Continuous Trajectories for Primal-Dual Potential-Reduction Methods

This article considers continuous trajectories of the vector fields induced by two different primal-dual potential-reduction algorithms for solving linear programming problems. For both algorithms, it is shown that the associated continuous trajectories include the central path and the duality gap converges to zero along all these trajectories. For the algorithm of Kojima, Mizuno, and Yoshise, there is a a surprisingly simple characterization of the associated trajectories. Using this characterization, it is shown that all associated trajectories converge to the analytic center of the primal-dual optimal face. Depending on the value of the potential function parameter, this convergence may be tangential to the central path, tangential to the optimal face, or in between.