Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. Both in the algorithm and in its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds both for large- and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely $O(\sqrt{n}\,\log\frac{n}{\e})$. For large-update methods the best obtained bound is $O(\sqrt{n}\,(\log n)\,\log\frac{n}{\e})$, which is up till now the best known bound for such methods.
Citation
TU Delft, Mekelweg 4, Delft, NL