A dynamical systems perspective for mathematical programming

Thesis. Karmarkar\u27s algorithm to solve linear programs has renewed interest in interior point methods to solve mathematical programs. Recently, it has been observed that these interior point methods for mathematical programming generate a dynamic trajectory through the feasible region. However, these observations have been mainly of a theoretical nature. After examining the qualitative properties such as existence, uniqueness and convergence for the system of differential equations that characterizes the trajectory, we describe a method to calculate the asymptotic behavior of this trajectory. This gives the optimal solution of the mathematical program. Scholarly significance. In this dissertation we rigorously establish the relation between mathematical programs and dynamical systems. In particular, we show the correspondence between the optimal solution of mathematical programs and the stability of equilibrium or rest points of dynamical systems. This is of great significance as it provides an entirely new avenue for numerical methods to solve mathematical programs. For example, mathematical programs can be solved by converting them to systems of differential equations which are then solved by the method of successive linearization described herein