A barrier on convex cones with parameter equal to the dimension

Transcription

1 A barrier on convex cones with parameter equal to the dimension Université Grenoble 1 / CNRS August 23 / ISMP 2012, Berlin

2 Outline 1 Universal barrier 2 Projective images of barriers Pseudo-metric on the product of projective spaces 3

3 Universal barrier Definition (Nesterov, Nemirovski 1994) Let K R n be a regular convex cone. A (self-concordant logarithmically homogeneous) barrier on K is a smooth function F : K o R on the interior of K such that F(αx) = ν logα+f(x) (logarithmic homogeneity) F (x) 0 (convexity) lim x K F(x) = + (boundary behaviour) F (x)[h, h, h] 2(F (x)[h, h]) 3/2 (self-concordance) for all tangent vectors h at x. The homogeneity parameter ν is called the barrier parameter.

4 Duality Universal barrier Theorem (Nesterov, Nemirovski 1994) Let K R n be a regular convex cone and F : K o R a barrier on K with parameter ν. Then the Legendre transform F is a barrier on K with parameter ν. the graph G F = {(x, F (x)) x K o } is a smooth n-dimensional submanifold of K ( K ) R n R n F is determined by this graph up to an additive constant

5 Universal barrier Universal barrier Theorem (Nesterov, Nemirovski 1994) There exists an absolute constant c > 0 such that F(x) = c log Vol{p K x, p < 1} is a (c n)-self-concordant barrier on K R n. Lemma (Güler 1996) F(x) = c logϕ(x) up to an additive constant, where ϕ(x) = e x,p dp K is the characteristic function of the cone.

6 Properties Universal barrier the universal barrier is invariant under the action of SL(R n ) fixed under unimodular automorphisms of K additive under the operation of taking products has barrier parameter O(n) not invariant with respect to duality

7 Projective images of cones Projective images of barriers Pseudo-metric on the product of projective spaces let RP n 1,RP n 1 be the primal and dual real projective space lines and hyperplanes through the origin of R n let K R n be a regular convex cone the canonical projection Π : R n \{0} RP n 1 maps K \{0} to a compact convex subset C RP n 1 K \{0} to the spherical boundary C RP n 1 the canonical projection Π : R n \{0} RP n 1 maps K \{0} to a compact convex subset C RP n 1 K \{0} to the spherical boundary C RP n 1

8 Product of projective spaces Projective images of barriers Pseudo-metric on the product of projective spaces between elements of RP n 1,RP n 1 there is no scalar product, but an orthogonality relation the set M = is dense in RP n 1 RP n 1 M = { } (x, p) RP n 1 RP n 1 x p { } (x, p) RP n 1 RP n 1 x p is a submanifold of RP n 1 RP n 1 of codimension 1

9 Product of projective spaces Projective images of barriers Pseudo-metric on the product of projective spaces between elements of RP n 1,RP n 1 there is no scalar product, but an orthogonality relation the set M = is dense in RP n 1 RP n 1 M = { } (x, p) RP n 1 RP n 1 x p { } (x, p) RP n 1 RP n 1 x p is a submanifold of RP n 1 RP n 1 of codimension 1

10 Images of conic boundaries Projective images of barriers Pseudo-metric on the product of projective spaces define the set δ K = {(Π(x),Π (p)) x K \{0}, p K \{0}, x p} M ( C C ) the projections of δ K on C and C are surjective if K is smooth, then δ K is homeomorphic to S n 2 call δ K the boundary frame corresponding to the cone K

11 Geometric interpretation Projective images of barriers Pseudo-metric on the product of projective spaces the boundary frame δ K consists of pairs z = (x, p) M where the line x contains a ray in K p is a supporting hyperplane at x

12 Images of barriers Projective images of barriers Pseudo-metric on the product of projective spaces let F : K o R be a barrier let M F be the projective image of the graph G F of the Legendre transform M F = { (Π(x),Π (F (x))) x K o} RP n 1 RP n 1 is a smooth (n 1)-dimensional submanifold of M

13 Geometric interpretation Projective images of barriers Pseudo-metric on the product of projective spaces the manifold M F consists of pairs (x, p) where x is a line through a point y K o p is parallel to the hyperplane which is tangent to the level surface of F at y if y ŷ K, then p tends to a supporting hyperplane at ŷ M F = δ K

14 Two-point function on M Projective images of barriers Pseudo-metric on the product of projective spaces let z = (x, p), z = (x, p ) M RP n 1 RP n 1 let y, y R n, s, s R n be representatives of x, x, p, p the quantity (z; z ) = (z ; z) := 1 y, s y, s y, s y, s R does not depend on the choice of the representatives

15 Pseudo-metric Projective images of barriers Pseudo-metric on the product of projective spaces Theorem (H., 2012) There exists a complete pseudo-metric g of neutral signature on M such that for all z, z M linked by a geodesic σ, if (z; z ) > 0, then the velocity vector of σ has positive square and d(z, z ) = arcsin (z; z ); if (z; z ) = 0, then σ is light-like; if (z; z ) < 0, then the velocity vector of σ has negative square and d(z, z ) = arc sinh (z; z ).

16 Interpretation of the barrier M F is a Riemannian submanifold of M bounded by δ K M

17 Minimal surface in Euclidean space

18 Definition Let M be a pseudo-riemannian manifold. Then M M is a minimal submanifold if M is a stationary point of the volume functional with respect to variations with compact support. a submanifold is minimal if and only if its mean curvature vanishes

19 Affine hyperspheres Theorem (H., 2011) Let K R n be a regular convex cone and F : K o R n a barrier on K. Then the submanifold M F M is minimal if and only if the level surfaces of F are affine hyperspheres.

20 Geometric interpretation the affine normal is the tangent to the curve made of the gravity centers of the sections an affine hypersphere is a hypersurface such that all affine normals meet in a point

21 Calabi conjecture Theorem (Fefferman 1976, Cheng-Yau 1986, Li 1990, and others) Let K R n be a regular convex cone. Then there exists a unique foliation of K o by a homothetic family of affine hyperspheres which are asymptotic to K.

22 Einstein-Hessian function Corollary Let K R n be a regular convex cone. Then there exists a unique (up to affine scaling) logarithmically homogeneous locally strongly convex function F on K such that M F is a minimal submanifold of M with M F = δ K. call F the Einstein-Hessian function characterized as convex solution of the PDE log det F = 2n ν F, F K = + already conjectured by O. Güler

23 Ricci curvature Theorem (Calabi, 1972) The Ricci curvature of an affine hypersphere is negative semi-definite. by a splitting theorem of [Loftin, 2002] we get Corollary The Ricci curvature of the Hessian metric F defined by the Einstein-Hessian function is negative semi-definite.

24 for ν = n non-positivity of the Ricci curvature amounts to 4F,ij F,uv F,rs F,iur F,jvs, u,v,r,s where F,ij = 2 F x i x j, F,ijk = 3 F x i x j x k, k F,ik F,kj = δ i j Theorem (H., 2012; independently by D. Fox) For ν = n the Einstein-Hessian function is self-concordant. call it the

25 Properties the is invariant with respect to the action of SL(R n ) fixed under unimodular automorphisms of K additive under the operation of taking products invariant with respect to duality has barrier parameter not greater than n

26 Power cone let p, q > 1 such that 1 p + 1 q = 1 and consider the power cone K p = {(x, y, z) R 3 x 0, y 0, z 2 x 1/p y 1/q} Theorem (H., 2011) The barrier parameter of the on the power cone K p is given by 3 max(p,q) max(p,q)+1.

27 dotted red: barrier parameter of the solid blue: lower bound on the barrier parameter

28 References The cross-ratio manifold: a model of centro-affine geometry. Intern. Elec. J. Geom., 4(2):32-62, Barriers on projective convex sets. AIMS Proc., Vol. 2011, pp , s on convex cones. Submitted to Math. Oper. Res., Centro-affine differential geometry, Lagrangian submanifolds of the reduced paracomplex projective space, and conic optimization. Differential geometry 2012, Będlewo, Poland.

29 Thank you