This paper deals with a class of inertial projection and contraction algorithms for solving a variational inequality problem involving quasimonotone and Lipschitz continuous mappings in Hilbert spaces. The algorithms incorporate inertial techniques and the Barzilai–Borwein step size strategy, moreover their line search conditions and some parameters are relaxed to obtain larger step sizes. The weak convergence of the algorithms is proved without the knowledge of the Lipschitz constant of the mappings. Meanwhile, the nonasymptotic convergence and the linear convergence of the algorithms are established. Some numerical experiments show that the proposed algorithms are more effective than some existing ones.

本文涉及一类惯性投影和收缩算法，用于解决涉及 Hilbert 空间中拟单调和 Lipschitz 连续映射的变分不等式问题。该算法结合了惯性技术和 Barzilai-Borwein 步长策略，而且它们的线搜索条件和一些参数被放宽以获得更大的步长。在不知道映射的 Lipschitz 常数的情况下证明了算法的弱收敛性。同时，建立了算法的非渐近收敛性和线性收敛性。一些数值实验表明，所提出的算法比现有的一些算法更有效。