Pictures of Karmarkar's linear programming algorithm.

Karmarkar's linear programming algorithm handles inequality constraints by changing variables to make constraints about equally distant; it moves in the steepest-descent direction seen by the new variables. This paper summarizes four variants of Karmarkar's linear programming algorithm (primal affine, primal projective, dual affine, and dual projective), discusses depicting polytopes (feasible regions), and presents pictures illustrating the latter three variants. These pictures give an algorithm's eye view of the variable changes and provide visual verification of some theoretical results.