# V-Prox-Regular Functions in Smooth Banach Spaces

Authors: Messaoud Bounkhel; Mostafa Bachar
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we continue the study of theV-prox-regularity that we have started recently for sets. We define an appropriate concept of the V-prox-regularity for functions in reflexive smooth Banach spaces by adapting the one given in Hilbert spaces. Our main goal is to study the relationship between the V-prox-regularity of a given l.s.c. f and the V-prox-regularity of its epigraph.


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## Body

## 1. Introduction and Preliminaries

Throughout this work,X will denote a reflexive smooth Banach space unless otherwise specified. We recall from [1] the concept of V-proximal subdifferential.Definition 1.
Letf:X⟶ℝ∪+∞ be a lower semicontinuous (l.s.c.) function and x∈X, where f is finite. We recall that the V-proximal subdifferential of f at x is defined as x∗∈∂πfx if and only if there exists σ>0 such that
(1)x∗,x′−x≤fx′−fx+σVJx,x′,∀x′nearx.Here,J is the normalised duality mapping on X, and V is the functional defined from X∗×X to 0,∞ by
(2)Vx∗,x=∥x∗∥2−2x∗;x+∥x∥2,foranyx∗,x∈X∗×X.TheV-proximal normal cone of a nonempty closed subset S in X at x∈S is defined as the V-proximal subdifferential of the indicator function of S, that is, NπS;x=∂πψSx.Another proximal subdifferential∂Gπfx is defined (see [5]) geometrically via the V-proximal normal cone of the epigraph as follows:
(3)∂Gπfx=x∗∈X∗:x∗;−1∈Nπepif;x,fx.It has been proved in [2] that in general, we have the inclusion ∂πfx⊂∂Gπfx. The generalized projection on a closed nonempty set S is defined as follows: x¯∈πSx∗ if and only if Vx∗,x¯=infx∈SVx∗,x (see [3] for convex sets and see [1] for nonconvex sets). We also recall (see for instance [2]) the definition of the Fréchet subdifferential and Fréchet normal cone as follows: x∗∈∂Ffx¯ if and only if for all ε>0, there exists δ>0 such that
(4)x∗;x−x¯≤fx−fx¯+ε∥x−x¯∥,∀x∈x¯+δB.The Fréchet normal coneNFS;x of a nonempty closed subset S in X at x¯∈S is defined as NFS;x¯=∂FψSx¯.


## 2. GeneralizedV-Prox-Regular Sets

In this section, we recall the concept of the generalizedV-prox-regularity of sets introduced and studied in [4] and other results needed in our work.Definition 2.
LetS be a nonempty closed set in a reflexive Banach space X and let x¯∈S. We will say that S is generalized V-prox-regular at x¯ if and only if there exist r>0 and ε>0 such that for all x∈S∩x¯+εB and for any x∗∈NπS;x∩εB∗, the point x is a generalized projection of Jx+rx∗ on S∩x¯+εB, that is, x∈πS∩x¯+εBJx+rx∗.We quote from [5, 6] the following two results.Lemma 3.
For anyx∈domf and any x,α∈epif (i.e., fx≤α), we have
(5)x∗,0∈Nπepif;x,α⇒x∗,0∈Nπepif;x,fx.Proposition 4.
LetX be an Asplund space and let f:X⟶ℝ∪∞ be a proper l.s.c. function around x¯∈domf. Then, for any x∗∈X∗ with x∗;0∈NFepif;x¯,fx¯, there exist sequences xk⟶x¯ with fxk⟶fx¯, λk⟶0+, and xk∗∈λk∂Ffxk such that ∥xk∗−x∗∥∗⟶0.We recall from [2] that X is V-proximal trustworthy provided that for any ε>0, any two functions f1,f2:X⟶ℝ∪∞ and any u∈X such that f1 is lower semicontinuous and f2 is Lipschitz around u the following fuzzy sum rule holds:
(6)∂Aπf1+f2u⊂⋃∂Aπf1u1+∂Aπf2u2:ui∈Ufiu,ε,i=1,2+εB∗.Here,Ufiu,ε≔x∈u+εBsuchthat∣fix−fiu∣<ε, and B∗ denotes the closed unit ball in X∗. The proof of the following proposition can be found in [2].Proposition 5.
LetX be a V-proximal trustworthy space. Let S be a closed subset of X with x∈S and let x∗∈NFS;x. Then, for any ε>0, there exists xε∈x¯+εB∩S such that x∗∈NπS;xε+εB∗.


## 3.V-Prox-Regular Functions

In Hilbert spaces setting, the authors in [7] extended and studied the concept of prox-regularity for functions which has been introduced for finite dimensional spaces in [8]. Many papers studied this concept in finite dimensional spaces and its applications to nonsmooth optimization (see for instance [9] and the references therein). Our aim here is to extend the concept of prox-regularity to reflexive smooth Banach spaces by using the concept of the V-proximal normal cone and the functional V. Another way to extend this concept has been proposed in [10].Definition 6.
Assume thatX is a reflexive smooth Banach space. A l.s.c. function f:X⟶ℝ∪∞ is said to be a V-prox-regular at x0∈domf for x0∗∈∂πfx0 if and only if there exist ε>0,r>0 such that ∀x,x∗∈∂Gπf with ∥x−x0∥≤ε, ∣fx−fx0∣<ε0 and ∥x∗−x0∗∥<ε0 such that
(7)fx′≥fx+x∗,x′−x−r2VJx,x′,∀x′∈x0+ε0B.We say thatf is V-prox-regular at x0 if it is V-prox-regular for any x0∗∈∂πfx0. Obviously, this concept coincides with the one studied in [7] in Hilbert spaces. Using this definition for functions, we can associate to it another definition of the V-prox-regularity for sets via the indicator function. We have the following definition:Definition 7.
LetX be a reflexive smooth Banach space. A closed nonempty set S is said to be V-prox-regular at x0∈S for x0∗∈NπS;x0 if and only if the indicator function ψS is V-prox-regular at x0 for x0∗∈∂πψx0.Proposition 8.
LetX be a reflexive smooth Banach space. A closed set S is generalized V-prox-regular at x0∈S in the sense of Definition 2, if and only if the indicator function ψS is V-prox-regular at x0 for x0∗=0 in the sense of Definition 6.Proof. Assume thatS is generalized V-prox-regular at x0∈S, then there exist r>0 and ε>0 such that for all x∈S∩x0+εB and for any x∗∈NπS;x∩εB∗, the point x is a generalized projection of Jx+rx∗ on S∩x0+εB, that is, x∈πS∩x0+εBJx+rx∗. By definition of generalized projection, we have
(8)VJx+rx∗;x≤VJx+rx∗;x′.∀x′∈S∩x0+εB.So, for anyx′∈S∩x0+εB, we have
(9)x2−2Jx+rx∗;x≤x′2−2Jx+rx∗;x′.Thus,(10)2rx∗;x′−x=2Jx+rx∗;x′−x−2Jx;x′−x=2Jx+rx∗;x′−x−2Jx;x′+2∥x∥2≤∥x′∥2−2Jx;x′+∥x∥2=VJx,x′,for any x′∈S∩x0+εB. Hence,
(11)x∗;x′−x≤12rVJx,x′,forallx′∈S∩x0+εBand so
(12)x∗;x′−x≤ψSx′−ψSx+12rVJx,x′,forallx′∈x0+εB.On the other hand, we havex∗∈NπS;x∩εB∗ which implies that x∗∈∂GπψSx with ∥x∗−x0∗∥<ε and x0∗=0. Thus, the inequality (12) ensures that ψS is V-prox-regular at x0 for x0∗=0. For the opposite direction, we follow the same lines as in the direct one.In the next theorem, we study the relationship between theV-prox-regularity of a function f and the V-prox-regularity of its epigraph.Theorem 9.
Letf:X⟶ℝ∪∞ be a function that is l.s.c. at x0∈domf. If f is V-prox-regular at x0 for x0∗∈∂πfx0, then the epigraph epif is V-prox-regular at x0,fx0forx0∗,−1∈Nπepif;x0,fx0. If, in addition, the space X is 2-uniformly convex, then the previous implication becomes an equivalence, that is, f is V-prox-regular at x0 for x0∗∈∂πfx0 if and only if the epigraph epif is V-prox-regular at x0,fx0 for x0∗,−1∈Nπepif;x0,fx0.Proof. Assume thatf is V-prox-regular at x0 for x0∗∈∂πfx0, fix ε0∈0,1 and r>0 such that ∀x,x∗∈∂Gπf with ∥x−x0∥≤ε, ∣fx−fx0∣<ε0 and ∥x∗−x0∗∥<ε0 such that
(13)fx′≥fx+x∗,x′−x−r2VJx,x′,∀x′∈x0+ε0B.By l.s.c. off, we choose ε∈0,1 with ε/1−ε1+∥x0∗∥<ε0 and such that for any x′∈x0+εB, we have fx0−ε<fx′. Fix any x,α∈epif, with ∥x,α−x0,fx0∥<ε and x∗,−λ∈Nπepif;x,α with ∥x∗,−λ−x0∗,−1∥<ε. Clearly, 0<1−ε<λ<1+ε, which ensures that α=fx and so x∗/λ∈∂Gπfx. Take now any x′,α′∈epif with ∥x′,α′−x0,fx0∥<ε. Notice that
(14)fx0−ε<fx′≤α′<fx0+ε0,∥x∗λ−x0∗∥≤1λ∥x∗−λx0∗∥=1λ∥x∗−x0∗−λx0∗+x0∗∥≤1λ∥x∗−x0∗∥+∥λx0∗−x0∗∥≤1λε+1−λ∥x0∗∥≤ελ+ε1−ε∥x0∗∥≤ε1−ε1+∥x0∗∥<ε0,which yields by (13)
(15)x∗λ,x′−x−α′−fx≤x∗λ,x′−x−fx′−fx≤r2VJx,x′,and hence,
(16)x∗,x′−x−λα′−α≤rλ2VJx,x′.On the other hand, we have(17)VX×BJx,α;x′−α′=VJx,x′+α′−α2.So,
(18)x∗,−λ;x′,α′−x,α≤rλ2VX×BJx,α;x′−α′.Thus, for anyx,α∈epif, with ∥x,α−x0,fx0∥<ε and x∗,−λ∈Nπepif;x,α with ∥x∗,−λ−x0∗,−1∥<ε, we have
(19)x∗,−λ;x′,α′−x,α≤r1+ε02VX×BJx,α;x′−α′,for any x′,α′∈epif with ∥x′,α′−x0,fx0∥<ε. This means that the epigraph epif is V-prox-regular at x0,fx0 for x0∗,−1∈Nπepif;x0,fx0.Assume now that the spaceX is a 2-uniformly convex space. Assume that the epigraph epif is V-prox-regular at x0,fx0 for x0∗,−1∈Nπepif;x0,fx0, choose ε0∈0,1 and r>0 such that for any x,α∈epif and any x∗,−λ∈Nπepif;x,α with ∥x,α−x0,fx0∥<ε0 and with ∥x∗,−λ−x0∗,−1∥<ε0, we have
(20)x∗,−λ;x′,α′−x,α≤r2VJx;x′+α′−α2,for all x′,α′∈epif with ∥x′,α′−x0,fx0∥<ε0. Using the fact that space X is a 2-uniformly convex space, we can find a positive constant β¯>0 depending only on the space X and the constant M≔∥x0∥+4ε0 such that
(21)VJx,x′≥β¯∥x−x′∥2,∀x′,x∈MB.Choose now a positive numberε∈0,minε0,1/3r1+∥x0∗∥,1/9βr1/p such that
(22)2ε1+∥x0∗∥+βεp1+β¯−1ε2+∥x0∗∥2<min1r,1r2and such that (according to the l.s.c. of f)
(23)fx>fx0−12r,∀x∈x0+εB.Here,β depends only on the space X and M. Then, for any x∈x0+εB, we have −ε<fx−fx0<ε and so
(24)fx−fx0+1r>fx0−12r−fx0+1r−ε=12r−ε>0.Also, we have by (23) one that has
(25)r1+β¯−1ε2+∥x0∗∥2<1r2ε1+∥x0∗∥+βεp,and hence since 1/r2ε1+∥x0∗∥+βεp>1 (by (23) again), we may fix some positive number σ such that
(26)max1,r1+β¯−1ε2+∥x0∗∥2<σ<1r2ε1+∥x0∗∥+βεp.Take now anyx,x′∈x0+ε/2B with ∣fx−fx0∣<ε/2 and ∣fx′−fx0∣<ε/2 and x∗∈∂πfx with x∗∈x0∗+ε/2B∗. Then, applying (20) with x∗,−1 instead of x∗,−λ, fx instead of α, and fx′ instead of α′, we obtain
(27)x∗;x′−x−fx′−fx≤r2VJx;x′+r2fx′−fx2,which is equivalent to (with ρ≔1/r)
(28)2ρx∗;x′−x−VJx;x′≤fx′−fx2+2ρfx′−fx=fx′−fx+ρ2−ρ2and so
(29)ρ2+2ρx∗;x′−x−VJx;x′≤fx′−fx+ρ2.Note that(30)ρ2+2ρx∗;x′−x−VJx;x′≥ρ2−2ρ∥x∗∥∥x′−x∥−βεp≥ρ2−2ρε1+∥x0∗∥−βεp≥ρ2−2ρρ3−ρ29=2ρ29>0.Thus,(31)fx′−fx+ρ≥ρ2+2ρx∗;x′−x−VJx;x′1/2.Now, we develop the expression(32)E≔ρ2+2ρx∗;x′−x−VJx;x′−ρ+x∗,x′−x−σVJx,x′2,E=−VJx;x′+2ρσVJx,x′−x∗,x′−x−σVJx,x′2=2ρσ−1VJx;x′−x∗,x′−x2+2σVJx,x′x∗,x′−x−σ2VJx,x′2.Observe that(33)x∗,x′−x≤∥x∗∥∥x′−x∥≤ε2+∥x0∗∥ε,VJx,x′≤βεp,x∗,x′−x2≤∥x∗∥2∥x′−x∥2≤ε2+∥x0∗∥2β¯−1VJx,x′.Thus,(34)E≥2ρσ−1VJx;x′−ε2+∥x0∗∥2β¯−1VJx,x′+2σVJx,x′x∗,x′−x−σ2VJx,x′2=VJx;x′2ρσ−1−ε2+∥x0∗∥2β¯−1−2ε2+∥x0∗∥εσ−σ2βεp.But asσ>1, one has
(35)2ρσ−1−2σε1+∥x0∗∥−ε2+∥x0∗∥2β¯−1−σ2βεp≥−1−ε2+∥x0∗∥2β¯−1+ρσ+σρ−2σε1+∥x0∗∥−σβεp.Using the choice ofσ in the inequality (26), we have
(36)σ>r1+β¯−1ε2+∥x0∗∥2,σ<1r2ε1+∥x0∗∥+βεp.which ensure, respectively,
(37)−1−ε2+∥x0∗∥2β¯−1+σr>0,1r−2σε1+∥x0∗∥−σβεp>0.Sinceρ=σ/r, we obtain
(38)−1−ε2+∥x0∗∥2β¯−1+ρσ+σρ−2σε1+∥x0∗∥−σβεp>0.Therefore, it follows from (35) that E>0 and so
(39)ρ2+2ρx∗;x′−x−VJx;x′>ρ+x∗,x′−x−σVJx,x′2>0,which ensures that
(40)ρ2+2ρx∗;x′−x−VJx;x′1/2>ρ+x∗,x′−x−σVJx,x′>0.Thus, by (31) and (35), we obtain
(41)fx′−fx+ρ≥ρ+x∗,x′−x−σVJx,x′.Now, it remains to show the conclusion forx′ satisfying ∣fx′−fx0∣≥ε. First choose, by the l.s.c. of f, some positive number
(42)η<ε41+∥x0∗∥,such that
(43)fx−fx0>−ε,∀x∈x0+ηB.Now, fix anyx,x′∈x0+ηB with only fx−fx0<η, and fix any x∗∈x0∗+ηB∗. Thus, if ∣fx′−fx0∣≥ε∣, then
(44)fx′−fx0≥εorfx−fx0≤−ε.But sincefx′−fx0>−ε (by (43)), we get only
(45)fx′≥ε+fx0.Therefore,(46)fx′=fx+fx0−fx+fx′−fx0≥fx−η+ε≥fx+ε2.Observe that(47)x∗,x′−x−σVJx,x′≤∥x∗∥∥x′−x∥≤1+∥x0∗∥η<ε2,and hence, we get
(48)fx′−fx+ρ≥ρ+x∗,x′−x−σVJx,x′.Finally, the relation (41) together with the last inequality conclude that f is V-prox-regular at x¯ for x0∗.Remark 10.
In the previous theorem, we proved the following: iff is V-prox-regular at x0 for x0∗∈∂πfx0, then we have the V-prox-regularity of epif at x0,fx0 for x0∗,−1∈Nπepif;x0,fx0. Unfortunately, we do not obtain the generalized V-prox-regularity of the epigraph in the sense of Definition 2 since epif is a generalized V-prox-regularity at x0,fx0 if and only if it is V-prox-regularity at x0,fx0 for 0,0∈Nπepif;x0,fx0 which cannot be the case (since x0∗,−1≠0,0) under the V-prox-regularity of f at x0. To ensure the generalized V-prox-regularity of the epigraph, we need another kind of regularity called the V-primal lower nice function introduced and studied in [5].Definition 11.
Letf:X⟶ℝ∪∞ be a l.s.c. function. We will say that f is V-primal lower nice at x¯∈domf, if there exist λ1,λ2>0,c>0,T>0 such that
(49)fy≥fx+x∗,y−x−t2VJx,y,whenever t≥T, x∈NVx¯,λ1,λ2,y∈x¯+λ1B,x∗∈∂Gπfx, and ∥x∗∥≤ct. Here,
(50)NVx¯,λ1,λ2≔x∈x¯+λ1BwithVJx¯,x≤λ22.This concept is stronger than theV-prox-regularity. Indeed, as mentioned in [8] for finite dimensional spaces, obviously, any function that is a V-p.l.n. function is V-prox-regular. However, the inverse is not true. For a V-p.l.n function, the condition (49) must hold for all V-proximal gradients with the linear growth condition, whereas for V-prox-regular functions, the condition (7) only has to hold for V-proximal subgradients close to a fixed x0∗∈∂πfx0 and just in a neighborhood of x0 with fx close to fx0. For instance (see [8]), the function f on ℝ with fx=1 for x>0 and fx=0 for x≤0 is easily seen to be V-prox-regular at x0=0 for x∗=0 but not V-p.l.n. for this x0. We have to notice that this definition of V-p.l.n. functions extends, to reflexive smooth Banach spaces, the definition of primal lower nice functions defined in finite dimension spaces in [11], and in Hilbert spaces in [12]. Our next proposition shows the generalized V-prox-regularity of the epigraph in the sense of Definition 2 of any V-primal lower nice functions.Proposition 12.
LetX be a V-proximal trustworthy space. If f is V-primal lower nice at x¯∈domf, then epif is generalized V-prox-regular at x¯,fx¯.Proof. By definition ofV-p.l.n. at x¯, we have positive numbers ε1>0,ε2>0,r>0,T>0 such that for any t≥T, any x∈Nx¯,ε1,ε2, and any x∗∈∂Gπfx∩rtB∗, we have
(51)x∗,y−x≤fy−fx+t2VJx,y,∀y∈x¯+ε1B.Takex,α∈epif and x∗,−λ∈Nπepif;x,α∩ε0BX×ℝ and x∈x¯+ε0B with ∣fx¯−α∣<ε0, where ε0 is taken in 0,ε1 such that
(52)VJx¯,z<ε2,∀z∈x¯+ε0B.Fix now anyx′,α′∈epif with ∥x′−x¯∥<ε0 and ∣α′−fx¯∣<ε0. Clearly, x′∈NVx¯;ε1,ε2.Case 1.
λ>0. In this case, we necessarily have α=fx. Then, x∗/λ,−1∈Nπepif;x,fx, so x∗/λ∈∂Gπfx and ∥x∗/λ∥≤ε0/λ≤rt for every t≥maxT,1/rλ. Hence, by (3.18), we obtain
(53)λ−1x∗,x′−x≤fx′−fx+t2VJx,x′,which entails
(54)x∗,x′−x≤λfx′−fx+tλ2VJx,x′≤λfx′−fx+tλ2VJx,x′+tλ2α′−α2.Sincefx′≤α′, we have ∀t′≥maxT/2,1/2r, and we have 2λ−1t′≥1/rλ and 2λ−1t′≥T/λ≥T and hence (54) entails with t=2λ−1t′(55)x∗,x′−x≤λfx′−fx+2λ−1t′λ2VJx,x′+2λ−1t′λ2α′−α2≤λfx′−fx+t′VJx,x′+t′α′−α2≤t′∥x∥2+t′∥x′∥2−2t′Jx;x′−x+λα′−α+t′α′−α2.Hence,(56)x∗+2t′Jx,x′−x≤t′∥x′∥2−t′∥x∥2+λα′−α+t′α′−α2.So,(57)t′∥x∥2−2t′Jx+x∗2t′,x≤t′∥x′∥2−2t′Jx+x∗2t′,x′+λα′−α+t′α′−α2.Dividing byt′>0 yields
(58)VJx+x∗2t′;x≤VJx+x∗2t′;x′+λt′α′−α+α′−α2.On the other hand, we have(59)α′−α+12t′λ2=α′−α2+λt′α′−α+λ24t′2=1t′t′α′−α2+λα′−α+λ24t′.So,(60)t′α′−α+12t′λ2−λ24t′=t′α′−α2+λα′−α.Thus, (58) becomes
(61)VJx+x∗2t′;x≤VJx+x∗2t′;x′+1t′t′α′−α2+λα′−α≤VJx+x∗2t′;x′+α′−α+12t′λ2−λ24t′2and so
(62)VJx+x∗2t′;x+λ24t′2≤VJx+x∗2t′;x′+α′−α+12t′λ2.Observe that(63)VX×ℝJx+x∗2t′,α+12t′−λ;x,α=VJx+x∗2t′;x+λ24t′2,andVX×ℝJx+x∗2t′,α+12t′−λ;x′,α′=VJx+x∗2t′;x′+α′−α+12t′λ2.Therefore,(64)VX×ℝJx+x∗2t′,α+12t′−λ;x,α≤VX×ℝJx+x∗2t′,α+12t′−λ;x′,α′,for any x′,α′∈epif∩x¯,fx¯+ε0BX×ℝ, that is,
(65)x,α∈πepif∩x¯,fx¯+ε0BX×ℝJx+x∗2t′,α+12t′−λ.Case 2.
λ=0. In this case, we have x∗;0∈Nπepif;x,α and so by Lemma 3, we obtain x∗,0∈Nπepif;x,fx. Using the fact that Nπepif;x,fx⊂NFepif;x,fx, we get x∗,0∈NFepif;x,fx and so by Proposition 4, there exists a sequence un⟶x with fun⟶fx, λn⟶0+, and un∗∈λn∂Ffun (i.e., un∗/λn,−1∈NFepif;un,fun such that ∥un∗−x∗∥∗⟶0. Using now Proposition 5, we choose for each n≥1(66)xn,αn∈un,fun+λn2BX×ℝ,xn,αn∈epif,yn∗,−μn∈Nπepif;xn,αn,∥yn∗−un∗∥<λn2and∣λn−μn∣<λn2.This ensures thatμn>0 and so αn=fxn. Consequently, yn∗∈μn∂Gπfxn with
(67)∥xn−x∥≤∥xn−un∥+∥un−x∥≤λn2+∥un−x∥⟶0asn⟶∞,fxn−fx≤αn−fun+fun−fx≤λn2+fun−fx⟶0asn⟶∞.Also, we have(68)0<μn<3λn2⟶0asn⟶∞.Letxn∗≔μn−1yn∗. Then, ∥μnxn∗−x∗∥≤∥yn∗−un∗∥+∥un−x∗∥⟶0asn⟶∞, that is, μnxn∗⟶x∗. For n large enough, (i.e., ∀n≥N0), we have ∥xn−x¯∥<ε0 and μn<1. Assume for a moment that x∗≠0. Let tn≔maxT/2μn,∥xn∗∥/2r∥x∗∥, ∀n≥N0, we see that
(69)2tn≥maxT,∥xn∗∥r,since1μn>1and1∥x∗∥>1and hence by definition of V-p.l.n. functions with y≔x′ and t≔2tn, we obtain
(70)xn∗,x′−xn≤fx′−fxn+tnVJxn,x′.Multiplying this inequality byμn, we get
(71)μnxn∗,x′−xn≤μnfx′−fxn+μntnVJxn,x′.Letρn≔maxT/2,μn∥xn∗∥/2r∥x∗∥. Clearly, ρn⟶ρ≔maxT/2,1/2r and tnμn=ρn. So, for any n≥N0, we have
(72)yn∗,x′−xn≤μnfx′−fxn+ρnVJxn,x′.Now, taking the limit asn⟶∞ that yields by continuity of V and J(73)x∗,x′−x≤ρVJx,x′≤ρVJx,x′+ρα′−α2and so
(74)x∗,x′≤x∗,x+ρ∥x∥2+ρ∥x′∥2−2ρJx,x′+ρα′−α2.Thus,(75)ρ∥x∥2−x∗+2ρJx,x≤ρ∥x′∥2−x∗+2ρJx,x′+ρα′−α2.Dividing byρ>0 gives
(76)∥x∥2−2Jx+12ρx∗,x≤∥x′∥2−2Jx+12ρx∗,x′+α′−α2and so
(77)VJx+12ρx∗,x≤VJx+12ρx∗,x′+α′−α2.This ensures forλ=0(78)VX×ℝJx+x∗2ρ,α+12ρ−λ;x,α≤VX×ℝJx+x∗2ρ,α+12ρ−λ;x′,α′,for any x′,α′∈epif∩x¯,fx¯+ε0BX×ℝ, that is,
(79)x,α∈πepif∩x¯,fx¯+ε0BX×ℝJx+x∗2ρ,α+12ρ−λ.Finally, we obtained from Case1 and Case 2 two positive numbers ε0 and r0≔1/2ρ such that for any x,α∈epif and x∗,−λ∈Nπepif;x,α∩ε0BX×ℝ and any x∈x¯+ε0B and ∣fx¯−α∣<ε0, we have
(80)x,α∈πepif∩x¯,fx¯+ε0BX×ℝJx,α+r0x∗,−λ.This means by Definition2 that the epigraph epif is generalized V-prox-regular at x¯,fx¯.



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*Source: 9465492-2020-11-25.xml*