Full Length Paper Series A

Mathematical Programming

, Volume 137, Issue 1, pp 131-153

Separating doubly nonnegative and completely positive matrices

  • Hongbo DongAffiliated withDepartment of Applied Mathematics and Computational Sciences, University of Iowa
  • , Kurt AnstreicherAffiliated withDepartment of Management Sciences, University of Iowa Email author 

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Abstract

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