A polynomial-time algorithm for linear optimization based on a new class of kernel functions

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the class of finite kernel functions by Y.Q. Bai, M.El Ghami and C. Roos [Y.Q. Bai, M. El Ghami, and C. Roos. A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (3) (2003) 766–782]. The proposed functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like the usual barrier functions. The goal of this paper is to investigate such a class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new arguments had to be used for the analysis. The iteration bound of large-update interior-point methods based on these functions and analyzed in this paper, is shown to be O ( n log n log n ϵ ) . For small-update interior-point methods the iteration bound is O ( n log n ϵ ) , which is currently the best-known bound for primal-dual IPMs. We also present some numerical results which show that by using a new kernel function, the best iteration numbers were achieved in most of the test problems.