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

Pravin M. Vaidya

Chapter Chapter 5 in Progress in Mathematical Programming, 1989, pp 79-90 from Springer

Abstract: 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.

Date: 1989
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DOI: 10.1007/978-1-4613-9617-8_5

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