一类变分不等式的惯性黄金比例算法的研究

Research on the Inertial Golden Ratio Algorithm for a Class of Variational Inequalities

摘要: 变分不等式是优化理论中的重要问题，在许多实际问题中都有广泛应用。传统求解变分不等式的方法有：投影外梯度法、镜像梯度法、邻近点算法、交替方向法、黄金比例算法等。黄金比例算法相较于投影梯度算法在每次迭代的过程中减少了一次投影的运算，因而提高了收敛速度。本文旨在研究一类变分不等式的黄金比例算法，在已有的一步惯性算法的基础上提出两步惯性黄金比例算法，并证明了在适当的参数条件下该算法的弱收敛性。最后通过数值实验比较了Malitsky的算法，Yang和Liu的算法，Chinedu和Yekini的算法以及我们所提出的算法在迭代时间和迭代步数上的性能差异，验证了本文所提出算法的优异性。

Abstract: Variational inequality is an important problem in optimization theory. The traditional methods for solving variational inequalities include: extragradient method, mirror gradient method, proximity point algorithm, alternating direction method, golden ratio algorithm, etc. Compared with the projection gradient algorithm, the golden ratio algorithm reduces the operation of one projection in the process of each iteration, so as to improve the convergence speed. The purpose of this thesis is to study a kind of golden ratio algorithm for variational inequality, and to propose a two-step inertial golden ratio algorithm based on the one-step ones. The weak convergence of the algorithm is proved under the appropriate parameter conditions. Finally, numerical experiments are carried out to compare the performance differences of Malitsky’s algorithm, Yang and Liu’s algorithms, Chinedu and Yekini’s algorithms, and the proposed algorithm in terms of iteration time and iteration steps, which verifies the superiority of the proposed algorithm.