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 … Read more
We consider the solution of nonconvex quadratic optimization problems using an outer approximation of the set-copositive cone that is iteratively strengthened with conic constraints and cutting planes. Our methodology utilizes an MILP-based oracle for a generalization of the copositive cone that considers additional linear equality constraints. In numerical testing we evaluate our algorithm on a … Read more
This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We intend the paper for readers new to the area, and hence the exposition is largely self-contained. We focus on examples having just a few variables … Read more
For a quadratic optimization problem (QOP) with linear equality constraints in continuous nonnegative variables and binary variables, we propose three relaxations in simplified forms with a parameter $\lambda$: Lagrangian, completely positive, and copositive relaxations. These relaxations are obtained by reducing the QOP to an equivalent QOP with a single quadratic equality constraint in nonnegative variables, … Read more