Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa = z, X ∈ Rm×n, m ≥ n, a ∈ R, z ∈ R. This problem is equivalent to the problem of determining a best LAD-hyperplane x → ax, x ∈ R on the basis of given data (xi, zi), xi = (x (i) 1 , . . . , x (i) n ) ∈ R, zi ∈ R, i = 1, . . . ,m, whereby the minimizing functional is of the form F (a) = ‖z− Xa‖1 = m ∑