A variant of Karmarkar's linear programming algorithm for problems with some unrestricted variables

A variant of Karmarkar’s projective linear programming algorithm that can be used on problems with some unrestricted variables is considered. The variant is derived in two ways. One derivation involves eliminating the unconstrained variables, and the other involves solving a constrained least squares problem. The results of Gonzaga are used to show that our algorithm converges in $O(nq)$ iterations where n is the number of nonnegative variables and q is the precision required in the objective function value.