MINIMIZATION OF CONSTRAINED QUADRATIC FORMS IN HILBERT SPACES

A common optimization problem is the minimization of a symmetric positive definite quadratic form < x, Tx > under linear constraints. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work at first we extend this result to infinite dimensional complex Hilbert spaces, where a generalization is given for positive operators not necessarily invertible, considering as constraint a singular operator. A new approach is proposed when T is positive semidefinite, where the minimization is considered for all vectors belonging to N(T)(perpendicular to).