依赖凝聚函数求解非线性互补问题的一种微分方程方法

利用凝聚函数一致逼近非光滑极大值函数的性质,将非线性互补问题转化为参数化光滑方程组.然后,对此方程组给出了一种微分方程解法,并且证明了非线性互补问题的解是微分方程系统的渐进稳定平衡点.在适当的假设条件下,证明了所给出的算法具有二次收敛速度.数值结果表明了此算法的有效性.     

求解张量互补问题的一类光滑模系矩阵迭代方法

本文提出了求解张量互补问题的一类光滑模系矩阵迭代方法.其基本思想是,先将张量互补问题转化为等价的模系方程组,然后引入一个逼近的光滑函数进行求解.我们分析了算法的收敛性,并通过数值实验验证了所提出算法的有效性.     

基于凝聚函数的互补问题的自调节内点算法

对于不可微的"极大值"形式的函数,可以利用凝聚函数对其进行光滑逼近.借助这个技术,给出了求解线性互补问题的一个具有自调节功能的内点算法.基于邻近度量和线性互补问题的标准中心化方程的关系,定义了一个新的邻近度量函数,并以极小化这个函数的最优性条件代替了该中心化方程.以此在摄动方程本身建立一种自调节的机制,从而使牛顿方向能够根据上次迭代点的信息做出自适应的调整.基于改造后的摄动方程组,建立了一个具有自调节功能的内点算法.通过一些考题对这个算法进行了数值试验,结果显示了算法的有效性和稳定性.     

混合互补问题的光滑算法及收敛性

利用Fischer-Burmeister函数将混合互补问题转化为非线性方程组,由光滑函数逼近FB函数来求解非线性方程组．文中将信赖域方法和梯度法相结合,提出了Jacobian光滑化方法．算法在一定条件下的全局收敛性得到了证明,数值试验表明算法切实有效,有一定的优越性．     

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．     

一类高阶牛顿迭代法及其在线性互补问题中的应用

通过等价转换,把线性互补问题转化为一个不可微的非线性方程组,进而采用光滑函数处理,得到一个光滑非线性方程组,利用高阶牛顿迭代法进行求解.该方法不再区分线性互补问题是否单调,因此扩大了线性互补问题的求解范围.计算结果表明,方法计算速度快,对线性互补问题求解较为有效.     

基于一个新的NCP函数的光滑牛顿法求解非线性互补问题

本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性.     

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.     

求解广义非线性互补问题的光滑化拟牛顿法

利用光滑对称扰动Fischer-Burmeister函数将广义非线性互补问题转化为非线性方程组,提出新的光滑化拟牛顿法求解该方程组.然后证明该算法是全局收敛的,且在一定条件下证明该算法具有局部超线性(二次)收敛性.最后用数值实验验证了该算法的有效性.     

非线性互补约束均衡问题的一个滤子SQP算法

提出了—个求解非线性互补约束均衡问题的滤子SQP算法.借助Fischer-Burmeister函数把均衡约束转化为—个非光滑方程组,然后利用逐步逼近和分裂思想,给出—个与原问题近似的一般的约束优化.引入滤子思想,避免了罚函数法在选择罚因子上的困难.在适当的条件下证明了算法的全局收敛性,部分的数值结果表明算法是有效的.     

A NEW SMOOTHING EQUATIONS APPROACH TO THE NONLINEAR COMPLEMENTARITY PROBLEMS

The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing Newton algorithm for the solution of the nonlinear complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm.     

Numerical comparisons based on four smoothing functions for absolute value equation

The system of absolute value equation, denoted by AVE, is a non-differentiable NP-hard problem. Many approaches have been proposed during the past decade and most of them focus on reformulating it as complementarity problem and then solve it accordingly. Another approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations. In this paper, we follow this path. In particular, we rewrite it as a system of smooth equations and propose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations. The main contribution of this paper focuses on numerical comparisons which suggest a better choice of smoothing function along with the smoothing-type algorithm.     

二阶锥规划一个超线性收敛的非内部连续化算法

基于非光滑向量值最小函数的一个新光滑函数, 建立了二阶锥规划一个超线性收敛的非内部连续化算法. 该算法的特点如下: 首先, 初始点任意; 其次, 每次迭代只需求解一个线性方程组即可得到搜索方向; 最后, 在无严格互补假设下, 获得算法的全局收敛性、强收敛性和超线性收敛性. 数值结果表明算法是有效的.     

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.

     

Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP

In this paper, we propose a general smoothing Broyden-like quasi-Newton method for solving a class of nonsmooth equations. Under appropriate conditions, the proposed method converges to a solution of the equation globally and superlinearly. In particular, the proposed method provides the possibility of developing a quasi-Newton method that enjoys superlinear convergence even if strict complementarity fails to hold. We pay particular attention to semismooth equations arising from nonlinear complementarity problems, mixed complementarity problems and variational inequality problems. We show that under certain conditions, the related methods based on the perturbed Fischer–Burmeister function, Chen–Harker–Kanzow–Smale smoothing function and the Gabriel–Moré class of smoothing functions converge globally and superlinearly.     

Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. Under suitable conditions, the method exhibits global and quadratic convergence properties. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.     

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.     

A self-adaptive trust region method for the extended linear complementarity problems

By using some NCP functions, we reformulate the extended linear complementarity problem as a nonsmooth equation. Then we propose a self-adaptive trust region algorithm for solving this nonsmooth equation. The novelty of this method is that the trust region radius is controlled by the objective function value which can be adjusted automatically according to the algorithm. The global convergence is obtained under mild conditions and the local superlinear convergence rate is also established under strict complementarity conditions. This work is supported by National Natural Science Foundation of China (No. 10671126) and Shanghai Leading Academic Discipline Project (S30501).     

A Smoothing Newton Method with Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

The second-order cone complementarity problem (SOCCP) is an important class of problems containing a lot of optimization problems. The SOCCP can be transformed into a system of nonsmooth equations. To solve this nonsmooth system, smoothing techniques are often used. Fukushima, Luo and Tseng (SIAM J. Optim. 12:436–460, 2001) studied concrete theories and properties of smoothing functions for the SOCCP. Recently, a practical computational method using the smoothed natural residual function to solve the SOCCP was given by Chen, Sun and Sun (Comput. Optim. Appl. 25:39–56, 2003). In the present paper, we propose an algorithm to solve the SOCCP by using the smoothed Fischer-Burmeister function. Some preliminary numerical results are given.