A polynomial-time algorithm for linear optimization based on a new simple kernel function

Abstract

We present a new barrier function, based on a kernel function with a linear growth term and an inverse linear barrier term. Existing kernel functions have a quadratic (or higher degree) growth term, and a barrier term that is either transcendent (e.g. logarithmic) or of a more complicated algebraic form. So the new kernel function has the simplest possible form compared with all existing kernel functions. It is shown that a primal–dual interior-point algorithm for linear optimization (LO) based on the new kernel function has the complexity bounds O(n) log(n/ϵ) and O(√n) log(n/ϵ) for large- and small-update methods, respectively. These complexity bounds are the same as those for the classical algorithm based on the logarithmic barrier function. *Y. Q. Bai is on leave from the Department of Mathematics, Shanghai University, Shanghai 200436, China. E-mail: y.bai@its.tudelft.nl