On the duality of quadratic minimization problems using pseudo inverses

Journal of Linear and Topological Algebra

مقاله 6، دوره 08، شماره 02، شهریور 2019، صفحه 133-143   

نوع مقاله: Research Paper

نویسندگان

D. Pappas^{* 1}؛ G. Domazakis²

¹Department of Statistics‎, Athens University of Economics and Business‎, 76 Patission Str 10434‎, Athens‎, Greece

²Department of Mathematics‎, ‎School of Applied Mathematics and Physical Sciences‎, ‎National Technical University of Athens‎, ‎Iroon Polytexneiou 9‎, ‎15780 Zografou‎, ‎Athens‎, ‎Greece

چکیده

‎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.

کلیدواژه‌ها

Moore-Penrose inverse؛ general normal equation؛ constrained optimization؛ Lagrange multipliers؛ duality principle؛ KKT conditions

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