Central Tehran Branch, Islamic Azad UniversityJournal of Linear and Topological Algebra (JLTA)2252-0201080220190601On the duality of quadratic minimization problems using pseudo inverses133143665181END. PappasDepartment of Statistics, Athens University of Economics and Business, 76 Patission Str 10434, Athens, Greece0000-0001-9415-1392G. DomazakisDepartment of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Iroon Polytexneiou 9, 15780 Zografou, Athens, GreeceJournal Article20180902In 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.http://jlta.iauctb.ac.ir/article_665181_6cbe7a9128b86e3f4cca4160c123b0ce.pdf