A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

求解线性互补问题的一种新算法

本文针对线性互补问题,提出了与其等价的非光滑方程的逐次逼近阻尼牛顿法,并在一定条件下证明了该算法具有的全局收敛性．同时给出了一些数值例子,得到很好的数值结果．     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

Generalized Newton’s method based on graphical derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton’s method for nonsmooth Lipschitzian equations.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

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.     

Some quadrature-based versions of the generalized Newton method for solving nonsmooth equations

In this paper, modifications of a generalized Newton method based on some rules of quadrature are studied. The methods considered are Newton-like iterative schemes for numerical solving systems of nonsmooth equations. Some mild conditions are given that ensure superlinear convergence to a solution. Moreover, a parameterized version of the midpoint version is presented. Finally, results of numerical tests are established.     

Globalizing a nonsmooth Newton method via nonmonotone path search

We give a framework for the globalization of a nonsmooth Newton method. In part one we start with recalling B. Kummer’s approach to convergence analysis of a nonsmooth Newton method and state his results for local convergence. In part two we give a globalized version of this method. Our approach uses a path search idea to control the descent. After elaborating the single steps, we analyze and prove the global convergence resp. the local superlinear or quadratic convergence of the algorithm. In the third part we illustrate the method for nonlinear complementarity problems.     

Tensor complementarity problems: the GUS-property and an algorithm

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.     

非线性互补问题非精确逐次逼近法的全局收敛性

本文针对非线性互补问题,提出了与其等价的非光滑方程的非精确逐次逼近算法,并在一定条件下证明了该算法的全局收敛性。     

A perturbed version of an inexact generalized Newton method for solving nonsmooth equations

In this paper, we present the combination of the inexact Newton method and the generalized Newton method for solving nonsmooth equations F(x)?=?0, characterizing the local convergence in terms of the perturbations and residuals. We assume that both iteration matrices taken from the B-differential and vectors F(x ^(k)) are perturbed at each step. Some results are motivated by the approach of C?tina? regarding to smooth equations. We study the conditions, which determine admissible magnitude of perturbations to preserve the convergence of method. Finally, the utility of these results is considered based on some variant of the perturbed inexact generalized Newton method for solving some general optimization problems.     

半定优化与半光滑牛顿算法

本文讨论半光滑牛顿算法的基本概念与其在求解半定优化问题中的应用．特别地，该算法可用于求解线性或非线性半定互补问题．本文同时综述最近在矩阵方程，增广拉格朗日公式和半定优化稳定性方面的、源于半光滑牛顿算法的理论成果．     

Nonsingularity of FB system and constraint nondegeneracy in semidefinite programming

For a KKT point of the linear semidefinite programming (SDP), we show that the nonsingularity of the B-subdifferential of Fischer-Burmeister (FB) nonsmooth system, the nonsingularity of Clarke’s Jacobian of this system, and the primal and dual constraint nondegeneracies, are all equivalent. Also, each of these conditions is equivalent to the nonsingularity of Clarke’s Jacobian of the smoothed counterpart of FB nonsmooth system, which particularly implies that the FB smoothing Newton method may attain the local quadratic convergence without strict complementarity assumption. We also report numerical results of the FB smoothing method for some benchmark problems.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

Local convergence of quasi-Newton methods for B-differentiable equations

We study local convergence of quasi-Newton methods for solving systems of nonlinear equations defined by B-differentiable functions. We extend the classical linear and superlinear convergence results for general quasi-Newton methods as well as for Broyden's method. We also show how Broyden's method may be applied to nonlinear complementarity problems and illustrate its computational performance on two small examples.     

求解非线性互补问题的逐次逼近阻尼牛顿法

针对非线性互补问题,提出了与其等价的非光滑方程的逐次逼近阻尼牛顿法,并 在一定条件下证明了该算法的全局收敛性.数值结果表明,这一算法是有效的.     

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.This work has been supported in parts by a grant from the German Scientific Foundation and by a grant from the Czech Academy of Sciences.     

Convergence analysis of nonsmooth equations for the general nonlinear complementarity problem

This paper deals with a general nonlinear complementarity problem, where the underlying functions are assumed to be continuous. Based on a nonlinear complementarity function, it is transformed into a system of nonsmooth equations. Then, two kinds of approximate Newton methods for the nonsmooth equations are developed and their convergence are proved. Finally, numerical tests are also listed.     

An interior point potential reduction method for constrained equations

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algorithm for solving several classes of convex programs. The work of this author was based on research supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and the Office of Naval Research under grant N00014-93-1-0228. The work of this author was based on research supported by the National Science Foundation under grant DMI-9496178 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

求解无限维非线性互补问题的光滑化牛顿法

研究一类无限维非线性互补问题的光滑化牛顿法.借助于非线性互补函数,将无限维非线性互补问题转化为一个非光滑算子方程.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化牛顿法具有超线性收敛性.