Newton methods for stationary points: an elementary view of regularity conditions and solution schemes

Abstract

In this article, we give an elementary view of Newton-type methods and related regularity conditions for a special class of nonsmooth equations arising from necessary optimality criteria for standard nonlinear programs. Different types of linearizations and parameterizations of these equations lead to different iteration schemes, where any abstract calculus of generalized derivatives for nonsmooth mappings is avoided. Based on a general local convergence result on (perturbed) Newton methods for solving Lipschitzian equations, we focus on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems for our special setting. We are particularly interested in certain parameterized Newton equations and in regularity conditions which are weaker than strong regularity.

Keywords:

• Generalized Newton methods
• Stationary solutions
• Nonlinear programs
• Regularity
• Particular realizations of Newton equations
• 90C31
• 49J40
• 49J52
• 26E25

Acknowledgement

The authors wish to thank Stephan Bütikofer (University of Zurich) and an anonymous referee for their critical comments.