on the duality of quadratic minimization problems using pseudo inverses

‎in this paper we consider the minimization of a positive semidefinite quadratic form‎, ‎having a singular corresponding matrix h‎. ‎we state the dual formulation of the original problem and treat both problems only using the vectors x∈n(h)⊥ instead of the classical approach of convex optimization techniques such as the null space method‎. ‎given this approach and based on the strong duality principle‎, ‎we provide a closed formula for the calculation of the lagrange multipliers lambda in the cases when (i) the constraint equation is consistent and (ii) the constraint equation is inconsistent‎, ‎using the general normal equation‎. ‎in both cases the moore-penrose inverse will be used to determine a unique solution of the problems‎. ‎in addition‎, ‎in the case of a consistent constraint equation‎, ‎we also give sufficient conditions for our solution to exist using the well known kkt conditions.