Improved Bounds and Containing Ellipsoids in Karmarkar's Linear Programming Algorithm

Karmarkar's projective algorithm for linear programming provides not only primal solutions but dual solutions giving bounds on the optimal value. Here we show how improved bounds can be obtained at the expense of solving a two-dimensional linear programming problem at every iteration, and also how an ellipsoid containing all dual optimal solutions can be generated from available information. We also give the results of limited computational experiments related to these topics.