Interior-point algorithm for linear optimization based on a new kernel function

Abstract

In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barrier term. Iteration bounds both for large- and small-update methods are derived, namely, O(nlog(n/ɛ)) and \(O(\sqrt n \log (n/\varepsilon ))\). This new kernel function has simple algebraic expression and the proximity function has not been used before. Analogous to the classical logarithmic kernel function, our complexity analysis is easier than the other primal-dual interior-point methods based on logarithmic barrier functions and recent kernel functions.