Pappas, D., Domazakis, G. (2019). On the duality of quadratic minimization problems using pseudo inverses. Journal of Linear and Topological Algebra (JLTA), 08(02), 133-143.

D. Pappas; G. Domazakis. "On the duality of quadratic minimization problems using pseudo inverses". Journal of Linear and Topological Algebra (JLTA), 08, 02, 2019, 133-143.

Pappas, D., Domazakis, G. (2019). 'On the duality of quadratic minimization problems using pseudo inverses', Journal of Linear and Topological Algebra (JLTA), 08(02), pp. 133-143.

Pappas, D., Domazakis, G. On the duality of quadratic minimization problems using pseudo inverses. Journal of Linear and Topological Algebra (JLTA), 2019; 08(02): 133-143.

On the duality of quadratic minimization problems using pseudo inverses

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

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

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.

[1] A. B. Israel, Generalized inverses and the Bott-Duffin network analysis, J. Math. Anal. Appl. 7 (1963), 428-435.

[2] A. Ben-Israel, T. N. E. Grenville, Generalized Inverses: Theory and Applications, Springer-Verlag, Berlin, 2002.

[3] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

[4] S. L. Campbell, C. D. Meyer, Generalized Inverses of Linear Transformations, Dover Publications Inc, New York, 1991.

[5] W. S. Dorn, Duality in quadratic programming, Quarterly of Applied Mathematics. 18 (2) (1960), 155-162.

[6] C. W. Groetsch, Generalized inverses of linear operators, Marcel Dekker Inc, New York, 1977.

[7] D. Luenberger, Optimization by Vector Space Methods, Wiley Publ, New York, 1969.

[8] R. K. Manherz, S. L. Hakimi, The generalized inverse network analysis and quadratic-error minimization problems, IEEE Trans. Circuit Theory. (1969), 559-562.

[9] J. Nocedal, S. Wright, Numerical Optimization, Springer, 2006.

[10] D. Pappas, Minimization of constrained quadratic forms in Hilbert spaces, Ann. Funct. Anal. (2) (1) (2011), 1-12.

[11] D. Pappas, Restricted linear constrained minimization of quadratic functionals, Linear and Multilinear Algebra. 61 (10) (2013), 1394-1407.

[12] D. Pappas, A. Perperoglou, Constrained matrix optimization with applications, J. Appl. Math. Comput. 40 (2012), 357-369.

[13] H. Schwerdtfeger, Introduction to Linear Algebra and the Theory of Matrices, Noordhoff, Groningen, 1950.

[14] P. Stanimirovic, D. Pappas, S. Miljkovic, Minimization of quadratic forms using the Drazin-inverse solution, Linear and Multilinear Algebra. 62 (2) (2014), 252-266.

[15] P. Stoica, A. Jakobsson, J. Li, Matrix optimization result with DSP applications, Dig. Signal Proc. 6 (1996), 195-201.