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. } Variational Analysis Sensitivity Analysis. (cones and subdifferentials) Numerical schemes. In particular, normal cones play an important role ! ! C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 2 / 37

Introduction Let us consider the mathematical programming problem ( P 1 ) min x ∈ X { f ( x ) | x ∈ K} . Suppose that ¯ x is a solution to ( P 1 ), then : 0 ∈ ∇ f (¯ x ) + N K (¯ x ) . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 3 / 37

Introduction Let us consider the calculus of variation problem �� b � ( P 2 ) min ℓ ( t , x ( t ) , ˙ x ( t )) dt | ( x ( a ) , x ( b )) ∈ K . x ∈AC [ a , b ] a Suppose that ¯ x ( · ) is a solution to ( P 2 ). In particular, there exists an arc p : [ a , b ] → R N such that the transversality condition holds : ( p ( a ) , − p ( b )) ∈ N K (¯ x ( a ) , ¯ x ( b )) . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 4 / 37

Against intuition... In full generality, many wild situation may occur ! ! Suppose K = { g ( x ) ≤ 0 } with g : S ⊆ R N → R a continuous function. x ∈ bdry ( K ) and suppose that g is differentiable at x , then Let ¯ � N K (¯ λ ∇ g (¯ x ) = x ) λ ≥ 0 Proposition (c.f. Baire’s Theorem) Let S ⊆ R N be a compact set. The set of continuous nowhere differentiable functions is dense in C ( S ) . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 5 / 37

What about almost differentiable functions ? Suppose K = { g ( x ) ≤ 0 } with g a L -Lipschitz function. x ∈ bdry ( K ) , then Let ¯ � N K (¯ x ) = λ∂ g (¯ x ) λ ≥ 0 Theorem (Borwein, Wang 1998) Let S ⊆ R N be a compact set. The set of 1-Lipschitz functions on S that satisfy ∀ x ∈ S ∂ g ( x ) = B is an open dense set in the metric space of 1-Lipschitz functions on S with the uniform metric. C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 6 / 37

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) C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 7 / 37

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 ⊆ R N is said to be stratifiable if there exists a locally finite collection {M i : i ∈ I} of submanifold of R N such that : � K = M i and M i ∩ M j = ∅ whenever i � = j i ∈I C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 8 / 37

Some examples K K K K C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 9 / 37

Regularity notions : pairwise conditions 1 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) 10 / 37

Submanifolds of R N Definition Let M ⊆ R N , we say that M is an (embedded) submanifold of R N provided for any x ∈ M , there exists Θ ⊆ R N , an open neighborhood of x so that Θ ∩ M = { y ∈ Θ : h ( y ) = 0 } where h : Θ → R N − 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 � v ∈ R N | �∇ h k ( x ) , v � = 0 , ∀ k = 1 , . . . , N − d � T M ( x ) = = ker ( Dh ( x )) � N − d � η ∈ R N | η = � α k ∇ h k ( x ) , α ∈ R N − d = im ( Dh ( x ) T ) . N M ( x ) = k = 1 C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 11 / 37

Some metrics Let S be a subset of R N , the distance function to S is dist S ( x ) = inf {| x − y | : y ∈ S} . Let S 1 and S 2 be two compact sets of R N , the Hausdorff distance between them is given by � � d H ( S 1 , S 2 ) = max sup dist S 1 ( x ) , sup dist S 2 ( x ) . x ∈S 2 x ∈S 1 C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 12 / 37

Metric space of Cones A set C ⊆ R N is a cone provided that λ C ⊆ C for any λ ≥ 0. The distance between two cones of R N , C 1 and C 2 , is given by D ( C 1 , C 2 ) = d H ( C 1 ∩ S , C 2 ∩ S ) . Definition (Limits) Let {C k } be a sequence of cones of R N , we say that it converges to C , another cone of R N , provided D ( C k , C ) → 0 as k → + ∞ . In such case we write C k → C or k → + ∞ C k = C . lim C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 13 / 37

Frontier condition Let M i and M j be two submanifolds of R N and let x ∈ M i ∩ M j . The pair ( M i , M j ) is said to satisfy the the frontier condition at x provided that M i ∩ M j � = ∅ , then M i ⊆ M j . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 14 / 37

Whitney regularity conditions Let M i and M j be two submanifolds of R N and let x ∈ M i ∩ M j . The pair ( M i , M j ) is said to satisfy the Whitney (a)-condition at x provided that for any sequence { x k } ⊆ M j with x k → x if T M j ( x k ) → T , then T M i ( x ) ⊆ T . The pair ( M i , M j ) is said to satisfy the Whitney (b)-condition at x provided that for any sequence { ( x k , y k ) } ⊆ M j × M i with x k , y k → x if T M j ( x k ) → T and the lines x k y k converge to ℓ , then ℓ ⊆ T . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 15 / 37

Comparing the Whitney conditions It turns out that the Whitney (b)-condition is stronger than the Whitney (a)-condition Proposition Let M i and M j be two submanifolds of R N and let x ∈ M i ∩ M j . Suppose that the pair ( M i , M j ) satisfies the Whitney (b)-condition at x, then it also satisfies the Whitney (a)-condition at x. ... but the converse is not true... C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 16 / 37

Does the Whitney (a)-condition always hold ? : No Let K = { x 2 = zy 2 } ⊆ R 3 . Consider the stratification of K : M i = { ( 0 , 0 , z ) | z ∈ R } and M j = K \ { ( 0 , 0 , z ) | x ∈ R } . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 17 / 37

Some consequences Proposition Let M i and M j be two submanifolds of R N . Suppose that the pair ( M i , M j ) satisfies the Whitney (b)-condition at some x ∈ M i ∩ M j . Then dim ( M i ) < dim ( M j ) . Let {M i : i ∈ I} be a partition of a closed set K into smooth manifolds, such that each pair ( M i , M j ) for which M i ⊆ M j � = ∅ satisfies the Whitney (b)-condition. Then the frontier condition holds. Definition A Whitney stratification is a stratification that satisfies the (b)-condition. C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 18 / 37

Semilinear sets Finite union of open polyhedra An open polyhedron is given by : � � � η k , x � = α k , � k = 1 , . . . , n , x ∈ R N � P = � � η k , x � < α k , k = n + 1 , . . . , m where η 1 , . . . , η m ∈ R N . Semilinear sets are close under Boolean operations : Let K 1 and K 2 be two semilinear sets, then K 1 ∪ K 2 , K 1 ∩ K 2 and R N \ K 1 are semilinear. The class of semilinear sets is stable under image and pre-image of affine mapping. C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 19 / 37

Semialgebraic sets A set K ⊆ R N is semialgebraic if it is a finite union of sets of the form � � � p k ( x ) = 0 , k = 1 , . . . , n , x ∈ R N � � p k ( x ) < 0 , k = n + 1 , . . . , m where all p 1 ( · ) , . . . , p m ( · ) are real polynomials. Semialgebraic sets are close under Boolean operations. The class of semialgebraic sets is stable under projection from R N into R N − 1 . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 20 / 37

Example of semialgebraic set : inflated tetrahedron Let us consider K = { x ∈ R 3 | A ( x ) is positive semi-defined } where   1 x 1 x 2  . A ( x ) = x 1 1 x 3  x 2 x 3 1 Figure: The E 3 elliptope (also known as “inflated tetrahedron”) is the yellow part (surface and interior) C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 21 / 37

O-minimal structures A collection S = {S n } where each S n is a set of subsets of R n is called an o-minimal structure provided that : All subalgebraic subsets of R n are in S n . For every n , S n is closed under Boolean operations. If A ∈ S m and B ∈ S n , then A × B ∈ S m + n . If π : R n + 1 → R n is the projection on the first n coordinates and A ∈ S n + 1 , then π ( A ) ∈ S n . The elements of S 1 are precisely the finite unions of points and intervals. The elements of S n are called the definable subsets of R n . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 22 / 37

Whitney stratification theorem Theorem Let K be a closet set in R N . 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 R N . C. Hermosilla (INRIA Saclay) Berlin (30/01/2014) 23 / 37