Abstract Primal–dual interior-point methods (IPMs) are the most efficient methods for a computational point of view. At present the theoretical iteration bound for small-update IPMs is better than the one for large-update IPMs. However, in practice, large-update IPMs are much more efficient than small-update IPMs. Peng et al. [14] , [15] proposed new variants of IPMs based on self-regular barrier functions and proved so far the best known complexity, e.g. O n log n log n ϵ , for large-update IPMs with some specific self-regular barrier functions. Recently, Roos et al. [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] proposed new primal–dual IPMs based on various barrier functions to improve the iteration bound for large-update methods from O n log n ϵ to O n log n log n ϵ . Motivated by their works we define a new barrier function and propose a new primal–dual interior point algorithm based on this function for linear optimization (LO) problems. We show that the new algorithm has O n log n log n ϵ iteration bound for large-update and O n log n ϵ for small-update methods which are currently the best known bounds, respectively.