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In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

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The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem. Theoretically, we first introduce an algorithm based on viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive operators. Under some suitable assumptions, a strong convergence theorem of the proposed algorithm is established. Subsequently, we utilize our proposed algorithm to solving a convex minimization problem of the sum of two convex functions. As an application, we apply and analyze our algorithm to image restoration problems. Moreover, we compare convergence behavior and efficiency of our algorithm with other well-known methods such as the forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm. By using image quality metrics, numerical experiments show that our algorithm has a higher efficiency than the mentioned algorithms.