Linear pencils and quadratic programming problems with a quadratic constraint

Abstract

Given bounded selfadjoint operators $A$ and $B$ acting on a Hilbert space $\mathcal{H}$, consider the linear pencil $P(\lambda )=A+\lambda B$, $\lambda \in \mathbb{R}$. The set of parameters $\lambda$ such that $P(\lambda )$ is a positive (semi)definite operator is characterized. These results are applied to solving a quadratic programming problem with an equality quadratic constraint (or a QP1EQC problem).

24 pages. Comments are welcomed!