Approach Method Of Quadratic Constraint Quadratic Programming Based On Generalized Eigenpairs

Quadratically constrained quadratic program has been worth study as a kind of optimization problem, because it could not only be used to solve practical problems such as engineering design,production scheduling and market economy, and a lot of Nonlinear problem can be converted into this kind of problem. So discussion on the quadratically constrained quadric programming problem, from both theory and practice, can not only help people deal with practical problems, but also contribute to theoretical research.In this paper we discuss the regularity of positive semidefinite matrix pencils, and derive several nonhomogeneous versions of S-lemma with equality for the quadratic programming with a single quadratic equality constraint. As an application, we propose a new descent method for the optimization problem by approximating a sequence of generalized eigenpairs of symmetric positive semidefinite matrix pencils. At the cost of storage of an extra iterate, a small size numerical example shows that this algorithm achieves better performance and cheaper computation than the classical descent methods. The algorithm will stop after one step in the two-dimensional quadratic case.