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We introduce a new barrier function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, its iteration bound, O √ n log n log n ε , is as good as it can be: it is the best known bound for large-update methods. The recently introduced notions of superconvexity and exponential convexity are crucial in the analysis.

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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 classical logarithmic function, the prototype selfregular function, and non-self-regular kernel functions as special cases. The analysis of the algorithms in the paper follows the same line of arguments as in Bai et al. (SIAM J. Optim. 15:101-128, 2004), where a variety of non-self-regular kernel functions were considered including the ones with linear and quadratic growth terms. However, the important case when the growth term is between linear and quadratic was not considered. The goal of this paper is to introduce such class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. They match the currently best known iteration bounds for the prototype self-regular function with quadratic growth term, the simple nonself-regular function with linear growth term, and the classical logarithmic kernel function. In order to achieve these complexity results, several new arguments had to be used.

We introduce a new parametric kernel function, which is a combination of the classic kernel function and a trigonometric barrier term, and present various properties of this new kernel function. A class of large- and small-update primal-dual interior-point methods for linear optimization based on this parametric kernel function is proposed. By utilizing the feature of the parametric kernel function, we derive the iteration bounds for large-update methods,O(n2/3log⁡(n/ε)), and small-update methods,O(nlog⁡(n/ε)). These results match the currently best known iteration bounds for large- and small-update methods based on the trigonometric kernel functions.