求解非单调变分不等式的一种二次投影算法北大核心CSCD

投影算法是求解变分不等式问题的主要方法之一.目前,有关投影算法的研究通常需要假设映射是单调且Lipschitz连续的,然而在实际问题中,往往不满足这些假设条件.该文利用线搜索方法,提出了一种新的求解非单调变分不等式问题的二次投影算法.在一致连续假设下,证明了算法产生的迭代序列强收敛到变分不等式问题的解.数值实验结果表明了该文所提算法的有效性和优越性.     

随机广义拟变分不等式的迭代解法及应用

为了获得Hilbert空间中一类随机广义拟变分不等式的迭代解法,证明了点到由具闭(凸)值的随机集值映射所刻画的变约束集上的投影算子的可测性.利用该可测性结果和可测选择定理,构造了求解随机广义拟变分不等式的随机迭代算法.在单调性及Lipschitz连续性条件下,获得了由算法生成的随机序列的收敛性.作为应用,给出了随机广义Nash博弈和随机Walrasian均衡问题的一些刻画性结果.     

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.     

一种新修正的求解变分不等式的双次梯度外梯度算法

当可行集为一光滑凸函数的下水平集时, 本文提出一种修正的双次梯度外梯度算法(MTSEGA)用于求解Hilbert空间中单调且Lipschitz连续的变分不等式. MTSEGA在每步迭代过程中仅需计算向半空间的两次投影及一次映射的值. 在与已知算法相同的假设条件下, 证明了新算法产生的序列能弱收敛到相关问题的一个解.     

无单调性集值变分不等式的一种投影算法

本文给出了求解无单调性集值变分不等式的一个新的投影算法,该算法所产生的迭代序列在Minty变分不等式解集非空且映射满足一定的连续性条件下收敛到解.对比文献[10]中的算法,本文中的算法使用了不同的线性搜索和半空间,在计算本文所引的两个数值例子时,该算法比文献[10]中的算法所需迭代步更少.     

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.     

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.     

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.     

一般单调变分不等式的近似邻近外梯度算法

近似邻近点算法是求解单调变分不等式的一个有效方法,该算法通过解决一系列强单调子问题,产生近似邻近点序列来逼近变分不等式的解,而外梯度算法则通过每次迭代中增加一个投影来克服一般投影算法限制太强的缺点,但它们均未能改变迭代步骤中不规则闭凸区域上投影难计算的问题.于是,本文结合外梯度算法的迭代格式,构造包含原投影区域的半空间,将投影建立在半空间上,简化了投影的求解过程,并对新的邻近点序列作相应限制,使得改进的算法具有较好的收敛性.     

一类广义非线性混合拟变分包含组

本文引入了一类新的带松弛单调映射和松弛Lipschitz映射的广义非线性混合拟变分包含组 ,构造了求解该类变分包含组的迭代算法 ,证明了该类变分包含组解的存在性以及由本文构造的迭代算法产生的迭代序列的强收敛性 .所得结果推广和改进了大量文献中的最新结果[1 5 ] .     

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.

     

Weak sharp solutions for variational inequalities in Banach spaces

In this paper, we introduce the notion of a weak sharp set of solutions to a variational inequality problem (VIP) in a reflexive, strictly convex and smooth Banach space, and present its several equivalent conditions. We also prove, under some continuity and monotonicity assumptions, that if any sequence generated by an algorithm for solving (VIP) converges to a weak sharp solution, then we can obtain solutions for (VIP) by solving a finite number of convex optimization subproblems with linear objective. Moreover, in order to characterize finite convergence of an iterative algorithm, we introduce the notion of a weak subsharp set of solutions to a variational inequality problem (VIP), which is more general than that of weak sharp solutions in Hilbert spaces. We establish a sufficient and necessary condition for the finite convergence of an algorithm for solving (VIP) which satisfies that the sequence generated by which converges to a weak subsharp solution of (VIP), and show that the proximal point algorithm satisfies this condition. As a consequence, we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of (VIP).     

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.     

Convergence of a subgradient extragradient algorithm for solving monotone variational inequalities

Numerical Algorithms - In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert...     

Monotone mixed variational inequalities

We consider and analyze some new splitting methods for solving quasi-monotone mixed variational inequalities by using the technique of updating the solution. The modified methods converge for quasi-monotone continuous operators. The new splitting methods differ from the existing splitting methods. Proof of convergence is very simple.     

An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems

In this paper, we introduce an inertial subgradient-type algorithm to find the common element of fixed point set of a family of nonexpansive mappings and the solution set of the single-valued variational inequality problem. Under the assumption that the mapping is monotone and Lipschitz continuous, we show that the sequence generated by our algorithm converges strongly to some common element of the fixed set and the solution set. Moreover, preliminary numerical experiments are also reported.     

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.