Linear programming by minimizing distances

To solve the linear program (LP): minimize c^Tl subject to Al+b≥0, for an n×d-matrix A, an n-vector b and a d-vector c, the positive orthant S and the plane E(t) are defined by S={(x1,x)∈ℝⁿ’+¹∈(x1,x)≥0∈, E(t)={(x1,x)∈ℝⁿ’+¹∈x1=¸-c^Tl+t, x=Al+b∈. First a geometric algorithm is given to determine d(E(t),S) for fixed t, where d(ë,ë) denotes euclidean distance. This algorithm is used to construct a second algorithm to find the minimal t with E(t)ℝSℝΘ, and thus solve LP. It is shown that all algorithms are finite.