On the duality of quadratic minimization problems using pseudo inverses

In this paper we consider the minimization of a positive semidenite quadratic form, having a singular corresponding matrix H. We state the dual formulation of the orig- inal problem and treat both problems only using the vectors x 2 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 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 sucient conditions for our solution to exist using the well known KKT conditions.