  


A bound on the Carathéodory number
Masaru Ito (ito.mmath.cst.nihonu.ac.jp) Abstract: The Carathéodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carathéodory's Theorem gives the bound k(K) <= dim (K). In this work we observe that this bound can be sharpened to k(K) <= l1, where l is the length of the longest chain of nonempty faces contained in K, thus tying the Carathéodory number with a key quantity that appears in the analysis of facial reduction algorithms. We show that this bound is tight for several families of cones, which include symmetric cones and the socalled smooth cones. We also give a family of examples showing that this bound can also fail to be sharp. In addition, we furnish a new proof of a result by Güler and Tunçel which states that the Carathéodory number of a symmetric cone is equal to its rank. Finally, we connect our discussion to the notion of cprank for completely positive matrices. Keywords: caratheodory number, longest chain of faces, symmetric cone, cprank Category 1: Linear, Cone and Semidefinite Programming Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF] Entry Submitted: 07/01/2016 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  