Accelerated Self-Adaptive Method for Solving Nonsmooth Convex Minimization Problem in Real Hilbert Spaces

In this manuscript, we propose a proximal gradient type algorithm together with a two step inertia method for approximating solution of convex minimization problem in real Hilbert spaces. The proposed proximal gradient type method is designed in such a way that it does not depend on the Lipschitz constant. Using a self-adaptive rule, we obtain a weak convergence result under the condition that the gradient function of one of the convex functions is uniformly continuous. Preliminary numerical results show that our proposed method has a better convergence in comparison to some other related results in the literature.