Limiting behavior of the affine scaling continuous trajectories for linear programming problems

The authors consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, they show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmic barrier families of problems, the authors show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called ‘centered’ optimal solution of the dual problem. They also present results related to asymptotic direction of the affine scaling trajectories. The authors briefly discuss how to apply the present results to linear programs formulated in formats different from the standard form. Finally, they extend the results to the primal-dual affine scaling algorithm.