A Note on the Behavior of Karmarkar's Potential Function in Linear Programming

Abstract

In this note we express Karmarkar's potential function in terms of the geometric mean of the decision variables of the linear programming problem, and obtain bounds on it. We also study the behaviour of the gradient and the hessian of the potential function at the center of the simplex and observe that the sum of all entries of the gradient and the hessian matrices at the center of the simplex are zero; and the center of the simplex is a saddle point for the potential function. Finally, we prove that the β-superlevel set of the function G(x) is a convex set.