Newton Method For Nonsmooth Equations Research Articles

Newton Differentiability of Convex Functions in Normed Spaces and of a Class of Operators

Newton differentiability is an important concept for analyzing generalized Newton methods for nonsmooth equations. In this work, for a convex function defined on an infinite-dimensional space, we discuss the relation between Newton and Bouligand differentiability and upper semicontinuity of its subdifferential. We also construct a Newton derivative of an operator of the form $(Fx)(p) = f(x,p)$ for general nonlinear operators $f$ that possess a Newton derivative with respect to $x$ and also for the case where $f$ is convex in $x$.

Convergence of the modified SOR–Newton method for non-smooth equations

Motivated by Chen [On the convergence of SOR methods for nonsmooth equations. Numer. Linear Algebra Appl. 9 (2002), pp. 81–92], in this paper, we further investigate a modified SOR–Newton (MSOR–Newton) method for solving a system of nonlinear equations F(x)=0, where F is strongly monotone and locally Lipschitz continuous but not necessarily differentiable. The convergence interval of the parameter in the MSOR–Newton method is given. Compared with that of the SOR–Newton method, this interval can be enlarged. Furthermore, when the B-differential of F(x) is difficult to compute, a simple replacement can be used, which can reduce the computational load. Numerical examples show that at the same cost of computational complexity, this MSOR–Newton method can converge faster than the corresponding SOR–Newton method by choosing a suitable parameter.

A parametrized newton method for nonsmooth equations with finitely many maximum functions

In this paper we propose a parametrized Newton method for nonsmooth equations with finitely many maximum functions. The convergence result of this method is proved and numerical experiments are listed.

Tame functions are semismooth

Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness” assumption. In this work we prove that locally Lipschitz functions definable in an o-minimal structure (in particular semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being γ-order semismooth, where γ is a positive parameter. As an application of this new estimate, we prove that the error at the kth step of the Newton method behaves like $$O(2^{-{(1+\gamma)}^k})$$.

An approximate Newton method for non-smooth equations with finite max functions

A new version of finite difference approximation of the generalized Jacobian for a finite max function is constructed. Numerical results are reported for the generalized Newton methods using this approximation.

A Globally Convergent Smoothing Newton Method for Nonsmooth Equations and Its Application to Complementarity Problems

The complementarity problem is theoretically and practically useful, and has been used to study and formulate various equilibrium problems arising in economics and engineerings. Recently, for solving complementarity problems, various equivalent equation formulations have been proposed and seem attractive. However, such formulations have the difficulty that the equation arising from complementarity problems is typically nonsmooth. In this paper, we propose a new smoothing Newton method for nonsmooth equations. In our method, we use an approximation function that is smooth when the approximation parameter is positive, and which coincides with original nonsmooth function when the parameter takes zero. Then, we apply Newton's method for the equation that is equivalent to the original nonsmooth equation and that includes an approximation parameter as a variable. The proposed method has the advantage that it has only to deal with a smooth function at any iteration and that it never requires a procedure to decrease an approximation parameter. We show that the sequence generated by the proposed method is globally convergent to a solution, and that, under semismooth assumption, its convergence rate is superlinear. Moreover, we apply the method to nonlinear complementarity problems. Numerical results show that the proposed method is practically efficient.

Solving variational inequality problems via smoothing-nonsmooth reformulations

It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent Newton methods for nonsmooth equations have been well established via the concept of semismoothness. When the constraint set of the variational inequality problem is a rectangle, several locally convergent Newton methods for the reformulated nonsmooth equations can also be globalized. In this paper, our main aim is to provide globally and locally high-order convergent Newton methods for solving variational inequality problems with general constraints. To achieve this, we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations, which have desirable properties for the design of Newton methods. We then reformulate the variational inequality problems as equivalent smoothing-nonsmooth equations and apply Newton-type methods to solve the latter systems, and so the variational inequality problems. Stronger convergence results have been obtained.

Eine Methode zur Verifizierung von Lösungen nicht-differenzierbarer Gleichungen

This paper proposes a verification method for the existence of solutions of nonsmooth equations. We generalize the Krawczyk operator to nonsmooth equations by using the mean-value theorem for nonsmooth functions. We establish a semi-local convergence theorem for the generalized Newton method for nonsmooth equations. The proposed method is a combination of the generalized Krawczyk operator and the semi-local convergence theorem.

A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.