对广义互补问题迭代方法收敛性分析的统一途径

1．介绍非线性互补问题，记作NCP（F0）；是指求XRn使得其中的：RNCP有各种不同形式的推广．例如，一种广泛的互补问题，记作NCP（F1，F2，…，FN），是指求xRn使得其中马：＊”＋＊”，马杜）是马（x）的第7个分量·NCP（凡）的发展及其应用在数学规划领域己经有很长的历...     

求解摩擦接触问题的一个非内点光滑化算法

给出了一个求解三维弹性有摩擦接触问题的新算法,即基于NCP函数的非内点光滑化算法．首先通过参变量变分原理和参数二次规划法,将三维弹性有摩擦接触问题的分析归结为线性互补问题的求解;然后利用NCP函数,将互补问题的求解转换为非光滑方程组的求解;再用凝聚函数对其进行光滑化,最后用NEWTON法解所得到的光滑非线性方程组．方法具有易于理解及实现方便等特点．通过线性互补问题的数值算例及接触问题实例证实了该算法的可靠性与有效性．     

非线性互补问题的一种不可行非内点连续算法

基于Chen-Harker—Kanzow-Smale光滑函数，对单调非线性互补问题NCP(f)给出了一种不可行非内点连续算法，该算法在每次迭代时只需求解一个线性等式系统，执行一次线搜索，算法在NCP(f)的解处不需要严格互补的条件下，具有全局线性收敛性和局部二次收敛性．     

广义非线性最小二乘问题的两个新方法

1．引言讨论如下的广义非线性最小二乘问题其中为常数（i＝1~m），W由于此问题的特殊形式，将此问题转化为如下两个子问题进行求解比较有效[1]子问题1．对每一固定的X，解得子问题2。对子问题1的解，解对两个子问题的求解，[1]中给出了一种有效的方法。然而在两个子问题的已有求解方法中，关于方法收敛速度的讨论非常少见，本文给出了求解这两个子问题的两个算法，并证明了算法的超线性收敛性．为书写简单，以下约定：一个符号在（，L）处的值略去（，L），如V‘F＝＊‘列X，L）等·一个具有上标k和＊的符号分别表示其在（x‘，t‘）和…     

广义Newton法求解非线性互补问题

１引言考虑非线性互补问题ＮＣＰ（ｆ）：的求解，即我们要寻求某ｘ∈Ｒｎ，使其满足（１．１）．其中映射ｆ：Ｒｎ→Ｒｎ为具有连续Ｆ－导数的非线性映射．众所周知，问题（１．ｌ）可以等价地转化为Ｂ－可微方程组：求解，其中：容易证明，由（１．３）定义的映射Ｇ处处Ｂ－可微，且其在点ｘ∈Ｒｎ处的Ｂ－导数ＢＧ（ｘ）为而对于问题（１．２）（１．３），我们希望直接用经典的广义Ｎｅｗｔｏｎ法进行求解．但是，由于由（１．３）定义映射Ｇ在（１．１）的解ｘ∈Ｒｎ处，没有可逆的强Ｆ－导数存在，因此，关于算法（１．５）（１．６）…     

解一般线性规划逆问题的一个O（n^3L）算法

本文讨论了一般线性规划逆问题在各种情况下的求解，并基于解凸二次规划的原对偶内点算法，给出了一个O（n3L）算法和一个实用算法．     

大型稀疏线性互补问题的行作用法

且引言考虑线性互补问题＊＊P（q，M）：求X二（X；，x。，…，x。厂E”使得x＞O，训x）E＊x＋g＞o，／U（X）一O（1）其中M一（m；。）为nXn矩阵（不必对称），q一切，q。，…，q。）rER“为给定常向量．通常情况下已有求解LCP（q，M）的若干著名算法［‘－’j．本文提出求解LCP（q，M）的一种新算法一行作用法，方法具有如下特点：（i）每次迭代只需n个简单的投影运算，每次投影只涉及矩阵M的一行；（n）生成新的迭代点x‘“‘时只利用前次迭代点／；（iii）对矩阵M不实施任何整体运算．因而适合于求解大型（巨型）稀疏问题，且…     

求解一类非单调线性互补问题的路径跟踪法及其计算复杂性

１．引言及记号 线性互补问题的一般形式是;求（ｘ,ｓ） 　　　　　　　 使其中 众所周知,当Ω＋非空时,单调线性互补问题可在多项式时间内求解,而且人们已经设计出了多种求解单调线性互补问题的有效的内点算法（见［１］和［７］）．然而,对于求解非单调线性互补问题的内点算法的研究可以说才刚刚开始．文［２］讨论了当Ｍ为Ｐ矩阵时问题（１）的中心路径的存在唯一性;文［３］给出了设计求解一类非单调线性互补问题的内点算法的一般框架;文［４］给出了求解一类非单调线性互补问题的一种势能函数约减法并讨论了其算法的计算复杂…     

单峰函数最优化问题的进化策略

１．引言 考虑无约束优化问题ｆ（ｘ）,其中ｆ（ｘ）为单峰函数．这类优化问题,不仅包含具有某种凸性的函数的极小化问题,而且包含其它许多问题,例如相容的和不相容的线性方程组的求解,也都可以归结为这类优化问题． 如果函数ｆ（ｘ）的性态良好,各类以梯度为基础的算法无疑是求解问题（Ｐ）的首选方法．假若问题（Ｐ）不可微,或者虽然可微,但 ｆ（ｘ）的 Ｈｅｓｓｉａｎ阵高度病态,则应该另辟蹊径． 近年来颇受人们重视的进化类算法,由于不使用梯度,计算过程对函数的性态依赖性较小,具有适应范围广、鲁棒性强的优点,而且特别…     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

A GLOBALal DERIVTIVE-FREE DESCENT METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

1. IntroductionConsider the nonlinear complementarity problem (NCP for short), which is to findan x E M" such thatwhere F: Wu - ac and the inequalities are taken componentwise. This problem havemany important applications in various fields. [13, 7, 5].Due to the less storage in computation, derivative--free descent method, which meansthe search direction used does not involye the Jacobian matrix of F, is popular infinding solutions of nonlinear complementarity Problems. We briefly view som…     

A penalty method for a finite-dimensional obstacle problem with derivative constraints

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.     

On construction of new NCP functions

We report a new method to construct complementarity functions for the nonlinear complementarity problem (NCP). Basic properties related to growth behavior, convexity and semismoothness of the newly discovered NCP functions are proved. We also present some variants, generalizations and other transformations of these NCP functions. Finally, we propose some interesting research directions that can be explored in the NCP research.     

A Globally Derivative-Free Descent Method for Nonlinear Complementarity Problems

Based on a class of functions, which generalize the squared Fischer-Burmeister NCP function and have many desirable properties as the latter function has, we reformulate nonlinear complementarity problem (NCP for short) as an equivalent unconstrained optimization problem, for which we propose a derivative-free descent method in monotone case. We show its global convergence under some mild conditions. If $F$, the function involved in NCP, is $R_0$-function, the optimization problems has bounded level sets. A local property of the merit function is discussed. Finally,we report some numerical results.     

Properties of Restricted NCP Functions for Nonlinear Complementarity Problems

In this paper, we study restricted NCP functions which may be used to reformulate the nonlinear complementarity problem as a constrained minimization problem. In particular, we consider three classes of restricted NCP functions, two of them introduced by Solodov and the other proposed in this paper. We give conditions under which a minimization problem based on a restricted NCP function enjoys favorable properties, such as equivalence between a stationary point of the minimization problem and the nonlinear complementarity problem, strict complementarity at a solution of the minimization problem, and boundedness of the level sets of the objective function. We examine these properties for three restricted NCP functions and show that the merit function based on the restricted NCP function proposed in this paper enjoys favorable properties compared with those based on the other restricted NCP functions.     

On stationary points of the implicit Lagrangian for nonlinear complementarity problems

Mangasarian and Solodov have recently introduced an unconstrained optimization problem whose global minima are solutions of the nonlinear complementarity problem (NCP). In this paper, we show that, if the mapping involved in NCP has a positive-definite Jacobian, then any stationary point of the optimization problem actually solves NCP. We also discuss a descent method for solving the unconstrained optimization problem.The authors are indebted to a referee for a helpful suggestion that led them to develop the descent method described in Section 3. They are grateful to Professor F. Facchinei, who kindly pointed out an error in the proof of Theorem 2.3 in an earlier version of the paper. The also thank Professor P. Tseng for a discussion on Theorem 3.1.     

Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only. This work is supported by the Research Grant Council of Hong Kong This paper is dedicated to Alex Rubinov on the occasion of his 65th Birthday     

NCP Functions Applied to Lagrangian Globalization for the Nonlinear Complementarity Problem

Based on NCP functions, we present a Lagrangian globalization (LG) algorithm model for solving the nonlinear complementarity problem. In particular, this algorithm model does not depend on some specific NCP function. Under several theoretical assumptions on NCP functions we prove that the algorithm model is well-defined and globally convergent. Several NCP functions applicable to the LG-method are analyzed in details and shown to satisfy these assumptions. Furthermore, we identify not only the properties of NCP functions which enable them to be used in the LG method but also their properties which enable the strict complementarity condition to be removed from the convergence conditions of the LG method. Moreover, we construct a new NCP function which possesses some favourable properties.     

The non-interior continuation methods for solving theP 0 function nonlinear complementarity problem

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP ₀ function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP ₀ function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP     

The non-interior continuation methods for solving theP0 function nonlinear complementarity problem

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving theP ₀ function nonlinear complementarity problem ( NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving theP ₀ function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP