The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP

^** Email: zhenghaihuang{at}yahoo.com.cn; huangzhenghai{at}hotmail.com In this paper, we propose a non-interior continuation algorithmfor solving the P₀-matrix linear complementarity problem (LCP),which is conceptually simpler than most existing non-interiorcontinuation algorithms in the sense that the proposed algorithmonly needs to solve at most one linear system of equations ateach iteration. We show that the proposed algorithm is globallyconvergent under a common assumption. In particular, we showthat the proposed algorithm is globally linearly and locallyquadratically convergent under some assumptions which are weakerthan those required in many existing non-interior continuationalgorithms. It should be pointed out that the assumptions usedin our analysis of both global linear and local quadratic convergencedo not imply the uniqueness of the solution to the LCP concerned.To the best of our knowledge, such a convergence result hasnot been reported in the literature.     

A continuation method for linear complementarity problems with P 0 matrix

In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P ₀ matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.     

An improved interior-type feasible QP-free algorithm for inequality constrained optimization problems

In this paper, an improved interior-type feasible QP-free algorithm for inequality constrained optimization problems is proposed. At each iteration, by solving three systems of linear equations with the same coefficient matrix, a search direction is generated. The algorithm is proved to be globally and superlinearly convergent under some mild conditions. Preliminary numerical results show that the proposed algorithm may be promising. Advantages of the algorithm include: the uniformly nonsingularity of the coefficient matrices without the strictly complementarity condition is obtained. Moreover, the global convergence is achieved even if the number of the stationary points is infinite.     

Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations

We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in Ref. 1 has a very nice global convergence property in that the whole sequence of iterates generated by this algorithm converges to a solution, if it exists. Superlinear convergence of this algorithm is obtained under a standard nonsingularity assumption. The nonsingularity condition implies that the problem has a unique solution; thus, for a problem with more than one solution, such a nonsingularity condition cannot hold. In this paper, we show that the superlinear convergence of this algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition. The local error-bound condition may hold even for problems with nonunique solutions. As an application, we obtain a Newton algorithm with very nice global and superlinear convergence for the minimum norm solution of linear programs.This research was supported by the Singapore-MIT Alliance and the Australian Research Council.     

Global Linear and Quadratic One-step Smoothing Newton Method for P 0-LCP

We propose a new smoothing Newton method for solving the P ₀-matrix linear complementarity problem (P ₀-LCP) based on CHKS smoothing function. Our algorithm solves only one linear system of equations and performs only one line search per iteration. It is shown to converge to a P ₀-LCP solution globally linearly and locally quadratically without the strict complementarity assumption at the solution. To the best of author's knowledge, this is the first one-step smoothing Newton method to possess both global linear and local quadratic convergence. Preliminary numerical results indicate that the proposed algorithm is promising.     

Superlinear／Quadratic One-step Smoothing Newton Methodf or P0-NCP

We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P0-function (P0-NCP) based on the smoothing symmetric perturbed Fisher function(for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P0-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions.     

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．     

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.     

A Smoothing Levenberg-Marquardt Type Method for LCP

In this paper, we convert the linear complementarity problem to a system of semismooth nonlinear equations by using smoothing technique. Then we use Levenberg-Marquardt type method to solve this system. Taking advantage of the new results obtained by Dan, Yamashita and Fukushima [11, 33], the global and local superlinear convergence properties of the method are obtained under very mild conditions. Especially, the algorithm is locally superlinearly convergent under the assumption of either strict complementarity or certain nonsingularity. Preliminary numerical experiments are reported to show the efficiency of the algorithm.