Abstract: An Analysis of Newton's Method for Equivalent Karush--Kuhn--Tucker Systems

In this paper we analyze the application of Newton's method to the solution of systems of nonlinear equations arising from equivalent forms of the first--order Karush--Kuhn--Tucker necessary conditions for constrained optimization. The analysis is carried out by using an abstract model for the original system of nonlinear equations and for an equivalent form of this system obtained by a reformulation that appears often when dealing with first--order Karush--Kuhn--Tucker necessary conditions. The model is used to determine the quantities that bound the difference between the Newton steps corresponding to the two equivalent systems of equations. The model is sufficiently abstract to include the cases of equality--constrained optimization, minimization with simple bounds, and also a class of discretized optimal control problems.