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Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection

Preservation of uniform prox-regularity of sets and application to constrained optimization

Florent Nacry Joint work with Samir Adly and Lionel Thibault, accepted for publication in SIOPT.

GDR MOA Université de Bourgogne

2015 December 2

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Plan

1

Uniform prox-regular sets in Hilbert setting Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

2

Uniform prox-regularity of constraint sets An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

3

Application to constrained optimization

4

Inverse image, intersection Intersection Inverse image

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Notations, definitions

Let H be a real Hilbert space, ·,· be the inner product and · =

• ·,· be the associated

norm. .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Notations, definitions

Let H be a real Hilbert space, ·,· be the inner product and · =

• ·,· be the associated

norm.

• BH = {x ∈ H : x ≤ 1}.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Notations, definitions

Let H be a real Hilbert space, ·,· be the inner product and · =

• ·,· be the associated

norm.

• BH = {x ∈ H : x ≤ 1}.
• For S ⊂ H , co S is the convex hull of S, bdry S is the boundary of S.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Notations, definitions

Let H be a real Hilbert space, ·,· be the inner product and · =

• ·,· be the associated

norm.

• BH = {x ∈ H : x ≤ 1}.
• For S ⊂ H , co S is the convex hull of S, bdry S is the boundary of S.
• For /

0 S ⊂ H , for all x ∈ H

dS(x) = inf

s∈Sx − s

and

ProjS(x) = {y ∈ S : dS(x) = x − y}.

For each x ∈ H , when ProjS(x) contains one and only one vector y ∈ H , we set PS(x) = y.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Chebyshev set

Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Chebyshev set

Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem:

• Assume that for all x ∈ H , ProjS(x) is a singleton (i.e., S is a Chebyshev set). Is the set S

convex ?

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Chebyshev set

Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem:

• Assume that for all x ∈ H , ProjS(x) is a singleton (i.e., S is a Chebyshev set). Is the set S

convex ?

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Chebyshev set

Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem:

• Assume that for all x ∈ H , ProjS(x) is a singleton (i.e., S is a Chebyshev set). Is the set S

convex ?

• T. Motzkin, L.N.H. Bunt, M. Kritikos (1930-1940): DimH < +∞ ⇒ S is convex.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Chebyshev set

Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem:

• Assume that for all x ∈ H , ProjS(x) is a singleton (i.e., S is a Chebyshev set). Is the set S

convex ?

• T. Motzkin, L.N.H. Bunt, M. Kritikos (1930-1940): DimH < +∞ ⇒ S is convex.
• DimH = +∞ ⇒ Still open. With additional assumptions (L.P

. Vlasov, E. Asplund, V. Klee, ...)

⇒ True

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Open enlargement

Let H be a real Hilbert space and r ∈]0,+∞] an extended real. For S a nonempty closed subset of H , one defines Ur (S) = {x ∈ H : dS(x) < r}, the r-open enlargement of S. Problem:

• What is the class of nonempty closed subsets S of H such that ProjS(x) is a singleton for all

x ∈ Ur (S) and the well-defined mapping PS : Ur (S) −→ H is continuous ?

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Clarke tangent cone, normal cone and subdifferential

Definition Let H be a real Hilbert space. The Clarke tangent cone (resp. Clarke normal cone) to a subset S ⊂ H at x ∈ S is the set T C(S;x) = {h ∈ X : ∀S ∋ xn → x,∀tn ↓ 0,∃X ∋ hn → h,xn + tnhn ∈ S,∀n ∈ N} (resp. NC(S;x) =

• v ∈ H ,v,h ≤ 0,∀h ∈ T C(S;x)
• ).

For x ∈ H , U an open neighborhood of x and f : U −→ R an extended real-valued function finite at x, the Clarke subdifferential of f at x is defined as the set

∂Cf(x) =

• v ∈ H : (v,−1) ∈ NC(epi f;(x,f(x)))
• .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Clarke subdifferential in finite dimension

Theorem Let m ∈ N, f : Rm −→ R be Lipschitz continuous near x ∈ Rm, △f the subset of Rm where ∇f exists, Df any subset of △f such that △f \ Df is Lebesgue negligible. Then, one has

∂Cf(x) = co

• lim

n→+∞∇f(xn) : Df ∋ xn → x

• .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Clarke subdifferential in finite dimension

Theorem Let m ∈ N, f : Rm −→ R be Lipschitz continuous near x ∈ Rm, △f the subset of Rm where ∇f exists, Df any subset of △f such that △f \ Df is Lebesgue negligible. Then, one has

∂Cf(x) = co

• lim

n→+∞∇f(xn) : Df ∋ xn → x

• .
• m = 1, f = −|·|, x = 0, △f = Df = R\{0}, ∂Cf(0) = [−1,1].

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Definition of uniform prox-regularity

Definition Let H be a real Hilbert space, S be a nonempty closed set 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 PS : Ur (S) −→ H is well-defined and norm-to-norm continuous.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Definition of uniform prox-regularity

Definition Let H be a real Hilbert space, S be a nonempty closed set 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 PS : Ur (S) −→ H is well-defined and norm-to-norm continuous.

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

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

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Characterizations of uniform prox-regular sets

Let r ∈]0,+∞]. Convention: 1

r = 0 whenever r = +∞.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Characterizations of uniform prox-regular sets

Let r ∈]0,+∞]. Convention: 1

r = 0 whenever r = +∞.

Theorem (R.A. Poliquin, R.T. Rockafellar, L. Thibault) Let H be a real Hilbert space, S be a nonempty closed subset of H , r ∈]0,+∞] be an extended

• real. The following assertions are equivalent.

(a) S is r-prox-regular. (b) For all x,x′ ∈ S, for all v ∈ NC(S;x), one has v,x′ − x ≤ 1

2r vx′ − x2.

(c) For all x ∈ S, for all ζ ∈ NC(S;x)∩BH and for all real t ∈]0,r], one has x ∈ ProjS(x + tζ). (d) For all x1,x2 ∈ S, for all i ∈ {1,2}, for all vi ∈ NC(S;xi)∩BH , one has

v1 − v2,x1 − x2 ≥ − 1

r x1 − x22 . (e) For all x,y ∈ S, for all t ∈ [0,1] such that (1− t)x + ty ∈ Ur (S), one has dS((1− t)x + ty) ≤ 1 2r t(1− t)y − x2 . (f) PS is well-defined on Ur (S) and locally Lipschitz continuous there. (g) The function d2

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

If in addition S is weakly closed in H , then one can add the following: (h) PS is well-defined on Ur (S).

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Examples

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Convex sets A nonempty closed set of a real Hilbert space is ∞-prox-regular if and only if it is convex.

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Examples

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Convex sets A nonempty closed set of a real Hilbert space is ∞-prox-regular if and only if it is convex. Complement of an open ball Let H be a real Hilbert space with H {0}. Then, the set S = H \ B(0,r) is r-prox-regular.

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

Examples

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Convex sets A nonempty closed set of a real Hilbert space is ∞-prox-regular if and only if it is convex. Complement of an open ball Let H be a real Hilbert space with H {0}. Then, the set S = H \ B(0,r) is r-prox-regular. Two sets which fail to be uniformly prox-regular

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Plan

1

Uniform prox-regular sets in Hilbert setting Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

2

Uniform prox-regularity of constraint sets An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

3

Application to constrained optimization

4

Inverse image, intersection Intersection Inverse image

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Constrained optimization

Let H be a real Hilbert space, C be a nonempty subset of H , I, J be finite sets, f,gi,hj : H −→ R (i ∈ I, j ∈ J) be functions. (P)

• Minimize f(x)

s.c. gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Constrained optimization

Let H be a real Hilbert space, C be a nonempty subset of H , I, J be finite sets, f,gi,hj : H −→ R (i ∈ I, j ∈ J) be functions. (P)

• Minimize f(x)

s.c. gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C S = {x ∈ H : gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C}

Argmin

S

f =

• x ∈ S : f(x) = inf

S f

• Florent Nacry, joint work with Samir Adly and Lionel Thibault

Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Constrained optimization

Let H be a real Hilbert space, C be a nonempty subset of H , I, J be finite sets, f,gi,hj : H −→ R (i ∈ I, j ∈ J) be functions. (P)

• Minimize f(x)

s.c. gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C S = {x ∈ H : gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C}

Argmin

S

f =

• x ∈ S : f(x) = inf

S f

• (P) is a convex minimization problem ⇒ S and Argmin

S

f are convex sets. Indeed, in such case S is the intersection of convex sets as well as

Argmin

S

f = S ∩

• x ∈ H : f(x) ≤ inf

S f

• Florent Nacry, joint work with Samir Adly and Lionel Thibault

Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Constrained optimization

Let H be a real Hilbert space, C be a nonempty subset of H , I, J be finite sets, f,gi,hj : H −→ R (i ∈ I, j ∈ J) be functions. (P)

• Minimize f(x)

s.c. gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C S = {x ∈ H : gi(x) ≤ 0, ∀i ∈ I, hj(x) = 0, ∀j ∈ J, x ∈ C}

Argmin

S

f =

• x ∈ S : f(x) = inf

S f

• (P) is a convex minimization problem ⇒ S and Argmin

S

f are convex sets. Indeed, in such case S is the intersection of convex sets as well as

Argmin

S

f = S ∩

• x ∈ H : f(x) ≤ inf

S f

• Problems: • Find sufficient conditions to ensure the uniform prox-regularity of a constraint set

(resp. the set of global solutions).

• Study the preservation of prox-regularity by intersection and inverse image.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

A little bit of literature

• Prox-regularity of constraint sets: (J.-P

. Vial (1983), J. Venel (2010)).

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

A little bit of literature

• Prox-regularity of constraint sets: (J.-P

. Vial (1983), J. Venel (2010)).

• Intersection of a subspace of R2 with a uniformly prox-regular set of R2 may fails to be

uniformly prox-regular (F . Bernard, L. Thibault and N. Zlateva (2005)).

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

A little bit of literature

• Prox-regularity of constraint sets: (J.-P

. Vial (1983), J. Venel (2010)).

• Intersection of a subspace of R2 with a uniformly prox-regular set of R2 may fails to be

uniformly prox-regular (F . Bernard, L. Thibault and N. Zlateva (2005)).

• Preservation of intersection and inverse image only under theoretical assumptions on normal

cones involved (G. Colombo and L. Thibault (2010)).

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

A little bit of literature

• Prox-regularity of constraint sets: (J.-P

. Vial (1983), J. Venel (2010)).

• Intersection of a subspace of R2 with a uniformly prox-regular set of R2 may fails to be

uniformly prox-regular (F . Bernard, L. Thibault and N. Zlateva (2005)).

• Preservation of intersection and inverse image only under theoretical assumptions on normal

cones involved (G. Colombo and L. Thibault (2010)).

• Preservation of inverse image in the framework of finite dimension and for a linear mapping (B.

Brogliato, C. Prieur and A. Tanwani (2014)).

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Sublevels of C ∞ functions

Consider f,g : R2 −→ R the functions defined by f(x,y) = xy and g(x,y) = ((x − 1)2 + y2 − 1)((x + 1)2 + y2 − 1) for all (x,y) ∈ R2 and set S = {(x,y) ∈ R2 : f(x,y) ≤ 0} and S′ = {(x,y) ∈ R2 : g(x,y) ≤ 0}.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Sublevels of C ∞ functions

Consider f,g : R2 −→ R the functions defined by f(x,y) = xy and g(x,y) = ((x − 1)2 + y2 − 1)((x + 1)2 + y2 − 1) for all (x,y) ∈ R2 and set S = {(x,y) ∈ R2 : f(x,y) ≤ 0} and S′ = {(x,y) ∈ R2 : g(x,y) ≤ 0}.

Figure: 0-sublevel of f Figure: 0-sublevel of g

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Sublevels of C ∞ functions

Consider f,g : R2 −→ R the functions defined by f(x,y) = xy and g(x,y) = ((x − 1)2 + y2 − 1)((x + 1)2 + y2 − 1) for all (x,y) ∈ R2 and set S = {(x,y) ∈ R2 : f(x,y) ≤ 0} and S′ = {(x,y) ∈ R2 : g(x,y) ≤ 0}. The sets S and S′ are not uniformly prox-regular.

Figure: 0-sublevel of f Figure: 0-sublevel of g

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Definition of semiconvexity

Definition Let H be a real Hilbert space, U a nonempty convex open subset of X, γ ∈ [0,+∞[ a real, f : U −→ R∪{+∞} an extended real-valued function. One says that f is γ-semiconvex on U whenever, for all x,y ∈ U, for all real t ∈]0,1[ f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y) + 1 2 γt(1− t)x − y2 .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Definition of semiconvexity

Definition Let H be a real Hilbert space, U a nonempty convex open subset of X, γ ∈ [0,+∞[ a real, f : U −→ R∪{+∞} an extended real-valued function. One says that f is γ-semiconvex on U whenever, for all x,y ∈ U, for all real t ∈]0,1[ f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y) + 1 2 γt(1− t)x − y2 .

• f is γ-semiconvex on U ⇔ f + γ

2 ·2 is convex on U.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Definition of semiconvexity

Definition Let H be a real Hilbert space, U a nonempty convex open subset of X, γ ∈ [0,+∞[ a real, f : U −→ R∪{+∞} an extended real-valued function. One says that f is γ-semiconvex on U whenever, for all x,y ∈ U, for all real t ∈]0,1[ f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y) + 1 2 γt(1− t)x − y2 .

• f is γ-semiconvex on U ⇔ f + γ

2 ·2 is convex on U.

• For any real γ ≥ 0, the function ln :]0,+∞[−→ R is not γ-semiconvex on ]0,+∞[.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Hypomonotonicity of Clarke subdifferential

Theorem Let H be a real Hilbert space, U a nonempty convex open subset of X, γ ∈ [0,+∞[ a real, f : U −→ R∪{+∞} an extended real-valued function locally Lipschitz on U. Then, f is

γ-semiconvex on U if and only if for all x,y ∈ U, for all x⋆ ∈ ∂Cf(x), for all y⋆ ∈ ∂Cf(y), x⋆ − y⋆,x − y ≥ −γ x − y2 .

i.e., ∂Cf : U ⇒ H is γ-hypomonotone.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Nonsmooth inequalities

Theorem Let H a Hilbert space, m ∈ N, g1,...,gm : H −→ R such that, the set C = {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0} is nonempty. Assume that there is an extended real ρ ∈]0,+∞] such that: (i) for all k ∈ {1,...,m}, gk is locally Lipschitz continuous on Uρ(C); (ii) there is a real γ ≥ 0 such that, for all k ∈ {1,...,m}, for all x1,x2 ∈ Uρ(C), and for all v1 ∈ ∂Cgk(x1) and all v2 ∈ ∂Cgk(x2)

v1 − v2,x1 − x2 ≥ −γ x1 − x22 .

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

ξ,v ≤ −δ.

(1) Then, C is r-prox-regular with r = min

• ρ, δ

γ

• .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Non prox-regularity by theorem of nonsmooth inequalities

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Smooth equalities

Theorem Let H be a real Hilbert space, and G : H → Y be a mapping from H into a real Banach space Y such that, the set C = {x ∈ H : G(x) = 0} is nonempty. Assume that, there exists an extended real ρ ∈]0,+∞] such that: (i) the mapping G is differentiable on Uρ(C); (ii) there is a real γ ≥ 0 such that, for all x1,x2 ∈ Uρ(C),

DG(x1)− DG(x2) ≤ γx1 − x2.

Assume also that there is some real δ > 0 such that

δBY ⊂ DG(x)

• BH
• for all x ∈ bdry C.

(2) Then, the set C is r-prox-regular with r = min

• ρ, δ

γ

• .

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

Mixed constraint set

Theorem Let H be a real Hilbert space, m,n ∈ N and gk : H → R with k ∈ {1,...,m + n} be functions such that, C := {x ∈ H : g1(x) ≤ 0,...,gm(x) ≤ 0,gm+1(x) = 0,...,gm+n(x) = 0} /

• 0. Assume

that there is an extended real ρ ∈]0,+∞] such that: (i) for all k ∈ {1,...,m + n} the functions gk are C1 on Uρ(C); (ii) there exists a real γ ≥ 0 such that, for all t ∈ I and for all x,y ∈ Uρ(C)

∇gk(x)−∇gk(y),x − y ≥ −γ x − y2

for all k ∈ {1,...,m} and

∇gk(x)−∇gk(y) ≤ γx − y

for all k ∈ {m + 1,...,m + n}. Assume also that there is a real δ > 0 such that for all x ∈ bdry C [−δ,δ]px × [−δ,δ]n ⊂ Ax(BH ) +Rpx

+ ×{0Rn},

where px = Card{k ∈ {1,...,m} : gk(x) = 0}, {i1,...,ipx } = {k ∈ {1,...,m} : gk(x) = 0} and Ax =

• Dgi1(x),...,Dgipx (x),Dgm+1(x),...,Dgm+n(x)
• Then, the set C is r-prox-regular with r = min{ρ, δ

γ }.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection

Plan

1

Uniform prox-regular sets in Hilbert setting Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

2

Uniform prox-regularity of constraint sets An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

3

Application to constrained optimization

4

Inverse image, intersection Intersection Inverse image

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection

Argmin of C2-constrained optimization problem fails to be prox-regular

Consider the C2-function f : R −→R x −→

• x6(1− cos( 1

x ))

if x 0

• therwise

and note that the constrained optimization problem Minimize f(x) subject to x ≥ 0 admits as set of solutions S = {0}∪

• 1

2kπ : k ∈ N

• which fails to be uniformly r-prox-regular for

any extended real r ∈]0,+∞].

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection

Prox-regularity of Argmin

Theorem Let m ∈ N, f0,...,fm : H −→ R be real-valued functions on a Hilbert space H and let the constrained optimization problem (P)

• Minimize f0(x)

s.c. f1(x) ≤ 0,...,fm(x) ≤ 0.

Set C = {x ∈ H : f1(x) ≤ 0,...,fm(x) ≤ 0}, µ = inf

C f0, K = {1,...,m}. Assume that µ ∈ R and

that S = Argmin

C

f0 /

• 0. Assume that:

(i) there are an extended real ρ ∈]0,+∞] and a real γ ≥ 0 such that f0,...,fm are of class C1 and

γ-semi-convex on Uρ(S);

(ii) there is a real δ > 0 such that for all x ∈ bdryS [−δ,δ]px +1 ⊂ Ax(BH ) +Rpx +1

+

,

where px = Card{k ∈ K : fk(x) = 0}, Ax = (Dfi1(x),...,Dfipx (x),Df0(x)) and

{i1,...,ipx } = {k ∈ K : fk(x) = 0}.

Then, the set Argmin

C

f is min

• ρ, δ

γ

• prox-regular.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Plan

1

Uniform prox-regular sets in Hilbert setting Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity

2

Uniform prox-regularity of constraint sets An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities

3

Application to constrained optimization

4

Inverse image, intersection Intersection Inverse image

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Intersection of two prox-regular sets

Theorem Let H be a real Hilbert space, C1, C2 be two r-prox-regular subsets of H with r ∈]0,+∞[ such that C1 ∩ C2 /

0.

Assume that there is a real s > 0 such that for all x ∈ bdry (C1 ∩ C2), there is a neighborhood U

• f x in H such that for all x1 ∈ U ∩ C1, for all x2 ∈ U ∩ C2

sBH ⊂ T C(C1;x1)∩BH − T C(C2;x2)∩BH . (3) Then, C1 ∩ C2 is rs

2 -prox-regular.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Inverse image

Theorem Let H , H ′ be two Hilbert spaces, D be a r-prox-regular subset of H

′ with r ∈]0,+∞] and

G : H → H ′ be a mapping such that C = G−1(D) /

• 0. Assume that there is an extended real

ρ ∈]0,+∞] such that:

(i) G is differentiable on Uρ(C); (ii) there is a real K > 0 such that for all x,y ∈ C with x − y ≤ 2ρ,

G(x)− G(y) ≤ K x − y;

(iii) there is a real γ ≥ 0 such that for all x,y ∈ Uρ(C),

DG(x)− DG(y) ≤ γ x − y;

(iv) there is a real s > 0 for which, for all x ∈ bdry C, there is a real η > 0 such that for all (x,y) ∈ (x +ηBH )×ηBH ′ with G(x)− y ∈ D, sBH ′ ⊂ DG(x)(BH )− T(D;G(x)− y). (4) Then, the set C is r′-prox-regular with r′ = min

• ρ,s
• K 2

r +γ

−1

.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Perspectives

• Algorithms in optimization for prox-regular sets ?

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Perspectives

• Algorithms in optimization for prox-regular sets ?
• Find an easy counter-example which shows that the intersection of two uniformly prox-regular

sets fails to be uniformly prox-regular.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Perspectives

• Algorithms in optimization for prox-regular sets ?
• Find an easy counter-example which shows that the intersection of two uniformly prox-regular

sets fails to be uniformly prox-regular.

• Improve the result on the preservation of prox-regularity by intersection.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

Perspectives

• Algorithms in optimization for prox-regular sets ?
• Find an easy counter-example which shows that the intersection of two uniformly prox-regular

sets fails to be uniformly prox-regular.

• Improve the result on the preservation of prox-regularity by intersection.
• Extend some results to a non-Hilbert setting.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Intersection Inverse image

• S. Adly, F

. Nacry, L. Thibault, Discontinuous sweeping process with prox-regular sets, submitted.

• S. Adly, F

. Nacry, L. Thibault, Preservation of uniformly prox-regular sets, Accepted for publication in SIAM Journal of Optimization. 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.

• H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491.

R.A. Poliquin, R.T. Rockafellar, L. Thibault, Local differentiability of distance functions,

• Trans. Amer. Math. Soc. 352 (2000), 5231-5249.
• A. Tanwani, B. Brogliato, C. Prieur, Stability and observer design for Lur’e systems with

multivalued, nonmonotone, time-varying nonlinearities and state jumps, SIAM J. Control

• Optim. 52 (2014), 3639-3672.
• 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.

Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Thank you for your attention