On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert Spaces

TL;DR

This paper investigates the existence of solutions for nonconvex quadratic programming problems in Hilbert spaces, providing sufficient conditions using properties of quadratic forms and operators, with special cases for linear constraints.

Contribution

It introduces new sufficient conditions for solution existence in nonconvex quadratic programming within Hilbert spaces, extending previous results to broader classes of problems.

Findings

Established sufficient conditions for solution existence

Derived results for problems with linear constraints

Utilized Legendre property and operator compactness

Abstract

In this paper, we consider the quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces. By using the Legendre property of quadratic forms or the compactness of operators in the presentations of quadratic forms, we establish some sufficient conditions for the solution existence of the considered problems. As special cases, we obtain some existence solution results for the quadratic programming problems under linear constraints in Hilbert spaces.