半定规划的非内点连续化方法

基于Fischer-Burmeister函数,本文将半定规划(SDP)的中心路径条件转化为非线性方程组,进而用SDCP的非内点连续化方法求解之.证明了牛顿方向的存在性,迭代点列的有界性.在适当的假设条件下,得到算法的全局收敛性及局部二次收敛率.数值结果表明算法的有效性.     

Superlinear Noninterior One-Step Continuation Method for Monotone LCP in the Absence of Strict Complementarity

We propose a noninterior continuation method for the monotone linear complementarity problem (LCP) by modifying the Burke–Xu framework of the noninterior predictor-corrector path-following method (Refs. 1–2). The new method solves one system of linear equations and carries out only one line search at each iteration. It is shown to converge to the LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution. Our analysis of the continuation method is based on a broader class of the smooth functions introduced by Chen and Mangasarian (Ref. 3).     

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

指数效用函数下投资组合问题的有效集方法

本在指数型效用函数条件下提出了求解高维数组合投资问题的有效集方法，实例说明：该方法是可行的、有效的．     

A Non-Interior Continuation Algorithm for the P0 or P* LCP with Strong Global and Local Convergence Properties

We propose a non-interior continuation algorithm for the solution of the linear complementarity problem (LCP) with a P₀ matrix. The proposed algorithm differentiates itself from the current continuation algorithms by combining good global convergence properties with good local convergence properties under unified conditions. Specifically, it is shown that the proposed algorithm is globally convergent under an assumption which may be satisfied even if the solution set of the LCP is unbounded. Moreover, the algorithm is globally linearly and locally superlinearly convergent under a nonsingularity assumption. If the matrix in the LCP is a P_* matrix, then the above results can be strengthened to include global linear and local quadratic convergence under a strict complementary condition without the nonsingularity assumption.     

A note on a characterization of P-matrices

In this note we show that the characterization results for P-matrices due to K.G. Murty and A. Tamir which state that a given square matrixM of ordern is a P-matrix if and only if the linear complementarity problem (q, M) has a unique solution for allq in a specified finite subset of ⁿ depending onM are incorrect whenn > 3.Research supported by Dr. K.S. Krishnan (DAE) fellowship for research in Mathematics and Computer Science, Bombay, India.     

A New Noninterior Predictor–Corrector Method for the P0 LCP

In this paper a new predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method, which is new in the predictor-corrector-type methods, and the corrector step is generated as in [3]. The method has the following merits: (i) any cluster point of the iteration sequence is a solution of the P₀ LCP; (ii) if the generalized Jacobian is nonsingular at a solution point, then the whole sequence converges to the (unique) solution of the P₀ LCP superlinearly; (iii) for the P₀ LCP, if an accumulation point of the iteration sequence satisfies the strict complementary condition, then the whole sequence converges to this accumulation point superlinearly. Preliminary numerical experiments are reported to show the efficiency of the algorithm.     

线性分式规划最优解集的求法

本文使用多面集的表示定理 ,导出了线性分式规划最优解集的结构 ,并给出确定全部最优解的计算步骤 .     

同伦算法在并联机器人运动学中的应用

讨论同伦算法求解并联机器人运动学正问题.通过参系数同伦法大大减少了跟踪路径的数目,从而提高了同伦算法的效率使之对求解一般非线性代数方程组更为可行.采用这一算法,求出了中科院沈阳自动化研究所机器人开放研究实验室的新型并联机器人模型的全部正解.为新模型的机构分析和实时控制提供了理论依据.     

On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach. This author’s work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10271002 and 10401038). This author’s work was partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.     

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.     

PSD方法的最优因子及效率分析

在系数矩阵是相容序2循环阵的情况下，本文给出了PSD方法的最优松弛参数和最优收敛因子，分析和讨论了它的实用性，并进而得到了一个新的迭代法，它的最优收敛因子与PSD方法一样，而迭代参数却只有一个．     

求解一类非线性规划的连续同伦方法(英文)

In this paper we present a homotopy continuation method for finding the Karush-Kuhn-Tucker point of a class of nonlinear non-convex programming problems. Two numerical examples are given to show that this method is effective. It should be pointed out that we extend the results of Lin et al. （see Appl. Math. Comput., 80（1996）, 209-224） to a broader class of non-convex programming problems.     

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.     

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.     

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 Truncated Newton Method for the Solution of Large-Scale Inequality Constrained Minimization Problems

A new active set Newton-type algorithm for the solution of inequality constrained minimization problems is proposed. The algorithm possesses the following favorable characteristics: (i) global convergence under mild assumptions; (ii) superlinear convergence of primal variables without strict complementarity; (iii) a Newton-type direction computed by means of a truncated conjugate gradient method. Preliminary computational results are reported to show viability of the approach in large scale problems having only a limited number of constraints.     

Augmented Lagrangian Active Set Methods for Obstacle Problems

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.     

一个新的求解二阶锥规划的非内部连续化算法

基于光滑Fischer-Burmeister函数,本文给出一个新的求解二阶锥规划的非内部连续化算法.算法对初始点的选取没有任何限制,并且在每一步迭代只需求解一个线性方程组并进行一次线性搜索.在不需要满足严格互补条件下,证明了算法是全局收敛且是局部超线性收敛的.数值试验表明算法是有效的.