Zero-one completely positive matrices and the A(R, S) classes

G. Dahl; T. A. Haufmann

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 296-304
  • ISSN: 2300-7451

Abstract

top
A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

How to cite

top

G. Dahl, and T. A. Haufmann. "Zero-one completely positive matrices and the A(R, S) classes." Special Matrices 4.1 (2016): 296-304. <http://eudml.org/doc/285553>.

@article{G2016,
abstract = {A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for \{0,1\}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.},
author = {G. Dahl, T. A. Haufmann},
journal = {Special Matrices},
keywords = {Completely positive matrix; (0, 1)-matrix; convex cone; completely positive matrix},
language = {eng},
number = {1},
pages = {296-304},
title = {Zero-one completely positive matrices and the A(R, S) classes},
url = {http://eudml.org/doc/285553},
volume = {4},
year = {2016},
}

TY - JOUR
AU - G. Dahl
AU - T. A. Haufmann
TI - Zero-one completely positive matrices and the A(R, S) classes
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 296
EP - 304
AB - A matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.
LA - eng
KW - Completely positive matrix; (0, 1)-matrix; convex cone; completely positive matrix
UR - http://eudml.org/doc/285553
ER -

References

top
  1. [1] A. Berman and N. Shaked-Monderer. Completely Positive Matrices. World Scientific Publishing Co. Pte Ltd., Singapore, 2003.  Zbl1030.15022
  2. [2] A. Berman and C. Xu. {0,1} Completely positive matrices. Linear Algebra Appl., 399:35–51, Apr 2005.  Zbl1072.15027
  3. [3] A. Berman and C. Xu. Uniform and minimal {0,1}-cp matrices. Linear and Multilinear Algebra, 55(5):439–456, Sep 2007.  Zbl1133.15012
  4. [4] R. A. Brualdi. Combinatorial Matrix Classes, volume 13. Cambridge University Press, Cambridge, 2006.  Zbl1106.05001
  5. [5] S. Bundfuss and M. Dür. An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim., 20(1):30–53, 2008.  Zbl1187.90187
  6. [6] S. Burer. On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program., 120(2):479–495, Apr 2008.  Zbl1180.90234
  7. [7] G. Dahl. A note on diagonally dominant matrices. Linear Algebra Appl., 317(1-3):217–224, Sep 2000.  Zbl0970.15017
  8. [8] M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springer, 1997.  Zbl0885.52001
  9. [9] M. Dür. Copositive Programming - A Survey. In Moritz Diehl, Francois Glineur, Elias Jarlebring, and Wim Michiels, editors, Recent Advances in Optimization and its Applications in Engineering, number 1, pages 3–21. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.  
  10. [10] A. V. Karzanov. Metrics and undirected cuts. Math. Program., 32(2):183–198, 1985.  Zbl0565.90016
  11. [11] F. Laburthe, M. Deza, and M. Laurent. The Hilbert basis of the cut cone over the complete graph on six vertices. Technical report, 1995.  
  12. [12] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, https://oeis.org, 2016.  
  13. [13] J. Zhong. Binary ranks and binary factorizations of nonnegative integermatrices. Electronic J. Linear Algebra, 23(June):540– 552, 2012.  Zbl1250.15019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.