Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization

• Authors: V. Jeyakumar;G. M. Lee;G. Y. Li
• Venue: SIAM Journal on Optimization
• Year: 2009

We establish alternative theorems for quadratic inequality systems. Consequently, we obtain Lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems. We present a generalization of Dine's theorem to a system of two homogeneous quadratic functions with a regular cone. The class of regular cones are cones $K$ for which $(K\cup-K)$ is a subspace. As a consequence, we establish a generalization of the powerful $S$-lemma, which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint. We then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint-range of three homogeneous quadratic functions using a regular cone. This yields Lagrange multiplier characterizations of global optimality for classes of trust-region type problems with two inequality constraints. Finally, we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving Z-matrices, which are matrices with nonpositive off diagonal elements, and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems.