变分不等式与不动点问题的惯性类Mann次梯度外梯度算法

Inertial Mann-Type Subgradient Extragradient Algorith for Variational Inequalities and Fixed Point Problems

摘要: 本文在实Hilbert空间中提出一种新的惯性次梯度外梯度算法，用于求解变分不等式与拟非扩张映射不动点问题的公共解。该算法融合三类技术策略：其一，采用惯性加速步对历史迭代信息进行外推，以提升收敛速率；其二，借助凸函数的次梯度信息构造包含可行集的半空间，将投影算子从复杂可行集转移至结构简单的半空间，有效降低每轮迭代的计算成本；其三，引入类Mann迭代格式生成新的迭代点，增强算法对拟非扩张映射的适应性。步长通过线性搜索过程自适应获取，无需预知映射的Lipschitz常数。在映射单调且Lipschitz连续、拟非扩张映射在零点处半闭等标准假设下，建立了算法的弱收敛性。与现有算法相比，本文方法在保持理论保证的同时，通过次梯度投影策略显著减少了计算负担。数值实验验证了该算法相较于文献中相关算法的有效性与竞争力。

Abstract: This paper proposes a novel inertial subgradient extragradient algorithm in real Hilbert spaces for finding a common solution to variational inequalities and fixed point problems of quasi-nonexpansive mappings. The algorithm integrates three key strategies: (i) an inertial extrapolation step that leverages historical iteration information to accelerate convergence; (ii) a half-space constructed via subgradient information of the convex function, which replaces the projection onto the original feasible set and substantially reduces computational cost per iteration; and (iii) a Mann-type update scheme that enhances adaptability to quasi-nonexpansive mappings. The step size is adaptively determined by a line search procedure, eliminating the need for prior knowledge of the Lipschitz constant. Under standard assumptions—including monotonicity and Lipschitz continuity of the mapping, and demiclosedness at zero of the quasi-nonexpansive mapping—weak convergence of the proposed algorithm is established. Compared with existing methods, the proposed approach significantly alleviates computational burden through the subgradient projection strategy while retaining theoretical guarantees. Numerical experiments demonstrate its effectiveness and competitiveness against recent algorithms in the literature.