线性二阶锥互补问题的一种非精确光滑算法

在光滑算法的框架下,就线性二阶锥互补问题,给出了一种非精确光滑算法. 在适当的条件下,证明了该算法具有全局收敛性. 数值试验表明该算法对高维线性二阶锥互补问题是有效的.     

一类非单调二阶锥互补问题解集的非空性与有界性

本文在二阶锥上引入一类新的映射,称之为笛卡尔P_*(κ)映射,它是单调映射的推广.文中讨论涉及这类映射的二阶锥互补问题的解的存在性和解集的有界性.主要结论为:如果所考虑的互补问题是严格可行的,那么它的解集是非空有界的.     

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.     

非线性二阶锥互补问题的低阶罚函数算法（英文）

本文研究非线性二阶锥互补问题的一般低阶罚函数算法.并将非线性二阶锥互补问题转化为序列非线性方程组.在一定条件下,当罚因子趋向于无穷时,获得序列非线性方程组的解序列以指数速度收敛于原始非线性二阶锥互补问题的解,推广了幂罚函数算法求解非线性二阶锥互补问题的结果.数值实验结果说明了算法的有效性.     

二阶锥约束随机变分不等式问题的数值方法研究

本文研究二阶锥约束随机变分不等式(SOCCSVI)问题,运用样本均值近似(SAA)方法结合光滑Fischer-Burmeister互补函数来求解该问题.首先,将SOCCSVI问题的Karush-Kuhn-Tucker系统转化为与之等价的方程组,并证明了该方程组的雅可比矩阵的非奇异性.其次,构造了光滑牛顿算法求解该方程组.最后,文章给出了两个数值实验证明了算法的有效性.     

混合互补问题的逐次逼近算法

针对混合互补问题 ,提出了与其等价的非光滑方程的逐次逼近算法 ,并在一定条件下证明了该算法的全局收敛性 .数值例子表明这一算法是有效的     

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.     

求解非线性互补问题的逐次逼近阻尼牛顿法

针对非线性互补问题,提出了与其等价的非光滑方程的逐次逼近阻尼牛顿法,并 在一定条件下证明了该算法的全局收敛性.数值结果表明,这一算法是有效的.     

求解非线性互补问题的逐次逼近拟牛顿法的收敛性分析

提出了求解非线性互补问题的一个逐次逼近拟牛顿算法。在适当的假设下,证明了该算法的全局收敛性和局部超线性收敛性。     

对称锥互补问题的一个惩罚NR函数(英文)

本文建立了一个对称锥互补问题的惩罚自然剩余函数,并且证明了单调情形下其相应势函数的水平有界性.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications

We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem. B.K. Pagnoncelli’s research was supported by CAPES and FUNENSEG. S. Ahmed’s research was partly supported by the NSF Award DMI-0133943. A. Shapiro’s research was partly supported by the NSF Award DMI-0619977.     

求解二阶锥规划问题的VU分解方法

给出解决二阶锥规划(SOCP)问题的VU-分解方法.问题首先被转化为非线性规划,并给出相应的精确罚函数的Clarke次微分结构及VU-空间分解．在某种条件下,可以计算出一个二阶连续可微的轨道,进而得到目标函数f在其上的二阶展开．最后给出一个具有超线性收敛速度的概念型算法．     

非线性互补问题的逐次逼近类Broyden算法

In this paper, we present a new form of successive approximation Broyden-like algorithm for nonlinear complementarity problem based on its equivalent nonsmooth equations. Under suitable conditions, we get the global convergence on the algorithms. Some numerical results are also reported.     

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.     

Feasible Semismooth Newton Method for a Class of Stochastic Linear Complementarity Problems

We consider a class of stochastic linear complementarity problems (SLCPs) with finitely many realizations. In this paper we reformulate this class of SLCPs as a constrained minimization (CM) problem. Then, we present a feasible semismooth Newton method to solve this CM problem. Preliminary numerical results show that this CM reformulation may yield a solution with high safety for SLCPs.     

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.     

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 Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998