A wide neighborhood interior-point algorithm based on the trigonometric kernel function

Abstract

In this paper, a wide neighborhood path-following interior point algorithm for linear optimization (LO) is proposed that uses a trigonometric kernel function to get search directions. The method treats the Newton direction as the sum of two other directions, according to the negative and positive parts of the right-hand-side based on the kernel function. By choosing different and appropriate step sizes for these two directions, the iterates stop in the Ai–Zhang’s wide neighborhood. By an elegant analysis, we show that the method enjoys the low iteration bound of $${\mathcal {O}}(\sqrt{n}\log \frac{(x^0)^Ts^0}{\varepsilon })$$ , where n is the dimension of the problem and $$(x^0, s^0)$$ the initial interior point with $$\varepsilon $$ the required precision. In our knowledge, this result is the first instance of a wide neighborhood interior point method for LO which involving the trigonometric kernel function.