Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

Authors:
Harish J. Palanthandalam-Madapusi;Tobin H. Van Pelt;Dennis S. Bernstein
Affiliations:
Department of Mechanical and Aerospace Engineering, Syracuse University, Syracuse, USA 13244;i3D Technologies, LLC, Mill Valley, USA 94941;Department of Aerospace Engineering, The University of Michigan, Ann Arbor, USA 48109
Venue:
Journal of Global Optimization
Year:
2009

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Abstract

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.