伪单调型向量F-互补问题可行集的最小元问题

本文引入了几类向量F-互补问题并给出了向量F-互补问题与广义向量变分不等式之间的关系.通过定义向量F-互补问题的可行集,研究了伪单调型向量F-互补问题的可行集的最小问题,推广了已有的一些结果.     

一类非单调互补问题的Canonical对偶问题研究

对于一类具有广泛应用背景的非单调互补问题,我们构建了这类问题的Canonical对偶问题。其对偶问题可以写成和原问题类似的互补问题。我们给出了对偶问题和原问题解之间的对偶关系,并且将对偶问题转化成一个一维优化问题,这不但可以方便的求解这类问题,也为研究这类问题性质提供了一个非常直观的研究工具。最后,本文给出了几个算例来演示对偶问题的性质。     

广义F-互补问题及其与变分不等式问题的等价性

在Banach空间中引入了一类广义F-互补问题,研究了它与一类变分不等式问题的等价性关系,给出了此类变分不等式问题的解的存在唯一性定理.     

论求解单调变分不等式的一些投影收缩算法

论求解单调变分不等式的一些投影收缩算法何炳生（南京大学）ＯＮＳＯＭＥＰＲＯＪＥＣＴＩＯＮＡＮＤＣＯＮＴＲＡＣＴＩＯＮＭＥＴＨＯＤＳＦＯＲＳＯＬＶＩＮＧＭＯＮＯＴＯＮＥＶＡＲＩＡＴＩＯＮＡＬＩＮＥＱＵＡＬＩＴＩＥＳ￥ＨｅＢｉｎｇ－ｓｈｅｎｇ（Ｎａｍｉｎ...     

关于线性互补问题的一个直接法

一类交分不等式问题通过有限差分法或有限元法离散可归为线性互补问题:     

关于一类变分不等式的LQP算法

本文研究一类ξ-单调的变分不等式问题.利用KKT条件将原问题转换为非线性互补问题(nonlinear complementarity problem,NCP)的方法,获得了基于logarithmic-quadratic proximal(LQP)的算法及其改进形式,推广了LQP算法的适用范围.     

求解变分不等式和互补问题的一种迭代法

最近,有限维的非线性变分不等式和互补问题的研究有了较快的发展,具体可见Harker和Pang的综述性文献。但是,在没有(强)单调性及可微性的条件下,却没有一个实用的算法。本文的主要兴趣是研究一种迭代算法—外梯度,在连续性和伪单调性条件下,证明了算法的全局收敛性(定理3.1)。     

互补问题算法的新进展

互补问题是一类重要的优化问题,在最近３０多年的时间里,人们为求解它而提出了许多算法,该文主要介绍１９９０－１９９７年之间出现的某些新算法,它们大致可归类为：（１）光滑方程法;（２）非光滑方程法;（３）可微无约束优化法;（４）ＧＬＰ投影法;（５）内点法;（６）磨光与非内点连续法,文中对每类算法及相应的收敛性结果做了描述与评论,并列出有关文献。     

求解单调变分不等式问题的一类迭代方法

１．引言 变分不等式问题在数学规划中起着重要作用,它最初作为研究偏微分方程的工具,首先由 Ｆｉｓｈｅｒａ和 Ｓｔａｍｐａｃｃｈｉａ等于六十年代初提出,可参看［１］及其参考文献,之后也被广泛用于研究经济学和运筹学等领域中的均衡模型,互补问题和凸规划问题都是变分不等式问题的特殊情形,文献［２］对有限维变分不等式问题和非线性互补问题的理论、算法及应用作了十分全面的综述．设 Ｃ是实有限维空间 Ｒｎ,的非空闲凸子集, Ｆ是 Ｒｎ → Ｒｎ的映射,本文讨论的变分不等式问题ＶＩ（Ｃ,Ｆ）是： 求向量ｒ＊∈Ｃ．使得：Ｆ（…     

无穷维Hilbert空间中的伪单调非线性互补问题

建立伪单调非线性互补问题在实Hilbert空间任意闭凸锥上的解的存在性理论,特别把互补问题在有限维实Hilbert空间上的一些重要理论推广到无限维实Hilbert空间,还证明了解的存在的可行性理论.     

Finite termination of a smoothing-type algorithm for the monotone affine variational inequality problem

In this paper, we consider the monotone affine variational inequality problem (AVIP for short). Based on a smooth reformulation of the AVIP, we propose a Newton-type method to solve the monotone AVIP, where a testing procedure is embedded into our algorithm. Under mild assumptions, we show that the proposed algorithm may find a maximally complementary solution to the monotone AVIP in a finite number of iterations. Preliminary numerical results are reported.     

关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注

Yair Censor,Aviv Gibali和Simeon Reich为求解变分不等式问题提出了 2-次梯度外梯度算法.关于此算法的收敛性,作者给出了部分证明,有一个问题:由算法产生的迭代点列能否收敛到变分不等式问题的一个解上,没有得到解决.此问题作为一个公开问题在文章"Extensions of Korpelevi...     

Proximal algorithm for solving monotone variational inclusion

In this paper, we introduce a contraction algorithm for solving monotone variational inclusion problem. To reach this goal, our main iterative algorithm combine Dong’s projection and contraction algorithm with resolvent operator. Under suitable assumptions, we prove that the sequence generated by our main iterative algorithm converges weakly to the solution of the considered problem. Finally, we give two numerical examples to verify the feasibility of our main algorithm.     

Convergence of a non-interior continuation algorithm for the monotone SCCP

It is well known that the symmetric cone complementarity problem（SCCP） is a broad class of optimization problems which contains many optimization problems as special cases.Based on a general smoothing function,we propose in this paper a non-interior continuation algorithm for solving the monotone SCCP.The proposed algorithm solves at most one system of linear equations at each iteration.By using the theory of Euclidean Jordan algebras,we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions.     

On the strong convergence of a projection-based algorithm in Hilbert spaces

In this paper, we introduce a new projection-based algorithm for solving variational inequality problems with a Lipschitz continuous pseudo-monotone mapping in Hilbert spaces. We prove a strong convergence of the generated sequences. The numerical behaviors of the proposed algorithm on test problems are illustrated and compared with previously known algorithms.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

Global linear convergence of a path-following algorithm for some monotone variational inequality problems

We consider a primal-scaling path-following algorithm for solving a certain class of monotone variational inequality problems. Included in this class are the convex separable programs considered by Monteiro and Adler and the monotone linear complementarity problem. This algorithm can start from any interior solution and attain a global linear rate of convergence with a convergence ratio of 1 ?c/√m, wherem denotes the dimension of the problem andc is a certain constant. One can also introduce a line search strategy to accelerate the convergence of this algorithm.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.