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 1 INSA 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 Notation and preliminaries 1 Aim and motivation Notation Prox-regular sets in Hilbert spaces Preservation of prox-regularity: state of the art 2 Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions Prox-regularity and generalized equations 3 Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F − 1 (0) Future works and references 4 Perspectives Bibliography Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Outline Notation and preliminaries 1 Aim and motivation Notation Prox-regular sets in Hilbert spaces Preservation of prox-regularity: state of the art 2 Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions Prox-regularity and generalized equations 3 Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F − 1 (0) Future works and references 4 Perspectives Bibliography Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Notation • All vector spaces will be real vector spaces. • Let X be a (real) normed space. B X := { x ∈ X : � x � ≤ 1 } , U := { x ∈ H : � x � < 1 } • For / 0 � S ⊂ X , for all x ∈ X d S ( x ) :=: d ( x , S ) := inf s ∈ S � x − s � and Proj S ( x ) := { y ∈ S : d S ( x ) = � x − y �} . For each x ∈ X , when Proj S ( x ) contains one and only one vector y ∈ X , we set proj S ( x ) := y . Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references 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 N P ( S ; x ) := { v ∈ H : ∃ r > 0 , x ∈ Proj S ( x + rv ) } . Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references 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 N P ( S ; x ) := { v ∈ H : ∃ r > 0 , x ∈ Proj S ( x + rv ) } . Figure: N P is often reduced to 0 Figure: N P ( C ; · ) = N ( C ; · ) for a Figure: N P fails to be closed. convex set C . Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Definition of uniform prox-regularity For any S ⊂ H and any r ∈ ]0 , + ∞ ], one sets U r ( S ) := { x ∈ H : d S ( x ) < r } . Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Definition of uniform prox-regularity For any S ⊂ H and any r ∈ ]0 , + ∞ ], one sets U r ( S ) := { x ∈ H : d S ( x ) < r } . Florent Nacry Preservation of prox-regularity

Notation and preliminaries Preservation of prox-regularity: state of the art Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Definition of uniform prox-regularity For any S ⊂ H and any r ∈ ]0 , + ∞ ], one sets U r ( S ) := { x ∈ H : d S ( 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 proj S : U r ( 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 Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references Definition of uniform prox-regularity For any S ⊂ H and any r ∈ ]0 , + ∞ ], one sets U r ( S ) := { x ∈ H : d S ( 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 proj S : U r ( 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 Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references 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 Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references 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 x 1 , x 2 ∈ S , for all i ∈ { 1 , 2 } , for all v i ∈ N P ( S ; x i ) ∩ B H , one has � v 1 − v 2 , x 1 − x 2 � ≥ − 1 r � x 1 − x 2 � 2 . S is C 1 , 1 (resp., C 1 , resp., Fréchet differentiable) on U r ( S ). ( c ) The function d 2 ( d ) ∂ P d S ( x ) � / 0 (resp., ∂ F d S ( x ) � / 0 ) for all x ∈ U r ( S ). ( e ) N P ( S ; x ) = N F ( S ; x ) = N L ( S ; x ) = N C ( S ; x ) for all x ∈ H . ( f ) the mapping proj S : U r ( 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 Notation Prox-regularity and generalized equations Prox-regular sets in Hilbert spaces Future works and references 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 Some natural questions on prox-regularity Preservation of prox-regularity: state of the art State of the art Prox-regularity and generalized equations Theoretical v.s. verifiable conditions Future works and references Outline Notation and preliminaries 1 Aim and motivation Notation Prox-regular sets in Hilbert spaces Preservation of prox-regularity: state of the art 2 Some natural questions on prox-regularity State of the art Theoretical v.s. verifiable conditions Prox-regularity and generalized equations 3 Metric regularity Prox-regularity of solution set of generalized equations An application of the prox-regularity of F − 1 (0) Future works and references 4 Perspectives Bibliography Florent Nacry Preservation of prox-regularity

Notation and preliminaries Some natural questions on prox-regularity Preservation of prox-regularity: state of the art State of the art Prox-regularity and generalized equations Theoretical v.s. verifiable conditions Future works and references 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 of a prox-regular set fails to be prox-regular Non prox-regular (sub)-level set Florent Nacry Preservation of prox-regularity

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