On the Solution Existence of Nonconvex Quadratic Programming Problems in   Hilbert Spaces
  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. 
Vu Van Dong, Nguyen Nang Tam
math.OC