Open problems in convex geometry
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1 Open problems in convex geometry 10 March 2017, Monash University Seminar talk Vera Roshchina, RMIT University Based on joint work with Tian Sang (RMIT University), Levent Tunçel (University of Waterloo) and David Yost (Federation University Australia). 0/18
2 Perceptron algorithm and its complexity Find an x R n such that a T i x > 0 i {1,..., m}, where a i R n, and we assume in addition that a i = 1 for all i. The perceptron algorithm 1. Let x 0 := 0, k = 0 2. Choose i such that a T i x k is minimal 3. Let x k+1 := x k + a i. 4. If a T i x k+1 > 0 i, halt. Otherwise, k := k + 1 and go to 2. If the problem is feasible, the perceptron algorithm terminates in finitely many steps, and the number of steps depends on geometry of the problem. 1/18
3 Spherical caps and complexity For a given p S n 1 and α [0, π] the spherical cap in S n 1 with centre at p and angular radius α is defined as p α cap(p, α) := {x S n 1 p T x cos α}. A smallest including spherical cap for a 1,..., a m S n 1 is such that the angle α is the smallest possible. The quantity cos α can be used to measures the difficulty of the problem a T i x > 0. If cos α > 0 (and hence the problem has a solution), the perceptron algorithm halts in 1 cos 2 α iterations. [P. Bürgisser and F. Cucker Condition, 2013] 2/18
4 Ellipsoid method C The ellipsoid method finds an interior point of a compact convex set C R n using a separation oracle. The algorithm starts with a ball B(y 0, R) of radius R centred at y 0 R n such that C B(y 0, R). 0. Let k = 0, E 0 = B(y 0, R). 1. If y k C, output y k and halt, otherwise find a separating half-space H and compute the minimal ellipsoid E k+1 E k H. 2. Let y k+1 be the centre of E k+1, set k = k+1 and go to step 1. 3/18
5 Ellipsoid method The ellipsoid method returns a point in the compact convex feasible set C. The number of iterations it performs on the input (C, R, y 0 ) is bounded by 2n ln V, ν where V is the volume of the initial ball B(y 0, R) that contains the set C, and ν is the volume of the set C. [L. Tunçel, Polyhedral and semidefinite programming methods in combinatorial optimization, 2010] 4/18
6 Interior-point methods Consider a pair of feasibility problems Ax = 0 x K (P) AT y K, (D) where K is a symmetric cone (self-dual and homogeneous) and A is an m n real matrix. An ipm decides which one of the systems is strictly feasible and halts in O( ν K ln(ν K C R (A))) iterations Here C R (A) is (possibly infinite) Renegar s condition number that measures the regularity of the intersection of the linear subspace with the boundary of the cone, and ν K is the complexity parameter of the barrier function for the symmetric cone K. [D. Amelunxen, P. Bürgisser, Intrinsic volumes of symmetric cones...] 5/18
7 Smale s 9 th problem Is there a polynomial time algorithm over the real numbers which decides the feasibility of the linear system of inequalities Ax b? Here the key is over the real numbers : this means that the algorithm should terminate after polynomially many algebraic operations. This polynomial bound should be in terms of the dimension of the problem. We just saw that the complexity of different methods for solving this kind of problems depends on the geometry of the problem. 6/18
8 Simplex method and the Klee-Minty cube Simplex method solves a problem of maximising a linear objective over a polytope defined by a system of inequalities by travelling from one vertex to another and hence increasing the value of the objective function until the optimal solution is reached. Klee-Minty cube refers to a range of examples that demonstrate the exponential inefficiency of the simplex method. min x n 0 x 1 1, εx i 1 x i 1 εx i 1, i 2 : n. [Gärtner, Henk, Ziegler, Randomized simplex algorithms...] 7/18
9 Polynomial Hirsch conjecture The Hirsch conjecture stated that for any d-dimensional polytope that is bounded by n inequalities, the shortest path between any two vertices is bounded by n d. This conjecture was disproved by Franciso Santos in He constructed a counterexample in 43 dimensional space of a polytope with 86 facets and combinatorial diameter at least 44. The quest now is to solve the polynomial Hirsch conjecture, i.e. to find out if there is a polynomial bound on the diameter of polytopes in terms of the number of facets and dimension. The best known bounds are as follows (there are better bounds of the same order): (n, d) (n d) log d and (n, d) 2d 2 3 n. [F. Santos, A counterexample to the Hirsch conjecture] 8/18
10 Hyperbolicity cones A homogeneous polynomial p R[x 1, x 2,..., x n ] is called hyperbolic in a direction e R n if p(e) 0 and the univariate polynomial t p(x + te) has only real roots for all x. A hyperbolicity cone is the connected component of the complement of {x p(x) = 0} which contains e. Examples of the set {x p(x) = 0} for elementary symmetric polynomials x 1 + x 2 + x 3, x 1 x 2 + x 1 x 3 + x 2 x 3 and x 1 x 2 x 3 : 9/18
11 Hyperbolicity cones The polynomial p(x, y, z) = 4xyz + xz 2 + yz 2 + 2z 3 x 3 3zx 2 y 3 3zy 2 is hyperbolic with respect to (0, 0, 1). [see Pablo Parrilo s notes Algebraic techniques and semidefinite optimization] The positive orthant R n ++ and the cone of positive definite symmetric matrices S n ++ (in the n(n 2)/2 dimensional space of n n symmetric matrices) are hyperbolicity cones. The generalised Lax conjecture states that a hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Many special cases were proven recently. 10/18
12 Facial reduction algorithm The aim is to convert the system Ax K b to Ax Fmin b, where F min is a face of K to make sure that the new problem satisfies constraint qualifications (there is a strictly feasible solution). b-ar n F min K [The original algorithm developed by J. Borwein and H. Wolkowicz] 0 11/18
13 f, u + v = f, u > 0 F {u} F. Proof It suffices to note the following 3 facts. IfFacial the secondreduction containment in (3.11) algorithm is strict, we shall say that (u, v) reduces the system Ax F b, or it is a reducing certificate. Next, we state an algorithm to construct F min ; it is a simplified version of the one given by Borwein and Wolkowicz Under appropriate [3]. assumptions... Facial Reduction Algorithm (A, b, K) Input: A, b, K. Output: t 0, u 0,..., u t K with F min = K {u u t }. Invariants: F min F i, F i = K {u u i }, Fi = tan(u u i, K ). Initialization: Let (u 0, v 0 ) = (0, 0), F 0 = K, i = 0. while F min F i Find (u i+1, v i+1 ) reducing Ax Fi b. Let F i+1 = F i {u i+1 }, i = i + 1. end while Output t = i, u 0,..., u t. [From G. Pataki, A Simple Derivation of a Facial Reduction Algorithm...] Theorem 3.2. The Facial Reduction Algorithm is finite, and correctly constructs 12/18 F min.
14 Chains of faces and dimensions Observe that the facial reduction algorithm is going from one face to another reducing the dimension on each step. This algorithm works particularly well with the cone of positive semidefinite matrices (SDP cone) because each face of this cone is linearly isomorphic to a lower dimensional SDP cone. Any nested chain of faces of S n + has length no more than n. The SDP cone is a peculiar example of a convex set that has very large gaps in the dimensions of its faces. Theorem 1. For any finite increasing sequence of integers d 1 < d 2 < < d k there exists a closed convex set C in R d k such that the faces of this set come in dimensions that form this sequence. [R, Sang, Yost, On the dimensions of faces of compact convex sets] 13/18
15 Nice and facially exposed cones One assumption that the facial reduction algorithm requires is that the underlying cone is facially dual complete (or nice). This means that the sum of the dual cone and the orthogonal complement of the span of each face is a closed set: F + K is closed for every face F of K. Niceness coincides with facial exposure in R 3, but in higher dimensions the two notions are essentially different. This is a slice of a four dimensional cone that is facially exposed but not nice. [R. Facially exposed cones are not always nice] 14/18
16 Primal characterisations of FDC Theorem 2. If a closed convex cone K is facially dual complete, then for every F K and every x F, we have T K (x) spanf = T F (x). (1) Here T C (x) is the tangent cone to the set C at a point x C. A tangent cone can be defined as the closure of the cone of feasible directions: dir C (x) = λ(c x), T C (x) = dir C (x). λ R + 0 x Equivalently T C (x) = {y R n {z k }, {t k } : z k + x C, t k 0 k, t k z k y}. 15/18
17 Strong tangential exposure We say that a closed convex set C in R n has tangential exposure property if T K (x) spanf = T F (x) F K, x F. (2) Tangential exposure is a stronger property than facial exposure. We say that a closed convex set is strongly tangentially exposed if it is tangentially exposed along with all its lexicographic tangents. This means that the property (2) holds recursively for all tangent cones constructed at all points of the original set. 16/18
18 Sufficient condition Theorem 3 (Sufficient condition). If a closed convex cone K R n is strongly tangentially exposed, then it is Facially Dual Complete. [Forthcoming paper with Levent Tunçel] 17/18
19 Thank you
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