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

  title={Generalized Newton and NCP-methods: convergence, regularity, actions},
  author={Bernd Kummer},
  journal={Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
  • B. Kummer
  • Published 2000
  • Mathematics, Computer Science
  • Discussiones Mathematicae. Differential Inclusions, Control and Optimization
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… 

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

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

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

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

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

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.

A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control

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

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

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.


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


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



Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis

This paper presents sufficient and necessary conditions for the convergence of Newton’s method based on generalized derivatives. These conditions require uniform injectivity of the derivatives as

Generalized Kojima–Functions and Lipschitz Stability of Critical Points

The main purpose is to point out how different concepts of generalized derivatives lead to characterizations of different Lipschitz properties of the critical point or the stationary solution set maps.

Metric regularity: characterizations, nonsmooth variations and successive approximation ∗

Metric regularity of (multi-) functions is characterized in terms of some uniform lower semicontinuity as well as by means of Ekeland points of related functionals. Specializations and consequences

On NCP-Functions

In this paper we reformulate several NCP-functions for the nonlinear complementarity problem (NCP) from their merit function forms and study some important properties of these NCP-functions. We point

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior and is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed non linear complementarity problem.

Strongly Regular Generalized Equations

A regularity condition is introduced for generalized equations and it is shown to be in a certain sense the weakest possible condition under which the stated properties will hold.

Solution of monotone complementarity problems with locally Lipschitzian functions

IfF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F) and the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.

A nonsmooth version of Newton's method

It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth, and the extended Newton's method can be used in the augmentedlagrangian method for solving nonlinear programs.

Strongly Stable Stationary Solutions in Nonlinear Programs.

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

\rm Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson's notion of strong regularity, as a criterion for the solution set to be a