Proximal Algorithms Analysis And Applications For Unconstrained Convex Optimization Problems

Until now, the problems of image restoration and signal processing have been extensively studied and applied. At present, most of these problems can be transformed into the minimization problem which involving two convex functions. Therefore, it is particular important to design convergent algorithms which are fast and efficient.After the study in recent years, the author finds that the proximity algorithms can get better results for solving the unconstrained convex optimization problem using the equality relationship between the proximity operator and the subdifferential of a convex function. However, the author notices that most of the proximity algorithms for solving the corresponding problems are weak convergent. Therefore, it is crucial to design strong convergence algorithms containing proximity operators.In this paper, the author intends to propose a series of algorithms based on proximity operators for solving the convex minimization problems, in which one function is convex and differentiable, and the other is convex and subdifferentiable. The author calls these algorithms respectively viscosity iterative algorithms and general viscosity iterative approximation for solving unconstrained convex optimization problems. Then, the main contents and results of the paper are listed below.Firstly, based on the existing weak convergence algorithm, the author combines viscosity iterative algorithms with the proximity operator, and transfer it to the strong convergence algorithm.Secondly, the author combines a sequence of contractive mappings with the proximal operator and proposes a viscosity approximation method for solving the unconstrained convex optimization problems. Meanwhile, the author gets the convergence point of the iterative method which is also the unique solution of some variational inequality problem. Further a numerical example will be given to demonstrate the effectiveness of our iterative scheme.Thirdly, inspired by Tian's general iterative algorithm the author obtains the generalized iterative algorithm for unconstrained convex optimization problems. Implicit and explicit iterative methods are provided while the proofs are given.