New inertial proximal gradient methods for unconstrained convex optimization problems

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.