Full-Newton step polynomial-time methods for linear optimization based on locally self-concordant barrier functions C. Roos ‡ H. Mansouri ∗‡ March 28, 2006 ‡ Department of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands e-mail: h.mansouri@ewi.tudelft.nl, c.roos@ewi.tudelft.nl Abstract Recently kernel function-based barrier functions (including so-called self-regular kernel functions) have been used to improve the iteration bound for large-update path-following interior-point methods from O(n log n ε ) to O( √ n(log n) log n ε ). It was observed (most of the time after a tedious analysis, for each kernel function separately) that the iteration bounds for small-update methods based on these barrier functions were always O( √ n log n ε ). In this paper this phenomenon is explained. This is achieved by localizing the notion of self- concordant barrier functions. It is shown that barrier functions based on kernel functions are not self-concordant on the whole domain of the problem under consideration, but they are locally self-concordant on the central path, and also in the small neighborhood of the central path where the iterates of a full-Newton step algorithm occur. Using the powerful tools for analyzing Newton’s method provided by the theory of self-concordant functions we show that under very weak conditions on the kernel function we can obtained the desired iteration bound. A surprising result is that for the kernel functions analyzed in this paper the complexity number on (or close to) the central path is smaller than the complexity number for the logarithmic barrier function. Keywords: Linear optimization, primal-dual method, self-concordance, kernel function, polynomial complexity. AMS Subject Classification: 90C05, 90C51 1 Introduction Since the seminal work of Karmarkar [10], many researchers have proposed and analyzed various Interior-point methods (IPMs) for Linear Optimization (LO) and a large amount of results have ∗ On leave from Department of Mathematical Science, Shahrekord University, P.O. Box 115, Shahrekord, Iran. 1