Preservation of prox-regularity Florent Nacry 1 Based on a joint - - PowerPoint PPT Presentation

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references

Preservation of prox-regularity

Florent Nacry 1 Based on a joint work with Samir Adly and Lionel Thibault, submitted in Journal de Mathématiques Pures et Appliquées (JMPA)

October 17-20 2017, University of Bordeaux GdR MOA & MIA

1INSA of Rennes (A.T.E.R.), florent.nacry@gmail.com Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references

Outline

1

Notation and preliminaries Aim and motivation Notation Prox-regular sets in Hilbert spaces

2

Preservation of prox-regularity: state of the art Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

3

Prox-regularity and generalized equations Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F−1(0)

4

Future works and references Perspectives Bibliography

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Outline

1

Notation and preliminaries Aim and motivation Notation Prox-regular sets in Hilbert spaces

2

Preservation of prox-regularity: state of the art Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

3

Prox-regularity and generalized equations Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F−1(0)

4

Future works and references Perspectives Bibliography

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Notation

• All vector spaces will be real vector spaces.
• Let X be a (real) normed space.

BX := {x ∈ X : x ≤ 1}, U := {x ∈ H : x < 1}

• For /

0 S ⊂ X, for all x ∈ X

dS(x) :=: d(x,S) := inf

s∈Sx − s

and

ProjS(x) := {y ∈ S : dS(x) = x − y}.

For each x ∈ X, when ProjS(x) contains one and only one vector y ∈ X, we set projS(x) := y.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Proximal normal cone

Definition Let S be a subset of H . One defines the proximal normal cone to S at x ∈ S as the set NP(S;x) := {v ∈ H : ∃ r > 0, x ∈ ProjS(x + rv)}.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Proximal normal cone

Definition Let S be a subset of H . One defines the proximal normal cone to S at x ∈ S as the set NP(S;x) := {v ∈ H : ∃ r > 0, x ∈ ProjS(x + rv)}.

Figure: NP is often reduced to 0 Figure: NP fails to be closed. Figure: NP(C;·) = N(C;·) for a convex set C.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Definition of uniform prox-regularity

For any S ⊂ H and any r ∈]0,+∞], one sets Ur (S) := {x ∈ H : dS(x) < r}.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Definition of uniform prox-regularity

For any S ⊂ H and any r ∈]0,+∞], one sets Ur (S) := {x ∈ H : dS(x) < r}.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Definition of uniform prox-regularity

For any S ⊂ H and any r ∈]0,+∞], one sets Ur (S) := {x ∈ H : dS(x) < r}. Definition Let S be a nonempty closed subset of H and r ∈]0,+∞] be an extended real. One says that S is r-prox-regular (or uniformly prox-regular with constant r) whenever the mapping

projS : Ur (S) → H is well-defined and norm-to-norm continuous.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Definition of uniform prox-regularity

For any S ⊂ H and any r ∈]0,+∞], one sets Ur (S) := {x ∈ H : dS(x) < r}. Definition Let S be a nonempty closed subset of H and r ∈]0,+∞] be an extended real. One says that S is r-prox-regular (or uniformly prox-regular with constant r) whenever the mapping

projS : Ur (S) → H is well-defined and norm-to-norm continuous.

• Notable contributors: H. Federer (1957); J.-P

. Vial (1983); A. Canino (1988); A. Shapiro (1994); F .H. Clarke, R.L. Stern, P .R. Wolenski (1995); R.A. Poliquin, R. T. Rockafellar, L. Thibault (2000).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Characterizations and properties of uniform prox-regular sets

Let r ∈]0,+∞]. Convention: 1

r = 0 whenever r = +∞.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Characterizations and properties of uniform prox-regular sets

Let r ∈]0,+∞]. Convention: 1

r = 0 whenever r = +∞.

Theorem

Let S be a nonempty closed subset of H , r ∈]0,+∞] be an extended real. Consider the following assertions. (a) S is r-prox-regular. (b) For all x1,x2 ∈ S, for all i ∈ {1,2}, for all vi ∈ NP(S;xi)∩BH , one has

v1 − v2,x1 − x2 ≥ −1

r x1 − x22 . (c) The function d2

S is C1,1 (resp., C1, resp., Fréchet differentiable) on Ur(S).

(d) ∂PdS(x) /

0 (resp., ∂FdS(x) / 0) for all x ∈ Ur(S).

(e) NP(S;x) = NF(S;x) = NL(S;x) = NC(S;x) for all x ∈ H . (f) the mapping projS : Ur(S) → H is well-defined. Then, one has (a) ⇔ (b) ⇔ (c) ⇔ (d) ⇒ (e). If in addition S is weakly closed, then (a) ⇔ (f).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Notation Prox-regular sets in Hilbert spaces

Prox-regular sets - examples and counter-examples

Nonempty closed convex

⇔ ∞-prox-regular

Lack of prox-regularity ("angle")

H \ B(0,r) is r-prox-regular

Lack of prox-regularity ("crushing")

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Outline

1

Notation and preliminaries Aim and motivation Notation Prox-regular sets in Hilbert spaces

2

Preservation of prox-regularity: state of the art Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

3

Prox-regularity and generalized equations Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F−1(0)

4

Future works and references Perspectives Bibliography

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Prox-regularity and preservation: counter-examples

The intersection of prox-regular sets fails to be prox-regular Non prox-regular union of two convex sets The projection along a vector space

• f a prox-regular set fails to be prox-regular

Non prox-regular (sub)-level set

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art I

• J.P

. Vial (1983): Study of the "weak convexity" of {f ≤ 0} and {f = 0} (Dim < ∞).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art I

• J.P

. Vial (1983): Study of the "weak convexity" of {f ≤ 0} and {f = 0} (Dim < ∞).

• J. Venel (2009): Study of prox-regularity of

m

• i=1

{fi ≤ 0} (fi C2, Dim < ∞).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art I

• J.P

. Vial (1983): Study of the "weak convexity" of {f ≤ 0} and {f = 0} (Dim < ∞).

• J. Venel (2009): Study of prox-regularity of

m

• i=1

{fi ≤ 0} (fi C2, Dim < ∞).

• F

. Bernard, L. Thibault and N. Zlateva (2010) :

◮ Study of inverse image under the condition

d(x,F−1(D)) ≤ γd(F(x),D).

◮ Counter-example/study of the intersection under the condition

d(x,

m

• k=1

Sk) ≤ γ

m

∑

k=1

d(x,Sk).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art I

• J.P

. Vial (1983): Study of the "weak convexity" of {f ≤ 0} and {f = 0} (Dim < ∞).

• J. Venel (2009): Study of prox-regularity of

m

• i=1

{fi ≤ 0} (fi C2, Dim < ∞).

• F

. Bernard, L. Thibault and N. Zlateva (2010) :

◮ Study of inverse image under the condition

d(x,F−1(D)) ≤ γd(F(x),D).

◮ Counter-example/study of the intersection under the condition

d(x,

m

• k=1

Sk) ≤ γ

m

∑

k=1

d(x,Sk).

• G. Colombo and L. Thibault (2010):

◮ Study of the inverse image under the condition

N(F−1(D);x)∩B ⊂ DF(x)⋆(N(D;F(x))∩γB).

◮ Study of the intersection under the condition

N(

m

• k=1

Sk;x)∩B ⊂ N(S1;x)∩γB +... + N(Sm;x)∩γB.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art II

• S. Adly, N., L. Thibault (2016)

Sufficient conditions guaranteeing the prox-regularity for:

◮ A set defined by equality constraints

C = {x ∈ H : G(x) = 0}, with G : H → Y under an openness condition sB ⊂ DG(x)(B).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art II

• S. Adly, N., L. Thibault (2016)

Sufficient conditions guaranteeing the prox-regularity for:

◮ A set defined by equality constraints

C = {x ∈ H : G(x) = 0}, with G : H → Y under an openness condition sB ⊂ DG(x)(B).

◮ A set defined by inequality/equality constraints gi : H → R {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0,gm+1(x) = 0,...,gm+n(x) = 0}

under an openness condition on the derivatives Dgi.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art II

• S. Adly, N., L. Thibault (2016)

Sufficient conditions guaranteeing the prox-regularity for:

◮ A set defined by equality constraints

C = {x ∈ H : G(x) = 0}, with G : H → Y under an openness condition sB ⊂ DG(x)(B).

◮ A set defined by inequality/equality constraints gi : H → R {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0,gm+1(x) = 0,...,gm+n(x) = 0}

under an openness condition on the derivatives Dgi.

◮ The intersection of two prox-regular sets S1,S2 under an openness condition

sBH ⊂ T(S1;x1)∩BH − T(S2;x2)∩BH ,

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Preservation: state of the art II

• S. Adly, N., L. Thibault (2016)

Sufficient conditions guaranteeing the prox-regularity for:

◮ A set defined by equality constraints

C = {x ∈ H : G(x) = 0}, with G : H → Y under an openness condition sB ⊂ DG(x)(B).

◮ A set defined by inequality/equality constraints gi : H → R {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0,gm+1(x) = 0,...,gm+n(x) = 0}

under an openness condition on the derivatives Dgi.

◮ The intersection of two prox-regular sets S1,S2 under an openness condition

sBH ⊂ T(S1;x1)∩BH − T(S2;x2)∩BH ,

◮ Inverse image of a prox-regular set G−1(D) with G : H → H ′ under the openness

condition sBH ′ ⊂ DG(x)(BH )− T(D;G(x)− y).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Vers une forme plus générale ?

• Questions. Peut-on unifier les problèmes de préservation ? Peut-on avoir des formes

d’ensembles plus générales que des ensembles de contraintes ?

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Vers une forme plus générale ?

• Questions. Peut-on unifier les problèmes de préservation ? Peut-on avoir des formes

d’ensembles plus générales que des ensembles de contraintes ?

• Une voie possible. Le concept d’équations généralisées (S.M. Robinson - 1979)

0 ∈ f(x) + F(x) x ∈ H

• ù f : H → H ′ est une application et F : H ⇒ H ′ une multi-application.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Vers une forme plus générale ?

• Questions. Peut-on unifier les problèmes de préservation ? Peut-on avoir des formes

d’ensembles plus générales que des ensembles de contraintes ?

• Une voie possible. Le concept d’équations généralisées (S.M. Robinson - 1979)

0 ∈ f(x) + F(x) x ∈ H

• ù f : H → H ′ est une application et F : H ⇒ H ′ une multi-application.

◮ Exemple 1. Ecriture d’un ensemble de contraintes sous forme d’une équation

généralisée :

{x ∈ H : f1(x) ≤ 0,...,fm(x) ≤ 0,fm+1(x) = 0,...,fm+n(x) = 0} = {x ∈ H : 0 ∈ f(x) + F(x)}

• ù f := (f1,...,fm+n) et F :≡ Rm

+ ×{0Rn}.

◮ Exemple 2. Ecriture d’une intersection d’ensembles sous forme d’une équation

généralisée :

m

• i=1

Si =

• x ∈ H : 0 ∈ (−x,−x) +

m

∏

i=1

Si

• .

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Un résultat ou deux avec cond theoriques *****

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Nonsmooth inequality constraints

Theorem (Adly, N., Thibault (2016)) Let H be a Hilbert space, m ∈ N, g1,...,gm : H → R such that C = {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0} /

0.

Assume that there exists ρ ∈]0,+∞] such that: (i) for each k ∈ {1,...,m}, gk is continuous on Uρ(C); (ii) there exists γ ≥ 0 such that for all k ∈ {1,...,m}, for all x1,x2 ∈ Uρ(C), for all v1 ∈ ∂Cgk(x1) and for all v2 ∈ ∂Cgk(x2)

v1 − v2,x1 − x2 ≥ −γ x1 − x22 .

Assume also that there exists δ > 0 such that for all x ∈ bdC, there exists v ∈ BH satisfying for all k ∈ {1,...,m} and for all ξ ∈ ∂Cgk(x),

ξ,v ≤ −δ.

Then, C is r-prox-regular with r = min

• ρ, δ

γ

• .

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Prox-régularité d’ensembles de contraintes - exemple

Soient f,g,h : R2 → R les fonctions définies par f(x,y) = y − ex + 1 2, g(x,y) = y − e−x + 1 2 et h(x,y) = −y pour tout (x,y) ∈ R2. Soit C l’ensemble C := {(x,y) ∈ R2 : f(x,y) ≤ 0,g(x,y) ≤ 0,h(x,y) ≤ 0}.

Alors, pour chaque réel ρ > 0, C est min

• ρ,

1

√

4e4+1e2(ln(2)+ρ)

• prox-régulier.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Interpretation of uniform Slater’s condition

C := {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0}. (S.U.)

∃δ > 0,∀x ∈ bdC,∃vx ∈ B,∀k ∈ {1,...,m},∀ξ ∈ ∂Cgk(x),ξ,vx ≤ −δ

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Interpretation of uniform Slater’s condition

C := {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0}. (S.U.)

∃δ > 0,∀x ∈ bdC,∃vx ∈ B,∀k ∈ {1,...,m},∀ξ ∈ ∂Cgk(x),ξ,vx ≤ −δ

Remarques.

• 1. 0 ∈ ∂Cgk(x) pour un x ∈ bdC ⇒ La condition (S.U.) n’est pas satisfaite.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

Interpretation of uniform Slater’s condition

C := {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0}. (S.U.)

∃δ > 0,∀x ∈ bdC,∃vx ∈ B,∀k ∈ {1,...,m},∀ξ ∈ ∂Cgk(x),ξ,vx ≤ −δ

Remarques.

• 1. 0 ∈ ∂Cgk(x) pour un x ∈ bdC ⇒ La condition (S.U.) n’est pas satisfaite.
• 2. Cas d’une contrainte lisse g ⇒ C = {g ≤ 0} et ∂Cg(x) = {∇g(x)}.

֒→ ∃δ > 0,∀x ∈ C avec g(x) = 0,∃vx ∈ B,∇g(x),vx ≤ −δ.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Outline

1

Notation and preliminaries Aim and motivation Notation Prox-regular sets in Hilbert spaces

2

Preservation of prox-regularity: state of the art Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

3

Prox-regularity and generalized equations Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F−1(0)

4

Future works and references Perspectives Bibliography

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Metric regularity (definition)

Definition Soient X,Y deux espaces vectoriels normés, M : X ⇒ Y une multi-application, (x,y) ∈ gph M := {(x,y) ∈ X × Y : y ∈ M(x)}. On dit que M est métriquement régulière en x pour y lorsqu’il existe un réel γ ≥ 0 et des voisinages U et V de x et y respectivement tels que d(x,M−1(y)) ≤ γd(y,M(x)) pour tout (x,y) ∈ U × V.

• Origine du concept → théorème de l’application ouverte ∼ 1930 (terme "régularité métrique"

dû à J.M. Borwein (1986)).

• Applications : calcul sous-différentiel, conditions nécessaires d’optimalité, bornes d’erreurs,

inclusions différentielles,...

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Théorème de Robinson-Ursescu (1975-1976)

Théorème (Robinson-Ursescu) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe convexe fermé, (x,y) ∈ gph M. On suppose qu’il existe c > 0 tel que y + cUY ⊂ M(x +BX ). Alors, pour tout x ∈ X, pour tout y ∈ y + cUY , on a d(x,M−1(y)) ≤ (c −y − y)−1(1 +x − x)d(y,M(x)).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Théorème de Robinson-Ursescu (1975-1976)

Théorème (Robinson-Ursescu) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe convexe fermé, (x,y) ∈ gph M. On suppose qu’il existe c > 0 tel que y + cUY ⊂ M(x +BX ). Alors, pour tout x ∈ X, pour tout y ∈ y + cUY , on a d(x,M−1(y)) ≤ (c −y − y)−1(1 +x − x)d(y,M(x)).

֒→ Affaiblir l’hypothèse de convexité sur le graphe de M :

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Théorème de Robinson-Ursescu (1975-1976)

Théorème (Robinson-Ursescu) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe convexe fermé, (x,y) ∈ gph M. On suppose qu’il existe c > 0 tel que y + cUY ⊂ M(x +BX ). Alors, pour tout x ∈ X, pour tout y ∈ y + cUY , on a d(x,M−1(y)) ≤ (c −y − y)−1(1 +x − x)d(y,M(x)).

֒→ Affaiblir l’hypothèse de convexité sur le graphe de M :

◮ A. Jourani (1996); H. Huang et R.X. Li (2011) : cas d’une multi-application M

paraconvexe.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Théorème de Robinson-Ursescu (1975-1976)

Théorème (Robinson-Ursescu) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe convexe fermé, (x,y) ∈ gph M. On suppose qu’il existe c > 0 tel que y + cUY ⊂ M(x +BX ). Alors, pour tout x ∈ X, pour tout y ∈ y + cUY , on a d(x,M−1(y)) ≤ (c −y − y)−1(1 +x − x)d(y,M(x)).

֒→ Affaiblir l’hypothèse de convexité sur le graphe de M :

◮ A. Jourani (1996); H. Huang et R.X. Li (2011) : cas d’une multi-application M

paraconvexe.

◮ X.Y. Zheng et K.F. Ng (2012) : graphe localement prox-régulier.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Théorème de Robinson-Ursescu (1975-1976)

Théorème (Robinson-Ursescu) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe convexe fermé, (x,y) ∈ gph M. On suppose qu’il existe c > 0 tel que y + cUY ⊂ M(x +BX ). Alors, pour tout x ∈ X, pour tout y ∈ y + cUY , on a d(x,M−1(y)) ≤ (c −y − y)−1(1 +x − x)d(y,M(x)).

֒→ Affaiblir l’hypothèse de convexité sur le graphe de M :

◮ A. Jourani (1996); H. Huang et R.X. Li (2011) : cas d’une multi-application M

paraconvexe.

◮ X.Y. Zheng et K.F. Ng (2012) : graphe localement prox-régulier. ◮ X.Y. Zheng, Q.H. He (2014) : graphe localement sous-lisse.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Robinson-Ursescu et prox-régularité

Théorème (Adly, N., Thibault (2017)) Soient X,Y des espaces de Banach, M : X ⇒ Y une multi-application de graphe fermé, Q une partie non vide de gph M. On suppose : (i) l’ensemble gph M est r-prox-régulier pour r ∈]0,+∞]. (ii) il existe α,β,ρ ∈]0,+∞[ avec

β > 3α ρ + 1

2r (1 + 1

ρ )(4α2 + (β − α ρ )2)

tels que pour tout (x,y) ∈ Q, y +βUY ⊂ M(x +αBX ); Alors, il existe un réel γ ∈

• 0,ρ[ tel que pour tout (x,y) ∈ Q, il existe un réel δ > 0 satisfaisant

pour tout x ∈ B(x,δ), pour tout y ∈ B(y,δ), d(x,M−1(y)) ≤ γd(y,M(x)).

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Metric regularity Prox-regularity of solution set of generalized equations

Prox-régularité d’ensembles de solutions d’équations généralisées

Théorème (Adly, N., Thibault (2017)) Soient H ,H ′ deux espaces de Hilbert, f : H → H ′ une application et F : H ⇒ H ′ une multi-application telles que S = {x ∈ H : 0 ∈ f(x) + F(x)} /

• 0. On suppose :

(i) gph F est r-prox-régulier avec r ∈]0,+∞]; (ii) f est différentiable sur H avec Df : H → H ′ γ-Lipschitz sur H avec γ ≥ 0 et il existe

ρ ∈]0,+∞], L ≥ 0 tels que pour tout x,y ∈ S avec x − y < 2ρ, f(x)− f(y) ≤ Lx − y;

(iii) il existe α,β,ρ > 0 satisfaisant β > 3α

ρ + 1

2r (1 + 1

ρ )(4α2 + (β − α ρ )2) tels que pour tout x ∈ S,

βU(H ×H ′)2 ⊂ −{

• (x,y),(x,y)
• : (x,y) ∈ (x,−f(x)) +αBH ×H ′} +gph F ×gph(−f).

Alors, l’ensemble S est r′-prox-régulier avec r′ = min

• ρ,

min{r, 1 γ } 4ρ(L2+1)

• .

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Perspectives Bibliography

Outline

1

Notation and preliminaries Aim and motivation Notation Prox-regular sets in Hilbert spaces

2

Preservation of prox-regularity: state of the art Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions

3

Prox-regularity and generalized equations Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F−1(0)

4

Future works and references Perspectives Bibliography

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Perspectives Bibliography

Perspectives

◮ Extend some results to some other class of sets (subsmooth, α-far, etc.). ◮ Coming back to the problem of intersection of prox-regular sets (number infinite of sets,

verifiable conditions, etc.).

◮ Weak the assumption of the graph prox-regularity in the study of prox-regularity of {x ∈ H : 0 ∈ f(x) + F(x)}. ◮ Prox-regularity of solution sets of variational inequality. ◮ Develop results in the framework of uniformly convex Banach spaces.

Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Prox-regularity and generalized equations Future works and references Perspectives Bibliography

• S. Adly, F

. Nacry, L. Thibault, Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization, SIAM J. Optim. 26 (2016), 448-473.

• S. Adly, F

. Nacry, L. Thibault, Prox-regularity approach to generalized equations and image projection, under revision in Journal de Mathématiques Pures et Appliquées. F . Bernard, L. Thibault, N. Zlateva, Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties, Trans. Amer. Math. Soc. 363 (2011), 2211-2247.

• G. Colombo, L. Thibault, Prox-regular sets and applications, Handbook of nonconvex

analysis and applications 99-182 (2010), Int. Press, Somerville, MA. 231-259.

• J. Venel, A numerical scheme for a class of sweeping processes, Numer. Math. 118 (2011),

367-400. J.-P . Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983), 231-259.

Thank you for your attention

Florent Nacry Preservation of prox-regularity