Applications of the Lorentz positive cone in nonconvex quadratic optimization
\(\)We consider the Lorentz positive cone of \(n \times m\) matrices that map the Lorentz cone in \(R^m\) into the Lorentz cone in \(R^n\). 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 \(R^n\) and \(R^m\). 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.