Local Behavior of the Newton Method on Two Equivalent Systems

Newton's method is a fundamental technique underlying many numerical methods for solving systems of nonlinear equations and optimization problems. However, it is often not fully appreciated that Newton's method can produce signi cantly di erent behavior when applied to equivalent systems, i.e., problems with the same solution but di erent mathematical formulations. In this paper, we investigate di erences in the local behavior of Newton's method when applied to two di erent but equivalent systems from linear programming: the optimality conditions of the logarithmic barrier function formulation, and the equations in the so-called perturbed optimality conditions. Through theoretical analysis and numerical results, we provide an explanation of why Newton's method performs more e ectively on the latter system.