A Locally Weil-Behaved Potential Function and a Simple Newton-Type Method for Finding the Center of a Polytope

Abstract

The center of a bounded full-dimensional polytope P = {x: Ax ≥ b} is the unique point ω that maximizes the strictly concave potential function $F(x) = \sum\nolimits_{i = 1}^m {\ln (a_i^T} x - {b_i})$ over the interior of P. Let x ₀ be a point in the interior of P. We show that the first two terms in the power series of F(x) at x ₀ serve as a good approximation to F(x) in a suitable ellipsoid around x ₀ and that minimizing the first-order (linear) term in the power series over this ellipsoid increases F(x) by a fixed additive constant as long as x ₀ is not too close to the center ω.