Applications of the Lorentz positive cone in nonconvex quadratic optimization

Abstract

We consider the Lorentz positive cone of $n\times m$ matrices that map the Lorentz cone in $\Rbb^m$ into the Lorentz cone in $\Rbb^n$. The Lorentz positive cone and its dual, the cone of Lorentz separable matrices, are shown to provide polynomial-time algorithms for the problem of minimizing a bilinear objective over variables contained in ellipsoids in $\Rbb^n$ and $\Rbb^m$. We also demonstrate how these cones can be used to strengthen SDP relaxations of other nonconvex quadratic optimization problems, including the two-trust-region subproblem.