Tractable Subcones and LP-based Algorithms for Testing Copositivity
Akihiro Tanaka, Akiko Yoshise · arXiv (Cornell University) · 2015
The authors in a previous paper devised certain subcones of the copositive cone and showed that one can detect whether a given matrix belongs to each of them by solving linear optimization problems (LPs) with O(n) variables and O(n 2 ) constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this paper, they investigate the properties of the subcones in more detail and explore better subcones of the copositive cone having desirable properties.They introduce a semidenite basis (SD basis) that is a basis of the space of n n symmetric matrices consisting of n(n + 1)=2 symmetric semidefinite matrices. Using the SD basis, they devise two new subcones for which detection can be done by solving LPs with O(n 2 ) variables and O(n 2 ) constraints. The new subcones are larger than the ones in the previous paper and inherit their nice properties. The authors also examine the efficiency of those subcones in numerical experiments. The results show that the subcones are promising for testing copositivity.