A Primal-Dual Interior-Point Algorithm Based on a Kernel Function with a New Barrier Term

In this paper, we propose a path-following interior-point method (IPM) for solving linear optimization (LO) problems based on a new kernel function (KF). The latter differs from other KFs in having an exponential-hyperbolic barrier term that belongs to the hyperbolic type, recently developed by I. Touil and W. Chikouche \cite{filomat2021,acta2022}. The complexity analysis for large-update primal-dual IPMs based on this KF yields an $\mathcal{O}\left( \sqrt{n}\log^2n\log \frac{n}{\epsilon }\right)$ iteration bound which improves the classical iteration bound. For small-update methods, the proposed algorithm enjoys the favorable iteration bound, namely, $\mathcal{O}\left( \sqrt{n}\log \frac{n}{\epsilon }\right)$. We back up these results with some preliminary numerical tests which show that our algorithm outperformed other algorithms with better theoretical convergence complexity. To our knowledge, this is the first feasible primal-dual interior-point algorithm based on an exponential-hyperbolic KF.