space X is smooth if and only if X∗ is strictly convex. Hence by (J3) and (J4), if X is a reflexive smooth Banach space, then J is a single valued mapping from X onto X∗. And, by (J7), if X is reflexive smooth strictly convex Banach space, then J-1=J∗. Let V:X∗×X→R be defined by (4)Vx∗,x=x∗2-2x∗,x+x2.First, we mention that, in Hilbert spaces (X∗=X), the functional V has the form V(x∗,x)=x∗-x2, ∀x,x∗∈X.We list now some important properties ofV needed in our proofs, when X is a reflexive smooth Banach space:(i)
V(x∗,x)≥0.(ii)
(x∗-x)2≤V(x∗,x)≤(x∗+x)2.(iii)
V(J(x),x)=0.(iv)
V(x∗,x) is continuous and V is convex with respect to x when x∗ is fixed and convex with respect to x∗ when x is fixed.(v)
V(x∗,x) is differentiable with respect to x when x∗ is fixed.(vi)
gradx⁡V(x∗,x)=2(J(x)-x∗). This property is true whenever the space X is smooth which is the case for uniformly convex spaces.(vii)
V(x∗,x)=0 if and only if x∗=J(x).Let f:X→R∪{+∞} be a function and x∈X where f is finite. We recall from [4] that the V-proximal subdifferential ∂πf(x) (called in [4] the analytical proximal subdifferential) is the set of all x∗∈X∗ for which there exists σ>0 such that(5)x∗,x′-x≤fx′-fx+σVJx,x′,for all x′ around x. Recall also [4] that the V-proximal normal cone (called in [4] the proximal normal cone) of a nonempty closed subset S in X at x∈S is defined by Nπ(S;x)=∂πψS(x), where ψS is the indicator function of S. It has been proved in [4] that Nπ(S;x) coincides with the normal cone in the sense of convex analysis N(S;x) given by N(S;x)={x∗∈X∗:〈x∗,y-x〉≤0,∀x∈S}.Based on the functionalV, a set πS(x∗) of generalized projections of x∗∈X∗ onto S is defined as follows (see [1]).Definition 1.
LetS be a nonempty subset of X and x∗∈X∗. If there exists a point x¯∈S satisfying (6)Vx∗,x¯=infx∈S⁡Vx∗,x,then x¯ is called a generalized projection of x∗ onto S. The set of all such points is denoted by πS(x∗).The following lemma is needed in our proofs and for its proof we refer to [1].Lemma 2.
IfE is a uniformly convex Banach space, then the inequality (7)VJx,y≥8C2δEx-y4Cholds for all x and y in E, where C=(x2+y2)/2.We end this section with the following important result proved in [4]. It proves the density of the set dom⁡(∂πf) in dom⁡f, that is, the set of points x in dom⁡f at which ∂πf(x)≠∅ is dense in dom⁡f.Theorem 3.
Letp≥2, let q∈(0,2], let X be a p-uniformly convex and q-uniformly smooth Banach space, and let f:X→R∪{+∞} be a lower semicontinuous function. Let x0∈dom⁡f, and let ϵ>0 be given. Then there exists a point y∈x0+ϵB satisfying ∂πf(y)≠∅ and f(x0)-ϵ≤f(y)≤f(x0). Consequently, dom⁡(∂πf) is dense in dom⁡f.


## 3. Minimization Principles in Banach Spaces

Given a lower semicontinuous functionf:X→R∪{+∞} that is bounded below and α>0, we define two functions fα:X→R and fα∗:X∗→R by(8)fαx≔infy∈X⁡fy+αVJy,x,(9)fα∗x∗≔infy∈X⁡fy+αVx∗,y.These functions fα and fα∗ coincide, in Hilbert spaces, with the inf-convolution of the function f and the function x↦αx2, which is due to the relation V(J(y),x)=V(J(x),y)=y-x2 in Hilbert spaces. In [4], the authors studied the function fα and they derived some minimization principles in p-uniformly convex and q-uniformly smooth Banach spaces. In this section, we establish similar results for the function fα∗ that will be used to prove our main theorem in this paper. We start with the following theorem proved in [4].Theorem 4.
Letp′≥2, let q′∈(1,2], let E be a p′-uniformly convex and q′-uniformly smooth Banach space, and let f:E→R∪{+∞} be a proper lower semicontinuous function. Suppose that f is bounded below by some constant c. Then fα is bounded below by c and is Lipschitz on each bounded subset of E. Furthermore, suppose that x∈E is such that ∂πfα(x) is nonempty. Then there exists a point y¯∈E satisfying the following: (i)
If{yi} is a minimizing sequence in E for the infimum in (8), then limi→+∞⁡yi=y¯.(ii)
The infimum in (8) is attained uniquely at y¯.LetX be a p-uniformly convex and q-uniformly smooth Banach space with p≥2 and q∈(0,2]. Then X is reflexive; that is, X∗∗=X, and J is one-to-one from X to X∗ with J-1=J∗. Thus, observe that the function fα∗ can be rewritten as follows: (10)fα∗x∗=infy∈X⁡fy+αVx∗,y=infy∗∈X∗⁡fJ∗y∗+αVx∗,J∗y∗=infy∗∈X∗⁡f∘J∗y∗+αV∗J∗y∗,x∗=infy∗∈X∗⁡Fy∗+αV∗J∗y∗,x∗=Fαx∗,where F:X∗→R is defined by F≔f∘J∗ and V∗:X∗∗×X∗→R is defined by (11)V∗J∗y∗,x∗=J∗y∗2-2J∗y∗;x∗+x∗2.Using this observation together with Theorem4 with E=X∗ and F playing the role of f we can prove the following theorem.Theorem 5.
Letp≥2, let q∈(1,2], let X be a p-uniformly convex and q-uniformly smooth Banach space, and let f:X→R∪{+∞} be a proper lower semicontinuous function which is bounded below by some constant c. Then fα∗ is bounded below by c and is Lipschitz on each bounded subset of X∗. Furthermore, for any x∗∈X∗ with ∂πfα∗(x∗)≠∅ there exists a point y¯∈X satisfying the following: i∗
If{yi} is a minimizing sequence in X for the infimum in (9), then limi→+∞⁡yi=y¯.ii∗
The infimum in (9) is attained uniquely at y¯∈X; that is, (12)fα∗x∗=fy¯+αVx∗,y¯.Proof.
LetE=X∗ and F=f∘J∗. Clearly F is a proper l.s.c. function on E and is bounded below by the same constant c. Then by Theorem 4 the function Fα=fα∗ is bounded below by c and is Lipschitz on each bounded subset in X∗. Furthermore, for any x∗∈X∗ with ∂πfα∗(x∗)≠∅, we have ∂πFα(x∗)≠∅ and so by Theorem 4 there exists a point z∗∈X∗ such that (i)
if{yn∗} is a minimizing sequence in X∗ for the infimum in(13)Fαx∗=infy∗∈X∗⁡Fy∗+αV∗J∗y∗,x∗,thenlimi→+∞⁡yn∗=z∗;(ii)
the infimum in (13) is attained uniquely at z∗∈X∗; that is, (14)Fαx∗=Fz∗+αV∗J∗z∗,x∗.The proof will be complete by taking y¯≔J∗(z∗) and by using the fact that J∗ is continuous in smooth Banach spaces.By taking different forms of the functionfα∗, we can obtain various types of minimization principles. We state here the two following types. The first one is Stegall’s minimization principle and the second one is a variant of the smooth Borwein-Preiss variational principle in Banach spaces (see [4] for different variants in Banach spaces and see [12] for those principles in Hilbert spaces).Theorem 6.
Letq∈(1,2] and p≥2, let X be a p-uniformly convex and q-uniformly smooth Banach space, and let f:X→R be a lower semicontinuous function. Suppose that f is bounded below on the bounded closed set S⊂X, with S∩dom⁡f≠∅. Then, for any ϵ>0, there exists x∗∈X∗ with x∗<ϵ such that the function y↦f(y)+〈x∗,y〉 attains a unique minimum over S.Proof.
Define onX∗ the function(15)gx∗≔infy∈X⁡fy+ψSy-12y2+12Vx∗,y,which is of the form hα∗ with h=f+ψS-1/2·2 and α=1/2. Furthermore, expression (15) for g can be rewritten as(16)gx∗=infy∈S⁡fy-x∗,y+12x∗2.Let ϵ>0 and let z∈S∩dom⁡f. Then for any x∗∈X∗ we have (17)gx∗≤fz-x∗,z+12x∗2<∞;that is, dom⁡g=X∗. Now, by the density theorem of the V-proximal subdifferential in Theorem 3, there exists y∗∈dom⁡∂πg; that is, ∂πg(y∗)≠∅ with y∗<ϵ and (18)infy∈S⁡fy-ϵ=g0-ϵ≤gy∗≤g0=infy∈S⁡fy.Using now Theorem 4(ii), we deduce that the infimum in (15) and (16) is attained at a unique point y¯∈S; that is, (19)gx∗=fy¯-x∗,y¯+12x∗2.Therefore, by taking x∗=-y∗, we obtain x∗<ϵ and the function y↦f(y)+〈x∗,y〉 attains a unique minimum over S at y¯∈S and so the proof is complete.The following theorem is a different variant of the smooth Borwein-Preiss variational principle in which the perturbation is given in terms of the functionalV.Theorem 7.
Letp≥2, let q∈(1,2], let X be a p-uniformly convex and q-uniformly smooth Banach space, and let f:X→R be a lower semicontinuous function bounded below, and ϵ>0. Suppose that x¯ is a point satisfying f(x¯)<infx∈X⁡f(x)+ϵ. Then for any λ>0 there exist points y¯∈X and z∗∈X∗ such that (i)
z∗-J(x¯)<λ, V(z∗,y¯)<λ, f(y¯)≤f(x¯),(ii)
f+(ϵ/λ)V(z∗;·) has a unique minimum at y¯.Proof.
Letϵ>0 be as in the statement of Theorem 7 and let λ>0. Put α=ϵ/λ and consider the function (20)fα∗x∗≔infy∈X⁡fy+ϵλVx∗,y.Since fα∗ is l.s.c. on X∗ and by the density result in Theorem 3, there exists z∗∈J(x¯)+λB satisfying fα∗(J(x¯))-ϵ≤fα∗(z∗)≤fα∗(J(x¯))≤f(x¯) with ∂πfα∗(z∗)≠∅. By Theorem 5 there is a unique point y¯∈X satisfying (21)fα∗z∗=fy¯+ϵλVz∗;y¯≤fx¯<infx∈X⁡fx+ϵand so (22)ϵλVz∗;y¯≤infx∈X⁡fx-fy¯+ϵ<ϵand so (23)Vz∗;y¯<λ.Thus, (24)fy¯+ϵλVz∗;y¯=fα∗z∗≤fx+ϵλVz∗;x,∀x∈X,and so the function x↦f(x)+(ϵ/λ)V(z∗;x) has a unique minimum at x=y¯.


## 4. Generalized Projections on Closed Nonconvex Sets

Let us start with the following example showing thatπK(φ) may be empty for nonconvex closed sets in uniformly convex and uniformly smooth Banach spaces.Example 8.
LetX=lp  (p≥1), let θ=(0,…,0,…)∈lp, and let S≔{e1,e2,…,en,…} with ej=(0,…,0,(j+1)/j,0,…). Then S is a closed nonconvex subset in X with πS(θ)=∅.Proof.
ClearlyS is closed and not convex. Let x be any element in {e1,e2,…,en,…}; that is, x=en for some n≥1, x=en=1+(1/n)>1. Then for any x∈S we have x>1 and so V(θ,x)=x2>1, ∀x∈S; that is, (25)1≤infx∈S⁡Vθ,x≤liminfn→∞⁡Vθ,en=liminfn→∞⁡1+1n=1and so (26)infx∈S⁡Vθ,x=1.This ensures that πS(θ)=∅.From the previous example, we see that even in uniformly convex and uniformly smooth Banach spaces the generalised projectionπK(x∗) may be empty for closed nonconvex sets and so there is no hope of getting the conclusion of Theorem  2.1 in [3] saying that πK(x∗)≠∅, ∀x∗∈X∗, whenever the set S is closed convex in reflexive Banach spaces. However, we are going to prove that, for closed nonconvex sets, the set of points x∗∈X∗ for which πS(x∗)≠∅ is dense in X∗. We are going to prove our main result in the following theorem. It is an analogue result to Lau’s theorem for metric projections in reflexive Banach spaces [13].Theorem 9.
Letp≥2 and q∈(1,2], let X be a p-uniformly convex and q-uniformly smooth Banach space, and let S be any closed nonempty set of X. Then there is a dense set of points in X∗ admitting unique generalised projection on S; that is, for any x∗∈X∗, there exists xn∗→x∗ with πS(xn∗)≠∅, ∀n.Proof.
Observe that(27)dSVx∗=infy∈X⁡ψSy+Vx∗,y,which means that dSV has the form fα∗ with α=1 and f=ψS. Since f is proper l.s.c. and is bounded below, we can apply Theorem 5 to get for any x∗∈X∗ with ∂πdSV(x∗)≠∅ the existence of some y¯∈X satisfying (28)dSVx∗=ψSy¯+Vx∗,y¯;that is, y¯∈S and dSV(x∗)=V(x∗,y¯), which means that y¯∈πS(x∗). Using now the density result in Theorem 3, to get the density of the set dom⁡∂πdSV(·) in X∗, that is, for any x∗∈X∗, there exists xn∗→x∗ with ∂πdSV(xn∗)≠∅. Therefore, by what precedes, there exists yn∈πS(xn∗)∀n; that is, πS(xn∗)≠∅, ∀n. This proves the conclusion of the theorem.


## 5. Applications to Nonconvex Variational Problems

Letp≥2 and q∈(1,2] and let X be a p-uniformly convex and q-uniformly smooth Banach space. Let F:X⇉X∗ be a set-valued mapping and let S⊂X be a nonempty closed set not necessarily convex. Our aim is to use the main result in the previous section to study the following nonconvex variational problem: (29)Findx¯∈Ssuch  thatNπS;x¯∩-Fx¯≠∅.First we show that in the convex case (29) coincides with the usual variational inequality(30)Findx¯∈S,y∗∈Fx¯such  thaty∗,y-x¯≥0,∀y∈S.Proposition 10.
WheneverS is a closed convex set, one has (29) ⇔ (30).Proof.
The proof follows from the fact thatNπ(S;x) coincides with the convex normal cone which can be characterised as N(S;x)={x∗∈X∗:〈x∗,s-x〉≤0,∀s∈S}.We suggest the following algorithm to solve the proposed problem (29) under some natural and appropriate assumptions on S and F.Algorithm 11.
Letδn↓0 with δ0 being too small: (i)
Selectx0∈S, y0∗∈F(x0), and ρ>0.(ii)
Forn≥0,
(a)
computezn+1≔J∗(J(xn)-ρyn∗);(b)
chooseun+1∈J∗(J(zn+1)+δnB∗) with πS(J(un+1))≠∅;(c)
computexn+1≔πS(J(un+1)) and yn+1∗∈F(xn+1).Since S is not necessarily convex, the generalised projection πS does not exist necessarily for any x∗∈X∗∖J(S). However, our previous algorithm is well defined as we will prove in the following proposition.Proposition 12.
Assume thatX is uniformly convex and uniformly smooth Banach space. The above algorithm is well defined.Proof.
Letn≥0 and let xn∈S with yn∗∈F(xn) be given. The point zn+1 is well defined since J and J∗ are well defined and one-to-one because the space X is assumed to be uniformly convex and uniformly smooth. Now, since the generalised projection of J(zn+1) is not ensured we use our main result in Theorem 9 to choose some point J(un+1)∈X∗ too close to J(zn+1) so that J(zn+1)-J(un+1)≤δn and πS(J(un+1))≠∅. Then by the same theorem we have the uniqueness of the generalised projection so we can take xn+1≔πS(J(un+1)) and then we are done.After proving the well definedness of the algorithm without any additional assumptions onS and F we add some natural assumptions on the data to prove the convergence of the sequence {xn}n to a solution of (29).In our analysis we need the following assumptions onS and F:Assumptions
A
(1)
The solution set of (29) is nonempty.
(2)
The setS is ball compact; that is, any bounded set in S is relatively compact.
(3)
F is bounded on S by some constant L>0.
(4)
F is β-Lipschitz on S; that is, (31)y1∗-y2∗≤βx1-x2,∀xi∈S,∀yi∈Fxi,i=1,2.
(5)
F is α-J-strongly monotone on S; that is, (32)J∗y1∗-y2∗;Jx1-Jx2≥αJx1-Jx22,∀xi∈S,∀yi∗∈Fxi,i=1,2.
(6)
There exist some constantsμ>0 and ξ>0 such that (33)πSu1∗-πSu2∗≤ξu1∗-u2∗,∀u1∗,u2∗∈JS+μB∗.
(7)
The constantsμ, δ0, α, c, ξ, and β satisfy (34)ξc>βξ-1,0<δ0<μ.Theorem 13.
Assume thatX is 2-uniformly smooth. Let {xn}n be a sequence generated by Algorithm 11. Assume that Assumptions A hold and that the parameter ρ satisfies the inequalities (35)αβ2-ϵ¯<ρ<min⁡μ-δ0L,αβ2+ϵ¯,where ϵ¯≔α2-β2(1-c/ξ2)/β2. Then there exists a subsequence of {xn}n converging to a solution x~ of (29).Proof.
Letx¯ be a solution of (29); that is, there exists y¯∗∈F(x¯) such that -y¯∗∈Nπ(S;x¯). Hence by definition of the V-proximal normal cone there exists r>0 such that x¯∈πS(J(x¯)-ry¯∗). Without loss of generality we may assume that ρ is too small so that ρ∈(0,r]. First we claim that x¯∈πS(J(x¯)-ρy¯∗); that is, V(J(x¯)-ρy¯∗,x¯)=infs∈S⁡V(J(x¯)-ρy¯∗,s). Let λ≔ρ/r∈(0,1]. Then for any s∈S we have (36)2Jx¯-ρy¯∗-Jx¯;s-x¯=2λJx¯-ry¯∗+1-λJx¯-Jx¯;s-x¯=2λJx¯-ry¯∗-Jx¯;s-x¯.We distinguish two cases.
Case  1. If 〈(J(x¯)-ry¯∗)-J(x¯);s-x¯〉≤0, then obviously we have (37)2Jx¯-ρy¯∗-Jx¯;s-x¯≤0≤VJx¯,s.
Case  2. If 〈(J(x¯)-ry¯∗)-J(x¯);s-x¯〉≥0, then since 0<λ≤1 we have (38)2λJx¯-ry¯∗-Jx¯;s-x¯≤2Jx¯-ry¯∗-Jx¯;s-x¯and so we obtain(39)2Jx¯-ρy¯∗-Jx¯;s-x¯≤2Jx¯-ry¯∗-Jx¯;s-x¯≤Jx¯-ry¯∗2-2Jx¯-ry¯∗;x¯+x¯2+2Jx¯-ry¯∗;s-Jx¯-ry¯∗2-s2+s2-2Jx¯;s-x¯-x¯2≤VJx¯-ry¯∗,x¯-VJx¯-ry¯∗,s+VJx¯,s≤infz∈S⁡VJx¯-ry¯∗,z-VJx¯-ry¯∗,s+VJx¯,s≤VJx¯,s.Therefore, in both cases, we have (40)2Jx¯-ρy¯∗-Jx¯;s-x¯≤VJx¯,s.Hence (41)2Jx¯-ρy¯∗-Jx¯;s-x¯-VJx¯,s≤0.On the other hand, simple decomposition yields (42)2Jx¯-ρy¯∗-Jx¯;s-x¯-VJx¯,s=VJx¯-ρy¯∗,x¯-VJx¯-ρy¯∗,s.Consequently, we have (43)VJx¯-ρy¯∗,x¯-VJx¯-ρy¯∗,s≤0,for  anys∈S,which means that x¯∈πS(J(x¯)-ρy¯∗). Set z¯≔J∗(J(x¯)-ρy¯∗). Since X is 2-uniformly smooth we have the 2-uniform convexity of the dual space X∗ and so δX∗(τ)≥2cτ2 for some constant c depending only on the space X∗. On the other hand, by Lemma 2, we have (44)V∗J∗x∗,y∗≥8C2δX∗x∗-y∗4C,∀x∗,y∗∈X∗,where C=(x∗2+y∗2)/2. Hence (45)V∗J∗x∗,y∗≥8C2δX∗x∗-y∗4C≥cx∗-y∗2,∀x∗,y∗∈X∗.Therefore, we obtain (46)cJzn+1-Jz¯2=cJxn-Jx¯-ρyn∗-y¯∗2≤V∗ρJ∗yn∗-y¯∗;Jxn-Jx¯≤ρ2yn∗-y¯∗2-2ρJ∗yn∗-y¯∗;Jxn-Jx¯+Jxn-Jx¯2.Using now the β-Lipschitz property and the α-J-strong monotony of F, we write (47)
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x
¯
2and so(48)Jzn+1-Jz¯≤c-11-2ρα+ρ2β2Jxn-Jx¯.Since xn and x¯ belong to S (by construction) we have J(xn), J(x¯)∈J(S) and so by our assumptions on the constants L, δ0, and μ and the choice of ρ we obtain (49)dJSJun≤dJSJzn+Jun-Jzn≤ρyn∗+δn<ρL+δ0<μ,dJSJz¯=dJSJx¯-ρy¯∗≤ρy¯∗<ρL<μwhich ensures that J(un) and J(z¯) belong to J(S)+μB∗. This yields with the Lipschitz assumption of the generalised projection on J(S)+μB∗ that (50)Jxn-Jx¯=JπSJun-JπSJz¯≤ξJun-Jz¯≤ξJzn-Jz¯+Jun-Jzn≤ξJzn-Jz¯+ξδn-1.And consequently inequality (48) becomes (51)Jzn+1-Jz¯≤ξc-11-2ρα+ρ2β2Jzn-Jz¯+ξδn-1c-11-2ρα+ρ2β2.Set ζ≔ξc-1(1-2ρα+ρ2β2), δ¯n≔ξδn-1c-1(1-2ρα+ρ2β2), and Φn≔J(zn)-J(z¯). Then for any n≥1 we have (52)Φn+1≤ζΦn+δ¯n.By mathematical induction we get (53)Φn+1≤∑k=0n-1ζkδ¯n-k+ζnΦ1≤∑k=0n-1ζk+ζnΦ1≤1-ζn1-ζ+ζnΦ1.Our Assumptions A and the choice of ρ ensure that 0<ζ<1 and hence the sequence Φn is bounded and so the sequences {zn}n and {xn}n are bounded and since the set S is ball compact then the sequence {xn}n is compact and hence there exists a subsequence {xnk}k converging to some limit x~∈S. By Lipschitz property of F we can check easily that the sequence {ynk}k is convergent to some limit y~∗ belonging to F(x~) and so limk⁡J(unk+1)=limk⁡J(znk+1)=J(x~)-ρy~∗. Set z~≔J∗(J(x~)-ρy~∗). To complete the proof we have to prove that x~ is a solution of (29). By Algorithm 11 the subsequence {xnk}k satisfies xnk+1=πS(J(unk+1)) and so by the Lipschitz property of the generalised projection on the set J(S)+μB∗ we can write (54)πSJz~-x~≤πSJz~-πSJunk+1+xnk+1-x~≤ξJunk+1-Jz~+xnk+1-x~⟶0and so x~=πS(J(z~))=πS(J(x~)-ρy~∗) which ensures by definition of the V-proximal normal cone that (55)0=ρ-1Jx~-ρy~∗-Jx~+y~∗∈NπS;x~+Fx~.Thus (56)NπS;x~∩-Fx~≠∅;that is, x~ is a solution of (29). Thus the proof is complete.



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*Source: 478437-2015-06-23.xml*