Minimization of constrained quadratic forms in Hilbert spaces

Abstract

A common optimization problem is the minimization of a symmetric positive‎ ‎definite quadratic form $\langle x,Tx\rangle$ 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 ‎$\mathcal{N}(T)^\perp$‎.