Abstract

In a real Hilbert space, let the VIP indicate a variational inequality problem with Lipschitzian, pseudomonotone operator, and let the FPP denote a fixed-point problem of a quasi-nonexpansive operator with a demiclosedness property. This article designs two quasi-inertial Tseng’s extragradient algorithms with adaptive stepsizes for finding a common solution of the VIP and FPP. The proposed algorithms are based on inertial Tseng’s extragradient approach with adaptive stepsizes, hybrid steepest-descent algorithm, and viscosity approximation technique. Under appropriate assumptions, it is proven that the sequences constructed by the proposed algorithms converge strongly to a common solution of the VIP and FPP. Finally, the main results are applied to deal with the VIP and FPP in an illustrating example.

摘要

在真实的希尔伯特空间中，让VIP表示伪单调算子Lipschitzian的变分不等式问题，让FPP表示具有非封闭性质的拟非扩张算子的不动点问题。本文设计了两种具有自适应步长的准惯性曾氏超梯度算法，以找到VIP和FPP的通用解决方案。所提出的算法基于具有自适应步长的惯性曾氏超梯度方法，混合最速下降算法和粘度逼近技术。在适当的假设下，证明了由所提出的算法构建的序列强烈地收敛于VIP和FPP的通用解决方案。最后，在一个说明性示例中，将主要结果应用于处理VIP和FPP。