The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

中文翻译：

---

求解无约束凸优化问题的新惯性近端梯度法

近端梯度法是解决复合凸优化问题的强大工具。本文首先提出了一种基于粘性逼近和近端比例梯度算法的不精确惯性加速方法，以加快算法的收敛速度。在合理的参数下，我们证明了我们的算法强烈收敛于该问题的某些解，这是变分不等式问题的唯一解。其次，提出了一种不精确的交替惯性近点算法。在适当的条件下，证明了弱收敛定理。最后，数值结果说明了我们算法的性能，并与相关算法进行了比较。我们的结果改进和扩展了许多作者最近报告的相应结果。