The Best Approximate Solution of the Inconsistent Linear System via a Quadratic Penalty Function

Abstract

In this paper, the inconsistent linear system of \(m\) equations in \(n\) unknowns is formulated as a quadratic programming problem, and the best approximate solution with the minimum norm for the inconsistent system of the linear equations is investigated using the optimality conditions of the quadratic penalty function (QPF). In addition, several algebraic characterizations of the equivalent cases of the QPF are given using the orthogonal decompositions of the coefficient matrices obtained from optimality conditions, and analytic results we obtained are satisfied with numerical examples.