P0函数非线性互补问题的非内部连续化算法

提出了一种新的光滑函数,它具有现存的一些光滑函数不具备的性质.基于此光滑函数,讨论了求解P₀函数非线性互补问题的光滑路径的存在性和连续性.在非线性互补问题的解集非空有界的假设下,利用新光滑函数的特性,研究了求解P₀函数非线性互补问题的非内部连续化算法得到的迭代序列的有界性.解集非空有界的条件弱于一些现存的求解非线性互补问题的连续化算法所要求的假设条件.     

P0函数非线性互补问题的一步非内点连续方法的收敛性

本文对于P0函数非线性互补问题提出了一个基于Kanzow光滑函数的一步非内点连续方法,在适当的假设条件下,证明了方法的全局线性及局部二次收敛性.特别,在方法的全局线性收敛性的分析中,不需要假定非线性互补问题的函数的Jacobi阵是Lipschitz连续的.文献中为了得到非内点连续方法的全局线性收敛性,这一假定是被广泛使用的.本文提出的方法在每一次迭代只须解一个线性方程式组.     

P_0函数非线性互补问题的一步非内点连续方法的收敛性

本文对于Ｐ０函数非线性互补问题提出了一个基于Ｋａｎｚｏｗ光滑函数的一步非内点连续方法,在适当的假设条件下,证明了方法的全局线性及局部二次收敛性．特别,在方法的全局线性收敛性的分析中,不需要假定非线性互补问题的函数的Ｊａｃｏｂｉ阵是Ｌｉｐｓｃｈｉｔｚ连续的．文献中为了得到非内点连续方法的全局线性收敛性,这一假定是被广泛使用的．本文提出的方法在每一次迭代只须解一个线性方程式组．     

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．     

一个求解P0函数非线性互补问题的非内部连续化算法

基于黄正海等2001年提出的光滑函数,本文给出一个求解P0函数非线性互补问题的非内部连续化算法.所给算法拥有一些好的特性.在较弱的条件下,证明了所给算法或者是全局线性收敛,或者是全局和局部超线性收敛.给出了所给算法求解两个标准测试问题的数值试验结果.     

二阶锥权互补问题的非精确非内点连续化算法

基于二阶锥权互补函数,将二阶锥权互补问题转化为一个方程组,运用非精确非内点连续化算法求解该方程组.该算法能以任意点作为初始点,且每次迭代时至多求解一个方程组.为节省算法求解方程组时的计算时间和内存,将非精确牛顿法引入到算法中.在适当假设下,证明了该算法是全局与局部二阶收敛的.最后数值实验表明了算法的良好性能.     

求解非线性互补问题的内点正算法

针对非线性互补问题,提出了与其等价的非光滑方程的内点正算法,并在一定条件下证明了该算法的收敛性定理。数值结果表明,该算法是十分有效的。     

求解非线性互补问题的一个下降算法

在[1]中,Soldov将非线性互补问题等价地转化成一个带非负约束的优化问题,基于这种转化形式,我们给出了一种求解非线性互补问题的下降算法,在映射为强单调时,证明了算法的全局收敛性。     

基于新光滑函数的P0映射非线性互补问题的光滑牛顿法(英文)

非线性互补问题(NCP)可以重新表述为一个非光滑方程组的解.通过引入一个新的光滑函数,将问题近似为参数化光滑方程组.基于这个光滑函数,我们提出了一个求解P₀映射和R₀映射非线性互补问题的光滑牛顿法.该算法每次迭代只求解一个线性方程和一次线搜索.在适当的条件下,证明了该方法是全局和局部二次收敛的.数值结果表明,该算法是有效的.     

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。     

Strict Feasibility Conditions in Nonlinear Complementarity Problems

Strict feasibility plays an important role in the development of the theoryand algorithms of complementarity problems. In this paper, we establishsufficient conditions to ensure strict feasibility of a nonlinearcomplementarity problem. Our analysis method, based on a newly introducedconcept of -exceptional sequence, can be viewed as a unified approachfor proving the existence of a strictly feasible point. Some equivalentconditions of strict feasibility are also developed for certaincomplementarity problems. In particular, we show that aP_*-complementarity problem is strictly feasible if and only ifits solution set is nonempty and bounded.     

A New Homotopy Method for Nonlinear Complementarity Problems

In this paper, we present a new homotopy method for the nonlinear complementarity problems. Without the regularity or non-singulary assumptions for▽F(x), we prove that our homotopy equations have a bounded solution curve. The numerical tests confirm the efficiency of our proposed method.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

奇异摄动问题最优阶一致收敛的间断有限元分析

采用非对称内罚间断有限元方法(以下简称NIPG方法)求解一维对流扩散型奇异摄动问题.理论上证明了采用拉格朗日线性元的NIPG方法在Bakhvalov-Shishkin网格上具有最优阶的一致收敛性,即在能量范数度量下其误差估计为O(N~(-1)),其中N为网格剖分中单元个数.数值算例验证了理论分析的正确性.     

Error Bounds for R0-Type and Monotone Nonlinear Complementarity Problems

The paper generalizes the Mangasarian–Ren (Ref. 1) error bounds forlinear complementarity problems (LCPs) to nonlinear complementarity problems(NCPs). This is done by extending the concept of R ₀-matrixto several R ₀-type functions, which include a subset ofmonotone functions as a special case. Both local and global error bounds areobtained for R ₀-type NCPs and some monotone NCPs.     

Stochastic Nonlinear Complementarity Problem and?Applications to?Traffic Equilibrium under?Uncertainty

The expected residual minimization (ERM) formulation for the stochastic nonlinear complementarity problem (SNCP) is studied in this paper. We show that the involved function is a stochastic R ₀ function if and only if the objective function in the ERM formulation is coercive under a mild assumption. Moreover, we model the traffic equilibrium problem (TEP) under uncertainty as SNCP and show that the objective function in the ERM formulation is a stochastic R ₀ function. Numerical experiments show that the ERM-SNCP model for TEP under uncertainty has various desirable properties. This work was partially supported by a Grant-in-Aid from the Japan Society for the Promotion of Science. The authors thank Professor Guihua Lin for pointing out an error in Proposition 2.1 on an earlier version of this paper. The authors are also grateful to the referees for their insightful comments.     

A power penalty approach to a Nonlinear Complementarity Problem

We propose a novel power penalty approach to a Nonlinear Complementarity Problem (NCP) in which the NCP is approximated by a nonlinear equation containing a power penalty term. We show that the solution to the penalty equation converges to that of the NCP at an exponential rate when the function involved is continuous and ξ-monotone. A higher convergence rate is also obtained when the function becomes Lipschitz continuous. Numerical results are presented to confirm the theoretical findings.     

非线性互补问题的水平值估计算法

本文研究非线性互补问题（NCP）的求解算法，先将NCP转化为约束全局优化问题（CGOP），然后直接移植求解问题（CGOP）的水平值估计算法^[4,5]来求解问题（NCP）．文章证明了算法对于NCP是收敛的，数值实验说明了算法的有效性．     

Cartesian P-property and Its Applications to the Semidefinite Linear Complementarity Problem

We introduce a Cartesian P-property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem (SDLCP). With this Cartesian P-property, we show that the SDLCP has GUS-property (i.e., globally unique solvability), and the solution map of the SDLCP is locally Lipschitzian with respect to input data. Our Cartesian P-property strengthens the corresponding P-properties of Gowda and Song [15] and allows us to extend several numerical approaches for monotone SDLCPs to solve more general SDLCPs, namely SDLCPs with the Cartesian P-property. In particular, we address important theoretical issues encountered in those numerical approaches, such as issues related to the stationary points in the merit function approach, and the existence of Newton directions and boundedness of iterates in the non-interior continuation method of Chen and Tseng [6]. This work is supported by the annual grant A2004/23 of University of Southampton.