混合互补问题牛顿型算法的二阶收敛性

在凸规划理论中,通过ＫＴ条件,往往将约束最优化问题归结为一个混合互补问题来求解。该文就正则解和一般解两种情形分别给出了求解混合互补问题牛顿型算法的二阶收敛性的充分性条件,并在一定条件下证明了非精确牛顿法和离散牛顿法所具有的二阶收敛性。     

互补约束均衡优化的一个共轭梯度投影法

讨论均衡约束最优化问题,利用一个互补函数和扰动技术将原问题转换为非线性等式约束最优化问题,然后利用共轭梯度投影算法的思想,给出了问题的一个求解算法,在适当的条件下,证明了算法的全局收敛性.     

半线性椭圆互补问题的上解与下解(英文)

本文针对半线性椭圆互补问题,提供几种获取该问题的一个上解和下解的方式.在求解过程中,仅仅需要求解一个线性互补问题或线性方程组.所得结果可应用于众多单调算法的初始化.     

求解特征值互补问题的一类ABS算法

ABS算法是20世纪80年代初,由Abaffy,Broyden和Spedicato完成的用于求解线性方程组的含有三个参量的投影算法,是一类有限次迭代直接法。目前,ABS算法不仅可以求解线性与非线性方程组,还可以求解线性规划和具有线性约束的非线性规划等问题。本文即是利用ABS算法求解特征值互补问题的一种尝试,构造了求解特征值互补问题的ABS算法,证明了求解特征值互补问题的ABS算法的收敛性。数值例子充分验证了求解特征值互补问题的ABS算法的有效性。     

一般互补约束问题的一种松弛逐步二次规划算法北大核心CSCD

1 引言 互补约束问题（简称MPCC）是一类具有特殊约束条件的约束最优化问题．不同于一般约束优化问题,其基本约束条件不仅包含等式约束和不等式约束,而且还包含比较复杂的互补约束．MPCC的一般形式如下：     

求解障碍问题自由边界的一种非精确光滑方法

利用差分原理将一类数学物理障碍问题转化为线性互补问题.给出了求解大规模线性互补问题的一种非精确光滑算法,证明了该算法的适定性和全局收敛性.数值试验表明该方法能很好地求解此类障碍问题.     

互补问题的光滑逼近法

提出求解互补问题的一个光滑逼近法,从而可直接利用各类光滑方程组成无约束可微优化算法求解线性和非线性互补问题,数值实验表明了方法的有效性。     

改进的分块模方法求解对角占优线性互补问题

为了高效求解中小型线性互补问题,本文提出了改进的分块模方法,并证明了关于严格对角占优（对角元素均为正数）线性互补问题的收敛性.对于广义对角占优线性互补问题,先将其转化为严格对角占优线性互补问题,再采用改进的分块模方法求解.数值结果表明,改进的分块模方法在求解广义对角占优线性互补问题时在内迭代次数和计算时间上均明显优于分块模方法.     

求解广义互补问题

1 引言 设为一闭凸锥,f是R~n到自身的一映射.广义互补问题,记作GCP(K,f),即找一向量x满足 GCP(K,f) x∈K,f(x)∈且x~Tf(x)=0,(1) 其中,是K的对偶锥(即对任一K中向量x,满足x~Ty≤0的所有y的集合).该问题首先 由Habetler和Price提出.当K=R_+~n(R~n空间的正卦限),此问题就是一般的互补问题.许多作者已经提出了很多求解线性或非线性互补问题的方法.例如:Dafermos,Fukushima,Harker和Price以及其它如参考文献所列.近年来,何针对单调线性变分不等式提出了一些投影收缩算法. Fang在函数是Lipschitz连续及强单调的条件下,在[3]给出一简单的迭代投影法,在[4]中给出一线性化方法去求解广义互补问题(1).在[3]中,他的迭代模式是     

线性互补问题与多目标优化

本文研究了求解线性互补问题的一类新方法:把线性互补问题转化为多目标优化问题,利用多目标优化有效解的定义,给出了零有效解的概念;进而获得多目标优化问题的零有效解就是线性互补问题的最优解.最后给出了有解、无解线性互补问题,并分别把这些问题转化为多目标优化,采用极大极小方法求解转化后的多目标优化问题.数值实验结果表明了该方法的正确性和有效性,完善了文献[19]的数值结果.     

On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian

Mangasarian and Solodov (Ref. 1) proposed to solve nonlinear complementarity problems by seeking the unconstrained global minima of a new merit function, which they called implicit Lagrangian. A crucial point in such an approach is to determine conditions which guarantee that every unconstrained stationary point of the implicit Lagrangian is a global solution, since standard unconstrained minimization techniques are only able to locate stationary points. Some authors partially answered this question by giving sufficient conditions which guarantee this key property. In this paper, we settle the issue by giving a necessary and sufficient condition for a stationary point of the implicit Lagrangian to be a global solution and, hence, a solution of the nonlinear complementarity problem. We show that this new condition easily allows us to recover all previous results and to establish new sufficient conditions. We then consider a constrained reformulation based on the implicit Lagrangian in which nonnegative constraints on the variables are added to the original unconstrained reformulation. This is motivated by the fact that often, in applications, the function which defines the complementarity problem is defined only on the nonnegative orthant. We consider the KKT-points of this new reformulation and show that the same necessary and sufficient condition which guarantees, in the unconstrained case, that every unconstrained stationary point is a global solution, also guarantees that every KKT-point of the new problem is a global solution.     

A Single Component Mutation Evolutionary Programming

In this paper, a novel evolutionary programming is proposed for solving the upper and lower bound optimization problems as well as the linear constrained optimization problems. There are two characteristics of the algorithm: first, only one component of the current solution is mutated in each iteration; second, it can solve the linear constrained optimization problems directly without converting it into unconstrained problems. By solving two kinds of the optimization problems, the algorithm can not only effectively find optimal or close to optimal solutions but also reduce the number of function evolutions compared with the other heuristic algorithms.     

New Constrained Optimization Reformulation of Complementarity Problems

We suggest a reformulation of the complementarity problem CP(F) as a minimization problem with nonnegativity constraints. This reformulation is based on a particular unconstrained minimization reformulation of CP(F) introduced by Geiger and Kanzow as well as Facchinei and Soares. This allows us to use nonnegativity constraints for all the variables or only a subset of the variables on which the function F depends. Appropriate regularity conditions ensure that a stationary point of the new reformulation is a solution of the complementarity problem. In particular, stationary points with negative components can be avoided in contrast to the reformulation as unconstrained minimization problem. This advantage will be demonstrated for a class of complementarity problems which arise when the Karush–Kuhn–Tucker conditions of a convex inequality constrained optimization problem are considered.     

New LP bound in multivariate Lipschitz optimization: Theory and applications

Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds.Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure.Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.The authors are indebted to the Editor-in-Chief of this journal for his valuable suggestions which have considerably improved the final version of this article.     

一种新的求解带约束的有限极大极小问题的精确罚函数

提出了一种新的精确光滑罚函数求解带约束的极大极小问题．仅仅添加一个额外的变量,利用这个精确光滑罚函数,将带约束的极大极小问题转化为无约束优化问题． 证明了在合理的假设条件下,当罚参数充分大,罚问题的极小值点就是原问题的极小值点．进一步,研究了局部精确性质．数值结果表明这种罚函数算法是求解带约束有限极大极小问题的一种有效算法．     

带等式约束的光滑优化问题的一类新的精确罚函数

罚函数方法是将约束优化问题转化为无约束优化问题的主要方法之一. 不包含目标函数和约束函数梯度信息的罚函数, 称为简单罚函数. 对传统精确罚函数而言, 如果它是简单的就一定是非光滑的; 如果它是光滑的, 就一定不是简单的. 针对等式约束优化问题, 提出一类新的简单罚函数, 该罚函数通过增加一个新的变量来控制罚项. 证明了此罚函数的光滑性和精确性, 并给出了一种解决等式约束优化问题的罚函数算法. 数值结果表明, 该算法对于求解等式约束优化问题是可行的.     

基于拟态物理学的约束多目标共轭梯度混合算法

在拟态物理学优化算法APO的基础上,将一种基于序值的无约束多目标算法RMOAPO的思想引入到约束多目标优化领域中.提出一种基于拟态物理学的约束多目标共轭梯度混合算法CGRMOAPA.算法采取外点罚函数法作为约束问题处理技术,并借鉴聚集函数法的思想,将约束多目标优化问题转化为单目标无约束优化问题,最终利用共轭梯度法进行求解.通过与CRMOAPO、MOGA、NSGA-II的实验对比,表明了算法CGRMOAPA具有较好的分布性能,也为约束多目标优化问题的求解提供了一种新的思路.     

求解MINLP的几类罚函数方法的分析和探讨

针对混合整数非线性约束优化问题(MINLP)的一般形式,通过罚函数的方法,给出了它的几种等价形式,并证明了最优解的等价性.将约束优化问题转化成更容易求解的无约束非线性优化问题,并把混合整数规划转化成非整数优化问题,从而将MINLP的求解简化为求解一个连续的无约束非线性优化问题,进而可用已有的一般无约束优化算法进行求解.     

随机R0张量互补问题的投影Levenberg-Marquardt方法

本文考虑一类离散型随机$R_0$张量互补问题,利用Fischer-Burmeister函数将问题转化为约束优化问题,并用投影Levenberg-Marquardt方法对其进行了求解。在一般的条件下得到了该方法的全局收敛性,相关的数值实验表明了该方法的有效性。     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.