A generic kernel function for interior point methods

In this paper, a new class of kernel functions is introduced. We show that most existing kernel functions belong to this class. All functions in the new class are eligible in the sense of Bai et al. (SIAM J Optim 15(1):101–128, 2004), and hence the analysis of the resulting interior-point methods can follow the scheme proposed in Bai et al. (2004). We introduce five new kernel functions and by using this scheme we show that primal-dual IPMs based on these functions enjoy the best known iteration bound for large-update methods, i.e., $$O(\sqrt{n}\log n\log \frac{n}{\epsilon }).$$ Finally, to demonstrate the efficiency of IPMs based on the new kernel functions, some numerical results are provided.