Linear pencils and quadratic programming problems with a quadratic constraint

TL;DR

This paper characterizes the set of parameters for which a linear operator pencil is positive semidefinite and applies these results to solve quadratic programming problems with a quadratic equality constraint.

Contribution

It provides a new characterization of parameter sets for positive semidefiniteness of operator pencils and applies this to quadratic programming with quadratic constraints.

Findings

Characterization of parameter sets for positive semidefiniteness

Application to quadratic programming with quadratic constraints

Method for solving QP1EQC problems

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).