A common problem encountered in the studies of the least squares problems is that of finding the minimum ℓ2 norm solution of a system of linear equations. In this paper, we consider an algorithm for computing the vector of minimum norm solution of a given system of linear inequalities. We replace the ℓ2 norm by $\ell^p\comma \, 1\lt p \lt\infty$ . Duality theorems and characterizations of the solution are given. The feasibility of the method is proved and some numerical experimentations are included.