求解非单调变分不等式的一种半空间投影算法

本文提出了一种求解非单调变分不等式的半空间投影算法,在映射是连续和对偶变分不等式解集非空的假设条件下证明了该算法生成的无穷序列是全局收敛的,并在局部误差界和Lipschitz连续条件下给出了收敛率分析.通过数值实验验证了所提出算法的有效性和可行性.     

求解拟单调变分不等式问题的交替惯性向前向后算法

该文结合线搜索方法,提出了改进的交替惯性向前向后算法求解拟单调变分不等式问题.该算法在每次迭代时只需计算一次到可行集上的投影,在一定的假设下证明了解集的弱收敛性定理.最后通过数值实验验证了算法的有效性.     

一种求解半定规划的邻近外梯度算法

本文提出了一种求解半定规划的邻近外梯度算法.通过转化半定规划的最优性条件为变分不等式,在变分不等式满足单调性和Lipschitz连续的前提下,构造包含原投影区域的半空间,产生邻近点序列来逼近变分不等式的解,简化了投影的求解过程.将该算法应用到教育测评问题中,数值实验结果表明,该方法是解大规模半定规划问题的一种可行方法.     

求解伪单调变分不等式的自适应加速外梯度算法

本文改进Tseng的外梯度算法,引入了一种新的求解伪单调变分不等式的投影算法.该算法的步长是自适应的,在Lipschitz常数未知的情况下通过一个简单的计算逐步更新.结合惯性加速技巧,在算子A是伪单调且Lipschitz连续的假设下,证明了该算法所产生的序列强收敛到变分不等式的解.进行的一些数值试验表明了所提出的算法比现有的一些算法具有竞争优势.     

一种新的求解拟单调变分不等式的压缩投影算法

2020年Liu和Yang提出了求解Hilbert空间中拟单调且Lipschitz连续的变分不等式问题的投影算法,简称LYA。本文在欧氏空间中提出了一种新的求解拟单调变分不等式的压缩投影算法,简称NPCA。新算法削弱了LYA中映射的Lipschitz连续性。在映射连续、拟单调且对偶变分不等式解集非空的条件下得到了NPCA所生成点列的聚点是解的结论。当变分不等式的解集还满足一定条件时,得到了NPCA的全局收敛性。数值实验结果表明NPCA所需的迭代步数少于LYA的迭代步数,NPCA在高维拟单调例子中所需的计算机耗时也更少。     

一类求解单调变分不等式的隐式方法

1．引言变分不等式是一个非常有趣。非常困难的数学问题［"］．它具有广泛的应用（例如，数学规划中的许多基本问题都可以归结为一个变分不等式问题），因而得到深入的研究并有了不少算法［1，2，5－8，17－21］．对线性单调变分不等式，我们最近提出了一系列投影收缩算法Ig－13］．本文考虑求解单调变分不等式其中0CW是一闭凸集，F是从正p到自身的一个单调算子，一即有我们用比（·）表示到0上的投影．求解单调变分不等式的一个简单方法是基本投影法［1，6］，它的迭代式为然而，如果F不是仿射函数，只有当F一致强单调且LIPSChitZ连续…     

求解变分不等式与不动点问题公共解的新Tseng型外梯度算法

该文在Hilbert空间中提出一种新Tseng型外梯度算法,用以求解一致连续伪单调映射的变分不等式问题与具有半封闭性拟非扩张映射的不动点问题的公共解.在一定的假设条件下,证明了算法所生成的序列的强收敛性.文章最后对算法进行数值实验,验证了算法的有效性.     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

一种新的求解变分不等式的惯性双次梯度外梯度算法（英文）

当可行集为一光滑凸函数的下水平集时,文献[Optimization,2020,69(6):1237-1253]提出了一种惯性双次梯度外梯度算法来求解Hilbert空间中的单调且Lipschitz连续的变分不等式问题.该算法在每次迭代中仅需向一个半空间计算两次投影,并得到了算法的弱收敛结果.本文通过使用黏性方法以及在惯性步采用新的步长来修正该算法.在适当的假设条件下证明了新算法所生成的序列能强收敛到变分不等式的一个解.此外,新算法在每次迭代中也仅需向半空间计算两次投影.     

非单调变分不等式黄金分割算法研究

该文考虑变分不等式的梯度投影算法,给出了一种非单调变分不等式的黄金分割算法,所给出的算法特点结合了惯性加速方法,无需知道映射的Lipschitz常数,且步长是非单调递减的.在一定的条件下,算法的收敛性被证明.最后给出数值实验结果.     

Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem

Many practical problems such as signal processing and network resource allocation are formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms to solve these problems have been proposed. This paper discusses a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping, which is called the triple-hierarchical constrained optimization problem, and presents an iterative algorithm for solving it. Strong convergence of the algorithm to the unique solution of the problem is guaranteed under certain assumptions.     

Modified hybrid projection methods for finding common solutions to variational inequality problems

In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.     

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

     

Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ??-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A?Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.     

A modified projection method with a new direction for solving variational inequalities

In this paper, we propose a new projection method for solving variational inequality problems, which can be viewed as an improvement of the method of Li et al. [M. Li, L.Z. Liao, X.M. Yuan, A modified projection method for co-coercive variational inequality, European Journal of Operational Research 189 (2008) 310-323], by adopting a new direction. Under the same assumptions as those in Li et al. (2008), we establish the global convergence of the proposed algorithm. Some preliminary computational results are reported, which illustrated that the new method is more efficient than the method of Li et al. (2008).     

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.     

The projection and contraction methods for finding common solutions to variational inequality problems

In this paper, we firstly introduce two projection and contraction methods for finding common solutions to variational inequality problems involving monotone and Lipschitz continuous operators in Hilbert spaces. Then, by modifying the two methods, we propose two hybrid projection and contraction methods. Both weak and strong convergence are investigated under standard assumptions imposed on the operators. Also, we generalize some methods to show the existence of solutions for a system of generalized equilibrium problems. Finally, some preliminary experiments are presented to illustrate the advantage of the proposed methods.     

Inertial projection and contraction algorithms for variational inequalities

In this article, we introduce an inertial projection and contraction algorithm by combining inertial type algorithms with the projection and contraction algorithm for solving a variational inequality in a Hilbert space H. In addition, we propose a modified version of our algorithm to find a common element of the set of solutions of a variational inequality and the set of fixed points of a nonexpansive mapping in H. We establish weak convergence theorems for both proposed algorithms. Finally, we give the numerical experiments to show the efficiency and advantage of the inertial projection and contraction algorithm.