Abstract
In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.
Highlights
The unconstrained minimization problem of the sum of two convex functions is modeled as the following form: min f (u) + g(u), (1)
Inspired by Cruz and Nghia [14], we suggest a new projected forward-backward algorithm for solving the constrained convex minimization problem, which is modeled as follows: min f (u) + g(u), (4)
We investigated inertial projected forward-backward algorithm using linesearches for constrained minimization problems
Summary
The unconstrained minimization problem of the sum of two convex functions is modeled as the following form: min f (u) + g(u), (1). Some works that relate to the forward-backward method for convex optimization problems can be investigated in [1,2,3,4,5,6]. This method covers the gradient method [7,8,9] and the proximal algorithm [10,11,12]. Combettes and Wajs [13] introduced the following relaxed forward-backward method
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