On the duality of quadratic minimization problems using pseudo inverses

Abstract

‎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 $x in mathcal{N}(H)^perp$ 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.