# Generalized Newton and NCP-methods: convergence, regularity, actions

@article{Kummer2000GeneralizedNA, title={Generalized Newton and NCP-methods: convergence, regularity, actions}, author={Bernd Kummer}, journal={Discussiones Mathematicae. Differential Inclusions, Control and Optimization}, year={2000}, volume={20}, pages={209-244} }

Solutions of several problems can be modelled as solutions of nonsmooth equations. Then, Newton-type methods for solving such equations induce particular iteration steps (actions) and regularity requirements in the original problems. We study these actions and requirements for nonlinear complementarity problems (NCP’s) and Karush–Kuhn–Tucker systems (KKT) of optimization models. We demonstrate their dependence on the applied Newton techniques and the corresponding reformulations. In this way…

## 29 Citations

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

- Mathematics
- 2007

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…

### Generalized Newton-type methods for nonsmooth equations in optimization and complementarity problems

- Mathematics
- 2008

Several problems in variational analysis for e.g. necessary optimality conditions for nonlinear programs, solutions of variational inequalities with explicit equality/inequality constraints and…

### Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems

- MathematicsNumerische Mathematik
- 2008

Based on its first order necessary optimality conditions, a semismooth Newton method is proposed and its fast local convergence in function space as well as a mesh-independence principle for appropriate discretizations are proved.

### Introduction to Nonsmooth Analysis and Optimization

- Mathematics
- 2020

These notes aim to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable…

### Novel Concepts for Nonsmooth Optimization and their Impact on Science and Technology

- Mathematics
- 2010

The notion of Newton differentiability combined with path following is of central importance and it will be demonstrated how these techniques are applicable to problems in mathematical imaging, and variational inequalities.

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- Mathematics, Computer Science
- 2006

This work presents an alternative approach to the analysis of Newton’s method for function space problems involving semi-smooth Nemyckii operators, and derives second order sufficient conditions for problems, where the underlying PDE has poor regularity properties.

### Mesh independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems

- Mathematics
- 2007

A class of mixed control-state constrained optimal control problems for elliptic partial differential equations arising, for example, in Lavrentiev-type regularized state constrained optimal control…

### Globalizing a nonsmooth Newton method via nonmonotone path search

- MathematicsMath. Methods Oper. Res.
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A framework for the globalization of a nonsmooth Newton method that uses a path search idea to control the descent and analyzes and proves the global convergence resp.

### NUMERICAL SOLUTION OF OPTIMAL CONTROL AND INVERSE PROBLEMS IN NON-REFLEXIVE BANACH SPACES

- Mathematics
- 2012

Convex duality is a powerful framework for solving nonsmooth optimal control problems. However, for problems set in non-re exive Banach spaces such as L(Ω) or BV(Ω), the dual problem is formulated in…

### NONSMOOTH ANALYSIS AND OPTIMIZATION

- Mathematics
- 2017

These lecture notes for a graduate course cover generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for nondifferentiable optimization problems…

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