Mathematical Programming

, Volume 137, Issue 1, pp 131–153

Separating doubly nonnegative and completely positive matrices

Full Length Paper Series A

DOI: 10.1007/s10107-011-0485-8

Cite this article as:
Dong, H. & Anstreicher, K. Math. Program. (2013) 137: 131. doi:10.1007/s10107-011-0485-8


The cone of Completely Positive (CP) matrices can be used to exactly formulate a variety of NP-Hard optimization problems. A tractable relaxation for CP matrices is provided by the cone of Doubly Nonnegative (DNN) matrices; that is, matrices that are both positive semidefinite and componentwise nonnegative. A natural problem in the optimization setting is then to separate a given DNN but non-CP matrix from the cone of CP matrices. We describe two different constructions for such a separation that apply to 5 × 5 matrices that are DNN but non-CP. We also describe a generalization that applies to larger DNN but non-CP matrices having block structure. Computational results illustrate the applicability of these separation procedures to generate improved bounds on difficult problems.

Mathematics Subject Classification (2000)

90C26 90C22 90C20 15B48 

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computational SciencesUniversity of IowaIowa CityUSA
  2. 2.Department of Management SciencesUniversity of IowaIowa CityUSA