On minimization of quadratic functions over closed convex sets in Hilbert spaces

Quadratic programming problems are of primary importance in various applications and arise as subproblems in many optimization algorithms. In this paper, we investigate quadratic programming problems in Hilbert spaces. By utilizing the Legendre property of quadratic forms and an asymptotically linear set with respect to a cone, we establish a sufficient condition for the existence of solutions to the considered problems through a Frank-Wolfe type theorem. The proposed condition is based on the special structure of Hilbert spaces, extending the applicability of quadratic programming methods. Finally, we provide a numerical example to illustrate the results obtained and demonstrate that existing approaches cannot be applied in certain cases.