The new full-Newton step interior-point algorithm for the Fisher market equilibrium problems based on a kernel function

Abstract

In this paper, we propose a full-Newton step interior-point method for the weighted linear complementarity problem (wLCP) model of the Fisher market equilibrium problem. The weighted complementarity problem (wCP) is a generalization of the complementarity problem (CP) with the nonnegative weight vector, where the zero on the right-hand side is replaced by the nonnegative weight vector. The importance of a wCP lies in the fact that many equilibrium problems in science, engineering, and economics can be reformulated as wCPs, which lead to the development of highly efficient algorithms. We extend a full-Newton step interior-point method (IPM) to the wLCP model of the Fisher problem. New search directions are obtained by using a kernel function in the scaled Newton system. The algorithm takes only full-Newton steps, thus, avoiding the calculation of the step size which is computationally advantageous. Under the suitable assumptions, the algorithm is shown to have global convergence and polynomial complexity. Some numerical results indicate the efficiency of the algorithm. To the best of our knowledge, this is the first full-Newton step IPM for Fisher problems, which uses the kernel function to obtain new search directions.