一致P-函数非线性互补问题的宽邻域不可行内点算法及其计算复杂性

1引言与记号单调线性互补问题和线性规划问题的原始-对偶路径跟踪算法,1989年的文献[1、2]分别首先提出。以后又出现了一些改进的算法。早期的原始-对偶路径跟踪算法及其改进算法的迭代点列大都是在包含中心路径C的一个2-范数的窄邻域里,这种可行内点算法通常理论上具有最好的迭代复杂性O(n~(1/2)L),但是由于窄邻域极大地限制了迭代步长,实     

基于一类新方向的宽邻域路径跟踪内点算法

基于一类带有参数theta的新方向, 提出了求解单调线性互补问题的宽邻 域路径跟踪内点算法, 且当theta=1时即为经典牛顿方向. 当取theta为与问题规模 n无关的常数时, 算法具有O(nL)迭代复杂性, 其中L是输入数据的长度, 这与经典宽邻 域算法的复杂性相同; 当取theta=sqrt{n/betatau}时, 算法具有O(sqrt{n}L)迭代复杂性, 这里的beta, tau是邻域参数, 这与窄邻域算法的复杂性相同. 这是首次研究包括经典宽邻域路径跟踪算法的一类内点算法, 给出了统一的算法框架和收敛性分析方法.     

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…     

基于代数等价路径的一致P-函数非线性互补问题的可行内点算法

对一致P-函数非线性互补问题，提出了一种新的基于代数等价路径的可行内点算法，并讨论了计算复杂性．该算法可以在任一内部可行点启动，并且全局收敛；当初始点靠近中心路径时，此算法便成为中心路径跟踪算法，特别对于单调线性互补问题，总迭代次数为O（√nL），其中L是问题的输入长度。     

求解一类非单调线性互补问题的宽邻域内点方法及其计算复杂性

对于一类非单调线性互补问题给出了一种新的算法——宽邻域内点算法，并讨论了其计算复杂性。     

基于1范数邻域的单调对称锥非线性互补问题的齐次算法复杂度分析

基于1范数邻域,研究了单调对称锥非线性互补问题的宽邻域齐次算法的复杂度.所获得的宽邻域齐次算法的复杂度与Yoshise所提出的窄邻域算法的复杂度一致.     

线性互补问题在宽邻域下的局部二次收敛算法

艾文宝(2004)的宽邻域算法弥补了内点法在理论和实践表现之间的差异.基于这个算法的优越性,将其推广到线性互补问题中.新算法在一次迭代中,采用两个方向的线性组合作为新方向,并以满步长到达下一个点.可以证明,该算法具有O(n~(1/2)L)的理论复杂度,这是迄今为止最好的复杂度结果.同时,在假设线性互补问题存在严格互补解的前提下,证明算法具有局部二次收敛性.最后,数值实验说明算法是有效的.     

一类新的求解P_*(κ)阵线性互补问题的多项式内点算法

利用核函数及其性质,对P_*(k)阵线性互补问题提出了一种新的宽邻域不可行内点算法.对核函数作了一些适当的改进,所以是不同于Peng等人介绍的自正则障碍函数.最后证明了算法具有近似O((1+2k)n3/4log(nμ~0)/ε)多项式复杂性,是优于传统的基于对数障碍函数求解宽邻域内点算法的复杂性.     

水平线性互补问题的广义中心路径跟踪算法

对水平线性互补问题提出了一种广义中心路径跟踪算法.任意的原始-对偶可行内点均可作为算法的初始点.每步迭代选择“仿射步”与“中心步”的凸组合为新的迭代方向,采用使对偶间隙尽可能减小的最大步长.算法的迭代复杂性为O(√nL).     

对称锥规划的邻域跟踪算法

本文把艾文宝的邻域跟踪算法推广到对称锥规划, 定义中心路径的宽邻域N(τ, β), 并证明该邻域的一个重要性质, 该性质在算法的复杂性分析中起到关键作用. 取宽邻域N(τ, β) 中一点为初始点并采用Nesterov-Todd (NT) 搜索方向, 则该算法的迭代复杂界为O(√r logε^-1), 其中, r是EuclidJordan 代数的秩, ε是允许误差. 这是对称锥规划的宽邻域内点算法最好的复杂界.     

Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem

We study the complexity of a noninterior path-following method for the linear complementarity problem. The method is based on the Chen–Harker–Kanzow–Smale smoothing function. It is assumed that the matrix M is either a P-matrix or symmetric and positive definite. When M is a P-matrix, it is shown that the algorithm finds a solution satisfying the conditions Mx-y+q=0 and in at most

A New Path-Following Algorithm for Nonlinear P*Complementarity Problems

Based on the recent theoretical results of Zhao and Li [Math. Oper. Res., 26 (2001), pp. 119—146], we present in this paper a new path-following method for nonlinear P* complementarity problems. Different from most existing interior-point algorithms that are based on the central path, this algorithm tracks the “regularized central path” which exists for any continuous P* problem. It turns out that the algorithm is globally convergent for any P* problem provided that its solution set is nonempty. By different choices of the parameters in the algorithm, the iterative sequence can approach to different types of points of the solution set. Moreover, local superlinear convergence of this algorithm can also be achieved under certain conditions. The research of the first author was supported by The National Natural Science Foundation of China under Grant No. 10201032 and Grant No. 70221001. The research of the second author was supported by Grant CUHK4214/01E, Research Grants Council, Hong Kong. An erratum to this article is available at .     

Numerical Comparisons of Path-Following Strategies for a Primal-Dual Interior-Point Method for Nonlinear Programming

An important research activity in primal-dual interior-point methods for general nonlinear programming is to determine effective path-following strategies and their implementations. The objective of this work is to present numerical comparisons of several path-following strategies for the local interior-point Newton method given by El-Bakry, Tapia, Tsuchiya, and Zhang. We conduct numerical experimentation of nine strategies using two central regions, three notions of proximity measures, and three merit functions to obtain an optimal solution. Six of these strategies are implemented for the first time. The numerical results show that the best path-following strategy is that given by Argáez and Tapia.     

P*(k)线性互补问题的Mehrotra型预估-校正算法复杂性分析

本文提出一种求解单调非线性互补问题的Mehrotra型预估-校正算法.新算法采用不同的自适应更新策略.在尺度化的Lipschitz条件下,证明了新算法的迭代复杂性为O(n2 log (x0)T s0/ε)),其中(x0,s0)为初始点,ε为精度.     

Infeasible-interior-point algorithm for a class of nonmonotone complementarity problems and its computational complexity

This paper presents an infeasible-interior-point algorithm for a class of nonmonotone complementarity problems, and analyses its convergence and computational complexity. The results indicate that the proposed algorithm is a polynomial-time one.     

一类奇摄动非线性激波问题

Using the method of matched asymptotic expansions,the shock solutions for a class of singularly perturbed nonlinear problems are discussed.The relation of the shock solutions and their boundary conditions is obtained.And the known results are generalized.     

A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems

We consider the standard linear complementarity problem (LCP): Find (x, y) R ²ⁿ such that y = M x + q, (x, y) 0 and x _i y _i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P ₀ LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in Newton iterations where is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods.