On the computation of weighted analytic centers and dual ellipsoids with the projective algorithm

The primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where a weighted Karmarkar potential function is used. This potential is defined with respect to a strict lower bound to the optimum. The minimization of htis potential when the lower bound is kept fixed, yields a primal and a dual feasible solution. The dual solution is the weighted analytic center of a certain dual polytope. Finally a pair of homothetic dual ellipsoids that extends results obtained by Sonnevend, Todd, Ye, Freund and Anstreicher are exhibited.