A fast viscosity forward-backward algorithm for convex minimization problems with an application in image recovery

Abstract

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.

A Novel Two-Step Inertial Viscosity Algorithm for Bilevel Optimization Problems Applied to Image Recovery

“…For the first of our main results, we draw upon the ideas of Jailoka et al [21] and Liang [24] and introduce a modified two-step inertial viscosity algorithm (MTIVA) for finding a common fixed point of a family of nonexpansive mappings {T n }, as follows:…”

“…where {β n } ∈ (0, 1), S is a contraction of C into itself, {T n } is a countable family of nonexpansive of C into itself, C is subset of a Banach space, and x 1 ∈ C. Takahashi proved the strong convergence of (9) to a common fixed point of T n . Jailoka et al [21] introduced a fast viscosity forward-backward algorithm (FVFBA) with the inertial technique for finding a common fixed point of a countable family of nonexpansive mappings. They proved a strong convergence result and applied it to solving a convex minimization problem of the sum of two convex functions.…”

Single-Valued Demicontractive Mappings: Half a Century of Developments and Future Prospects

“…We end this section by mentioning another important application of the above results to convex minimization over the fixed-point set of demicontractive mappings (a topic which is not reviewed here); see Maingé [21], Mewomo [128], Okeke et al [109], Chang et al [101], Jailoka et al [129], Arfat et al [31], etc.…”

A Novel Inertial Viscosity Algorithm for Bilevel Optimization Problems Applied to Classification Problems

“…Fixedpoint problems with nonexpansive mappings have been investigated by many authors using the method of viscosity approximation [21][22][23][24]. This method provides a strong convergence result and it is defined by the following:…”