Stratification and Regularity Concepts Cristopher Hermosilla - - PowerPoint PPT Presentation

Stratification and Regularity Concepts

Cristopher Hermosilla

Commands, INRIA Saclay Île-de-France UMA, ENSTA ParisTech

YRW SADCO-WIAS Berlin, January 30th, 2014

• C. Hermosilla (INRIA Saclay)

Berlin (30/01/2014) 1 / 37

Introduction : Optimization point of view

In Optimization Theory we are usually interested in :

Optimality conditions. Value functions. Sensitivity Analysis. Numerical schemes.

}

Variational Analysis

(cones and subdifferentials)

In particular, normal cones play an important role ! !

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Berlin (30/01/2014) 2 / 37

Introduction

Let us consider the mathematical programming problem min

x∈X {f (x) | x ∈ K} .

(P1) Suppose that ¯ x is a solution to (P1), then : 0 ∈ ∇f (¯ x) + NK(¯ x).

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Berlin (30/01/2014) 3 / 37

Introduction

Let us consider the calculus of variation problem min

x∈AC[a,b]

b

a

ℓ(t, x(t), ˙ x(t))dt | (x(a), x(b)) ∈ K

• .

(P2) Suppose that ¯ x(·) is a solution to (P2). In particular, there exists an arc p : [a, b] → RN such that the transversality condition holds : (p(a), −p(b)) ∈ NK(¯ x(a), ¯ x(b)).

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Against intuition... In full generality, many wild situation may occur ! !

Suppose K = {g(x) ≤ 0} with g : S ⊆ RN → R a continuous function. Let ¯ x ∈ bdry(K) and suppose that g is differentiable at x, then NK(¯ x) =

• λ≥0

λ∇g(¯ x)

Proposition (c.f. Baire’s Theorem)

Let S ⊆ RN be a compact set. The set of continuous nowhere differentiable functions is dense in C(S).

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What about almost differentiable functions ?

Suppose K = {g(x) ≤ 0} with g a L-Lipschitz function. Let ¯ x ∈ bdry(K), then NK(¯ x) =

• λ≥0

λ∂g(¯ x)

Theorem (Borwein, Wang 1998)

Let S ⊆ RN be a compact set. The set of 1-Lipschitz functions on S that satisfy ∂g(x) = B ∀x ∈ S is an open dense set in the metric space of 1-Lipschitz functions on S with the uniform metric.

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Main motivation

...“general topology” was developed by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes... (Grothendieck’s Sketch of a Programme) ⇓ Full generality often considers unnecessarily broad classes of sets and functions. ⇓ Well structured objects (Tame optimization)

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Stratifications

...A tame set need not be a manifold, but it can always be decomposed into a locally finite collection of manifolds which fit each other in a nice way.... (Ioffe’s Invitation to Tame optimization)

Definition

A closed set K ⊆ RN is said to be stratifiable if there exists a locally finite collection {Mi : i ∈ I} of submanifold of RN such that : K =

• i∈I

Mi and Mi ∩ Mj = ∅ whenever i = j

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Some examples

K

K K K

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1

Regularity notions : pairwise conditions Some notions of Differential Geometry Frontier Condition Whitney Conditions Classes of stratifiable sets

2

Regularity notions : "stratum-wise" conditions Quick review about Variational Analysis Relative Wedgedness Bounded curvature

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Submanifolds of RN

Definition

Let M ⊆ RN, we say that M is an (embedded) submanifold of RN provided for any x ∈ M, there exists Θ ⊆ RN, an open neighborhood of x so that Θ ∩ M = {y ∈ Θ : h(y) = 0} where h : Θ → RN−d is smooth with rank(Dh(y)) = N − d, ∀y ∈ Θ. In such case, the tangent and normal space to M at x are given by TM(x) =

• v ∈ RN | ∇hk(x), v = 0, ∀k = 1, . . . , N − d
• = ker(Dh(x))

NM(x) =

• η ∈ RN | η =

N−d

• k=1

αk∇hk(x), α ∈ RN−d

• = im(Dh(x)T).
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Some metrics

Let S be a subset of RN, the distance function to S is distS(x) = inf{|x − y| : y ∈ S}. Let S1 and S2 be two compact sets of RN, the Hausdorff distance between them is given by dH(S1, S2) = max

• sup

x∈S2

distS1(x), sup

x∈S1

distS2(x)

• .
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Metric space of Cones

A set C ⊆ RN is a cone provided that λC ⊆ C for any λ ≥ 0. The distance between two cones of RN, C1 and C2, is given by D(C1, C2) = dH(C1 ∩ S, C2 ∩ S).

Definition (Limits)

Let {Ck} be a sequence of cones of RN, we say that it converges to C, another cone of RN, provided D(Ck, C) → 0 as k → +∞. In such case we write Ck → C

• r

lim

k→+∞ Ck = C.

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Frontier condition

Let Mi and Mj be two submanifolds of RN and let x ∈ Mi ∩ Mj. The pair (Mi, Mj) is said to satisfy the the frontier condition at x provided that Mi ∩ Mj = ∅, then Mi ⊆ Mj.

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Whitney regularity conditions

Let Mi and Mj be two submanifolds of RN and let x ∈ Mi ∩ Mj. The pair (Mi, Mj) is said to satisfy the Whitney (a)-condition at x provided that for any sequence {xk} ⊆ Mj with xk → x if TMj(xk) → T , then TMi(x) ⊆ T . The pair (Mi, Mj) is said to satisfy the Whitney (b)-condition at x provided that for any sequence {(xk, yk)} ⊆ Mj × Mi with xk, yk → x if TMj(xk) → T and the lines xkyk converge to ℓ, then ℓ ⊆ T .

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Comparing the Whitney conditions

It turns out that the Whitney (b)-condition is stronger than the Whitney (a)-condition

Proposition

Let Mi and Mj be two submanifolds of RN and let x ∈ Mi ∩ Mj. Suppose that the pair (Mi, Mj) satisfies the Whitney (b)-condition at x, then it also satisfies the Whitney (a)-condition at x. ... but the converse is not true...

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Does the Whitney (a)-condition always hold ? : No

Let K = {x2 = zy 2} ⊆ R3. Consider the stratification of K : Mi = {(0, 0, z) | z ∈ R} and Mj = K \ {(0, 0, z) | x ∈ R}.

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Some consequences

Proposition

Let Mi and Mj be two submanifolds of RN. Suppose that the pair (Mi, Mj) satisfies the Whitney (b)-condition at some x ∈ Mi ∩ Mj. Then dim(Mi) < dim(Mj). Let {Mi : i ∈ I} be a partition of a closed set K into smooth manifolds, such that each pair (Mi, Mj) for which Mi ⊆ Mj = ∅ satisfies the Whitney (b)-condition. Then the frontier condition holds.

Definition

A Whitney stratification is a stratification that satisfies the (b)-condition.

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Semilinear sets

Finite union of open polyhedra

An open polyhedron is given by : P =

• x ∈ RN
• ηk, x = αk,

k = 1, . . . , n, ηk, x < αk, k = n + 1, . . . , m

• where η1, . . . , ηm ∈ RN.

Semilinear sets are close under Boolean operations : Let K1 and K2 be two semilinear sets, then K1 ∪ K2, K1 ∩ K2 and RN \ K1 are semilinear. The class of semilinear sets is stable under image and pre-image of affine mapping.

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Semialgebraic sets

A set K ⊆ RN is semialgebraic if it is a finite union of sets of the form

• x ∈ RN
• pk(x) = 0,

k = 1, . . . , n, pk(x) < 0, k = n + 1, . . . , m

• where all p1(·), . . . , pm(·) are real polynomials.

Semialgebraic sets are close under Boolean operations. The class of semialgebraic sets is stable under projection from RN into RN−1.

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Example of semialgebraic set : inflated tetrahedron

Let us consider K = {x ∈ R3 | A(x) is positive semi-defined} where A(x) =   1 x1 x2 x1 1 x3 x2 x3 1   .

Figure: The E3 elliptope (also known as “inflated tetrahedron”) is the yellow part (surface and interior)

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O-minimal structures

A collection S = {Sn} where each Sn is a set of subsets of Rn is called an

• -minimal structure provided that :

All subalgebraic subsets of Rn are in Sn. For every n, Sn is closed under Boolean operations. If A ∈ Sm and B ∈ Sn, then A × B ∈ Sm+n. If π : Rn+1 → Rn is the projection on the first n coordinates and A ∈ Sn+1, then π(A) ∈ Sn. The elements of S1 are precisely the finite unions of points and intervals. The elements of Sn are called the definable subsets of Rn.

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Whitney stratification theorem

Theorem

Let K be a closet set in RN. If K is semialgebraic (or definable) then K admits a finite Whitney stratification (not only locally finite). Moreover the strata are semialgebraic (resp. definable) subsets of RN.

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Subanalytic sets

A set K ⊆ RN is semianalytic provided that each point of RN admits a neighborhood Θ for which K ∩ Θ has the form

• x ∈ RN
• fk(x) = 0,

k = 1, . . . , n, fk(x) < 0, k = n + 1, . . . , m

• where all f1(·), . . . , fm(·) are real analytic functions.

A set K ⊆ RN is subanalytic provided that each point of RN admits a neighborhood Θ such that K ∩ Θ =

• x ∈ RN | (x, y) ∈ S
• where S is a bounded semianalytic subset of RN × Rm for some m ≥ 1.

Subanalytic sets are close under Boolean operations.

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Berlin (30/01/2014) 24 / 37

Whitney stratification theorem

Theorem

Let K be a subanalytic closet set in RN. Then K admits a Whitney

• stratification. Moreover the strata are subanalytic subsets of RN.
• C. Hermosilla (INRIA Saclay)

Berlin (30/01/2014) 25 / 37

1

Regularity notions : pairwise conditions Some notions of Differential Geometry Frontier Condition Whitney Conditions Classes of stratifiable sets

2

Regularity notions : "stratum-wise" conditions Quick review about Variational Analysis Relative Wedgedness Bounded curvature

• C. Hermosilla (INRIA Saclay)

Berlin (30/01/2014) 26 / 37

Normal cones

Definition

Let S ⊆ RN be a locally closed set and x ∈ S given.

1 The Proximal normal cone to S at x, denoted by N P

S (x), is the set of

all η ∈ RN such that σ|x − y|2 ≥ η, y − x ∀y ∈ S, for some σ = σ(x, η) ≥ 0.

2 The Limiting normal cone to S at x, denoted by N L

S(x), is given by

N L

S(x) :=

• lim

n→∞ ηn : ∃{xn} ⊆ S with xn → x, ∃ηn ∈ N P S (xn)

• .

3 The Clarke normal cone to S at x, denoted by N C

S (x), is exactly the

convex closed hull of the N L

S(x).

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Graphically

R R S

η N P

S (¯

x) x2

2 = x1

x2 = x1 δ ¯ x + η ¯ x = (0, 0) S ∩ B(¯ x + δη, δ) = {¯ x}

R R S

N C

S (¯

x) N P

S (¯

x) = ∅ x5

2 = x3 1

¯ x = (0, 0)

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Closure of Manifold

Proposition

Let M be a d−dimensional submanifold of RN. for any x ∈ M there exists N d ⊆ Rd, a nonempty cone such that, after a change of coordinates, the Clarke normal cone can be written as N C

M(x) = N d × RN−d.

Moreover, if x ∈ M then N d = {0}d.

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Relative Wedgedness

Definition

A submanifolds of RN, say M, is relatively wedged around ¯ x ∈ M \ M provided ∃Θ open with ¯ x ∈ Θ and ∃A : Θ ∩ M → SO(N) a continuous map such that Ax

• N C

M(x)

• = N d

x × RN−d.

N d

x pointed in Rd.

In particular, dim(T C

M(x)) = d if dim(M) = d. Special orthonormal matrix SO(N) denoted the set of orthonormal matrix with det(A) = 1.

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Graphically

AT

x({0} × RN−d)

AT

x(N d x × {0})

AT

x({0} × RN−d)

N C

M(¯

x1) N C

M(¯

x2) AT

x(N d x × {0})

M

¯ x1 ¯ x2

Figure: Example : relatively wedged

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Curvature of a submanifold of RN

Let M be a submanifold of RN. Note that for every x ∈ M, there exists δ = δ(x) > 0 such that |η| 2δ |x − y|2 ≥ η, y − x ∀η ∈ NM(x), ∀y ∈ M.

Definition

Let M be a submanifold of RN.

1 The radius of curvature of M at x ∈ M, denoted by κ(x), is given by

κ(x) = sup 2η, y − x |y − x|2 : η ∈ NM(x) ∩ S, y ∈ M \ {x}

• .

2 M is said to have constant curvature if there is a constant κ0 ∈ R

such that κ(x) ≤ κ0 for any x ∈ M.

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Berlin (30/01/2014) 32 / 37

Recall

R R S

η N P

S (¯

x) x2

2 = x1

x2 = x1 δ ¯ x + η ¯ x = (0, 0) S ∩ B(¯ x + δη, δ) = {¯ x}

Figure: Example : proximal normal

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Final words

Is the notion of relative wedgedness appropriate. Relation between pairwise condition (Whitney conditions) and "stratum-wise" conditions.

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References I

• A. Ioffe

An invitation to tame optimization. SIAM J. Optimization Vol. 19, No. 4, pp. 1894–1917 J.M. Borwein & X. Wang Lipschitz functions with maximal subdifferentials are generic.

• Proc. Amer. Math. Soc. 128 (2000), 3221–3229.
• J. Mather

Notes on topological stability. Bulletin of the american mathematical society Vol. 49, No 4, pp. 475–506

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Berlin (30/01/2014) 35 / 37

References I

• M. Coste

An introduction to o-minimal geometry. Pisa : Istituti editoriali e poligrafici internazionali

• L. Van den Dries & C. Miller

Geometric categories and o-minimal structures. Duke Math. J, Vol. 84, No 2, pp. 497-540

• V. Kaloshin

A geometric proof of the existence of Whitney stratifications.

• Mosc. Math. J, Vol.5, No 1, pp. 125-133
• L. Nicolaescu

An invitation to Morse theory. Second edition, Springer 2011.

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Berlin (30/01/2014) 36 / 37

Thanks for your attention !

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