Large-step predictor-corrector interior point method for sufficient linear complementarity problems based on the algebraic equivalent transformation

Abstract

We introduce a new predictor-corrector interior-point algorithm for solving $P_{⁎}\left(\kappa \right)$-linear complementarity problems which works in a wide neighbourhood of the central path. We use the technique of algebraic equivalent transformation of the centering equations of the central path system. In this technique, we apply the function $\phi \left(t\right)=\sqrt{t}$ in order to obtain the new search directions. We define the new wide neighbourhood $D_{\phi }$. In this way, we obtain the first interior-point method, where not only the central path system is transformed, but the definition of the neighbourhood is also modified taking into consideration the algebraic equivalent transformation technique. This gives a new direction in the research of interior-point algorithms. We prove that the interior-point method has $O\left(\right(1+\kappa )n\log \left(\frac{{\left(x^0\right)}^T{s}^0}{ϵ}\right))$ iteration complexity. Furthermore, we show the efficiency of the proposed predictor-corrector algorithm by providing numerical results. To our best knowledge, this is the first predictor-corrector interior-point algorithm which works in the $D_{\phi }$ neighbourhood using $\phi \left(t\right)=\sqrt{t}$.