کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5773145 | 1631063 | 2017 | 18 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
A bound on the Carathéodory number
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله

چکیده انگلیسی
The Carathéodory number κ(K) of a pointed closed convex cone K is the minimum among all the κ for which every element of K can be written as a nonnegative linear combination of at most κ elements belonging to extreme rays. Carathéodory's Theorem gives the bound κ(K)â¤dimâ¡K. In this work we observe that this bound can be sharpened to κ(K)â¤âKâ1, where âK 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 so-called smooth cones. We also give a simple example 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 cp-rank for completely positive matrices.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 532, 1 November 2017, Pages 347-363
Journal: Linear Algebra and its Applications - Volume 532, 1 November 2017, Pages 347-363
نویسندگان
Masaru Ito, Bruno F. Lourenço,