单调线性互补问题的Mehrotra型预估-校正算法的迭代复杂性

Mehrotra型预估-校正算法是很多内点算法软件包的算法基础,但它的多项式迭代复杂性直到2007年才被Salahi等人证明.通过选择一个固定的预估步长及与Salahi文中不同的校正方向,本文把Salahi等人的算法拓展到单调线性互补问题,使得新算法的迭代复杂性为O(n log((x0)T s0/ε)),同时,初步的数值实验证明了新算法是有效的.     

A Hybrid Smoothing Method for Mixed Nonlinear Complementarity Problems

In this paper, we describe a new, integral-based smoothing method for solving the mixed nonlinear complementarity problem (MNCP). This approach is based on recasting MNCP as finding the zero of a nonsmooth system and then generating iterates via two types of smooth approximations to this system. Under weak regularity conditions, we establish that the sequence of iterates converges to a solution if the limit point of this sequence is regular. In addition, we show that the rate is Q-linear, Q-superlinear, or Q-quadratic depending on the level of inexactness in the subproblem calculations and we make use of the inexact Newton theory of Dembo, Eisenstat, and Steihaug. Lastly, we demonstrate the viability of the proposed method by presenting the results of numerical tests on a variety of complementarity problems.     

A Smoothing Newton Method for General Nonlinear Complementarity Problems

Smoothing Newton methods for nonlinear complementarity problems NCP(F) often require F to be at least a P ₀-function in order to guarantee that the underlying Newton equation is solvable. Based on a special equation reformulation of NCP(F), we propose a new smoothing Newton method for general nonlinear complementarity problems. The introduction of Kanzow and Pieper's gradient step makes our algorithm to be globally convergent. Under certain conditions, our method achieves fast local convergence rate. Extensive numerical results are also reported for all complementarity problems in MCPLIB and GAMSLIB libraries with all available starting points.     

垂直线性互补问题的一步全局线性和局部二次收敛光滑Newton法

基于凝聚函数,提出一个求解垂直线性互补问题的光滑Newton法.该算法具有以下优点:(i)每次迭代仅需解一个线性系统和实施一次线性搜索;(ⅱ)算法对垂直分块P0矩阵的线性互补问题有定义且迭代序列的每个聚点都是它的解.而且,对垂直分块P₀+R₀矩阵的线性互补问题,算法产生的迭代序列有界且其任一聚点都是它的解;(ⅲ)在无严格互补条件下证得算法即具有全局线性收敛性又具有局部二次收敛性.许多已存在的求解此问题的光滑Newton法都不具有性质(ⅲ).     

A Smoothing Method for a Mathematical Program with P-Matrix Linear Complementarity Constraints

We consider a mathematical program whose constraints involve a parametric P-matrix linear complementarity problem with the design (upper level) variables as parameters. Solutions of this complementarity problem define a piecewise linear function of the parameters. We study a smoothing function of this function for solving the mathematical program. We investigate the limiting behaviour of optimal solutions, KKT points and B-stationary points of the smoothing problem. We show that a class of mathematical programs with P-matrix linear complementarity constraints can be reformulated as a piecewise convex program and solved through a sequence of continuously differentiable convex programs. Preliminary numerical results indicate that the method and convex reformulation are promising.     

基于凝聚函数的互补问题的光滑化算法

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光滑方程组算法.首先是将互补问题转化为等价的非光滑方程组,再利用凝聚函数进行光滑逼近,从而转化为光滑方程组的求解问题.通过一些考题对这个算法进行了数值试验,结果显示了该算法的有效性和稳定性.     

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.     

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.     

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.     

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.     

求解线性互补问题的同伦方法

It is well known that a linear complementarity problem (LCP) can be formulated as a system of nonsmooth equations F(x) = 0, where F is a map from Rninto itself. Using the aggregate function, we construct a smooth Newton homotopy H(x,t) = 0. Under certain assumptions, we prove the existence of a smooth path defined by the Newton homotopy which leads to a solution of the original problem, and study limiting properties of the homotopy path.     

Modulus-Based GSTS Iteration Method for Linear Complementarity Problems

In this paper, a modulus-based generalized skew-Hermitian triangular splitting (MGSTS) iteration method is present for solving a class of linear complementarity problems with the system matrix either being an $H_+$-matrix with non-positive off-diagonal entries or a symmetric positive definite matrix. The convergence of the MGSTS iteration method is studied in detail. By choosing different parameters, a series of existing and new iterative methods are derived, including the modulus-based Jacobi (MJ) and the modulus-based Gauss-Seidel (MGS) iteration methods and so on. Experimental results are given to show the effectiveness and feasibility of the new method when it is employed for solving this class of linear complementarity problems.     

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

多值单调算子的隐补问题

在Banach空间中引入了多值算子的隐补问题和相补问题的新概念，并证明了多值单调算子隐补问题和相补问题解的存在性定理。     

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．     

线性互补问题的宽邻域预估校正算法

对线性互补问题提出了一种新的宽邻域预估校正算法,算法是基于经典线性规划路径跟踪算法的思想,将Maziar Salahi关于线性规划预估校正算法推广到线性互补问题中,给出了算法的具体迭代步骤并讨论了算法迭代复杂性,最后证明了算法具有多项式复杂性为O(ηlog(X~0)~Ts~0/ε)。     

Extended LQP Method for Monotone Nonlinear Complementarity Problems

To solve nonlinear complementarity problems (NCP), the logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations at each iteration. In this paper, the iterates generated by the original LQP method are extended by explicit formulas and thus an extended LQP method is presented. It is proved theoretically that the lower bound of the progress obtained by the extended LQP method is greater than that by the original LQP method. Preliminary numerical results are provided to verify the theoretical assertions and the effectiveness of both the original and the extended LQP method.     

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P ₀ functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of China (Grant 10171030).     

A Self-Adaptive Projection and Contraction Method for Linear Complementarity Problems

In this paper we develop a self-adaptive projection and contraction method for the linear complementarity problem (LCP). This method improves the practical performance of the modified projection and contraction method in [10] by adopting a self-adaptive technique. The global convergence of our new method is proved under mild assumptions. Our numerical tests clearly demonstrate the necessity and effectiveness of our proposed method.     

非线性互补问题光滑化拟牛顿算法的收敛性分析

给出求解p_0函数非线性互补问题光滑化拟牛顿算法,在p_0函数非线性互补问题有非空有界解集且F'是Lipschitz连续的条件下,证明了算法的全局收敛性.全局收敛性的主要特征是不需要提前假设水平集是有界的.