Optimal Solution Properties of an Overdetermined System of Linear Equations

Abstract

The paper considers the solution properties of an overdetermined system of linear equations in a given norm. The problem is observed as a minimization of the corresponding functional of the errors. Presenting the main results of $p$ norm it is shown that the functional is convex. Following the convex properties we examine minimization properties showing that the problem possesses regression, scale, and affine equivariant properties. As an example we illustrated the problem of finding weighted mean and weighted median of the data.

The Component Weighted Median Absolute Deviations Problem

“…where med(w, z) is called the Weighted Median of Data [4,8]. In general, the global minimum of f is not always unique, what confirms the well-known LAD regression properties of a possibility of a multiple solution existence.…”

“…Considering the minimization of the function f , which implies that the LAD regression can be observed as a problem of determining the appropriate global minimum of f . Furthermore, the convexity properties of the function f , can also be taken into consideration of the observed problem [4,5]. In that sense, we may conclude that the global minimum of f always exists, which indicates the existence of the LAD solution [10], and thus it can be written that…”