Newton's method is a fundamental technique for approximating solutions of nonlinear equations. However, it is often not fully appreciated that the method can produce significantly different behavior when applied to equivalent systems. In this thesis, we investigate differences in local and global behavior of two well-known methods for constrained optimization: the Newton logarithmic barrier function method and the Newton primal-dual interior-point method. As we shall show, these two methods can be viewed as applying Newton's method to two different but equivalent systems. Through theoretical analysis and numerical experimentation, we show the Newton primal-dual method performs more effectively.