Minimization of quadratic forms using the Drazin-inverse solution

Abstract

We analyse the problem of constrained minimization of the real quadratic functional, subject to the inconsistent system of linear equations, where is a positive definite or positive semidefinite matrix. Both cases are analysed separately, and respective relationships have been established between the solution of the original problem and the Drazin-inverse solution of the equation. In the special case when is a positive definite and, we show that the solution of the problem can be represented as a particular-inverse solution.

Keywords:

• quadratic functional
• constrained optimization
• Moore–Penrose inverse
• Drazin inverse
• outer inverse

AMS Subject Classification:

• 47A05
• 47N10
• 15A09

Acknowledgments

The first author gratefully acknowledges the support from the Research Project 174013 of the Serbian Ministry of Science.