文章针对实Hilbert空间中的单调变分不等式和不动点连续映射的凸可行性问题,提出了一种非单调步长算法来求解。该算法利用可行集的信息构造特殊半空间,以及结合外梯度方法构造半空间。每次向两个半空间作投影。同时结合惯性加速技巧与M...文章针对实Hilbert空间中的单调变分不等式和不动点连续映射的凸可行性问题,提出了一种非单调步长算法来求解。该算法利用可行集的信息构造特殊半空间,以及结合外梯度方法构造半空间。每次向两个半空间作投影。同时结合惯性加速技巧与Mann迭代方法,在一定条件下,建立了所提算法的弱收敛性定理。最后,我们进行了一些计算测试,以证明所提算法的效率和优点,并与现有算法进行了比较。This paper presents a new inertial subgradient extragradient algorithm designed to solve variational inequalities and fixed point problems in real Hilbert spaces. Integrating the Mann iteration method with the subgradient extragradient approach and employing inertial acceleration techniques, the algorithm constructs a half-space using subgradient information and projects onto it. Step lengths are determined via a line search procedure, eliminating the need to compute the Lipschitz constant of the mapping. The algorithm’s weak convergence is established under assumptions like the pseudo-nonexpansiveness of the mappings. Finally, Numerical experiments additionally illustrate the algorithm’s advantages over existing approaches in the literature.展开更多
半定锥约束变分不等式问题是目前研究热门话题——锥约束变分不等式问题的其中一部分,在优化理论、经济学和工程应用中具有重要意义。在本文中,我们给出了半定锥约束变分不等式问题的二阶充分性条件。我们首先回顾了变分不等式的基本理...半定锥约束变分不等式问题是目前研究热门话题——锥约束变分不等式问题的其中一部分,在优化理论、经济学和工程应用中具有重要意义。在本文中,我们给出了半定锥约束变分不等式问题的二阶充分性条件。我们首先回顾了变分不等式的基本理论概念及其在半定锥约束下的具体形式,然后通过构造相应的拉格朗日函数,将问题的解与二阶充分性条件关联起来,最后我们给出了相关定理的具体内容及其证明,确保在给定约束条件下解的存在性与最优性。The semi-definite cone-constrained variational inequality problem is currently a hot topic of research and is a part of the broader cone-constrained variational inequality issues, which hold significant importance in optimization theory, economics, and engineering applications. In this paper, we present the second-order sufficient conditions for the semi-definite cone-constrained variational inequality problem. We first review the fundamental theoretical concepts of variational inequalities and their specific forms under semi-definite cone constraints. Then, by constructing the corresponding Lagrangian function, we establish a connection between the solutions of the problem and the second-order sufficient conditions. Finally, we provide the specific content and proof of the relevant theorems to ensure the existence and optimality of the solutions under the given constraints.展开更多
文摘文章针对实Hilbert空间中的单调变分不等式和不动点连续映射的凸可行性问题,提出了一种非单调步长算法来求解。该算法利用可行集的信息构造特殊半空间,以及结合外梯度方法构造半空间。每次向两个半空间作投影。同时结合惯性加速技巧与Mann迭代方法,在一定条件下,建立了所提算法的弱收敛性定理。最后,我们进行了一些计算测试,以证明所提算法的效率和优点,并与现有算法进行了比较。This paper presents a new inertial subgradient extragradient algorithm designed to solve variational inequalities and fixed point problems in real Hilbert spaces. Integrating the Mann iteration method with the subgradient extragradient approach and employing inertial acceleration techniques, the algorithm constructs a half-space using subgradient information and projects onto it. Step lengths are determined via a line search procedure, eliminating the need to compute the Lipschitz constant of the mapping. The algorithm’s weak convergence is established under assumptions like the pseudo-nonexpansiveness of the mappings. Finally, Numerical experiments additionally illustrate the algorithm’s advantages over existing approaches in the literature.
文摘半定锥约束变分不等式问题是目前研究热门话题——锥约束变分不等式问题的其中一部分,在优化理论、经济学和工程应用中具有重要意义。在本文中,我们给出了半定锥约束变分不等式问题的二阶充分性条件。我们首先回顾了变分不等式的基本理论概念及其在半定锥约束下的具体形式,然后通过构造相应的拉格朗日函数,将问题的解与二阶充分性条件关联起来,最后我们给出了相关定理的具体内容及其证明,确保在给定约束条件下解的存在性与最优性。The semi-definite cone-constrained variational inequality problem is currently a hot topic of research and is a part of the broader cone-constrained variational inequality issues, which hold significant importance in optimization theory, economics, and engineering applications. In this paper, we present the second-order sufficient conditions for the semi-definite cone-constrained variational inequality problem. We first review the fundamental theoretical concepts of variational inequalities and their specific forms under semi-definite cone constraints. Then, by constructing the corresponding Lagrangian function, we establish a connection between the solutions of the problem and the second-order sufficient conditions. Finally, we provide the specific content and proof of the relevant theorems to ensure the existence and optimality of the solutions under the given constraints.
