Abstract
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.
Highlights
The convex minimization problem is one of the important problems in mathematical optimization
We focus on the forward-backward splitting algorithm based on the viscosity approximation method [21, 34] as follows
We introduce a new modification of Linesearches A and B and present a double forward-backward algorithm based on the viscosity approximation method by using an inertial technique for solving Problem (1) with Assumptions (AI) and (AII)
Summary
The convex minimization problem is one of the important problems in mathematical optimization. Various optimization methods for solving the convex minimization problem have been introduced and developed by many researchers, see [1, 3,4,5, 7,8,9, 11, 14, 16,17,18,19, 23, 26, 28] for instance. If ∇h1 is Lipschitz continuous with a coefficient L > 0 and α ∈ (0, 2/L), the forward-backward operator FBα is nonexpansive. In this case, we can employ fixed point approximation methods for the class of nonexpansive operators to solve (1). One of the popular methods is known as the forward-backward splitting (FBS) algorithm [8, 18]
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