Mathematical programming for the sum of two convex functions with applications to lasso problem, split feasibility problems, and image deblurring problem

Chih Sheng Chuang,Zenn-Tsun Yu,Lai-Jiu Lin

doi:10.1186/s13663-015-0388-0

Chih Sheng Chuang, Zenn-Tsun Yu + Show 1 more

Open Access

https://doi.org/10.1186/s13663-015-0388-0

Copy DOI

Abstract

AbstractIn this paper, two iteration processes are used to find the solutions of the mathematical programming for the sum of two convex functions. In infinite Hilbert space, we establish two strong convergence theorems as regards this problem. As applications of our results, we give strong convergence theorems as regards the split feasibility problem with modified CQ method, strong convergence theorem as regards the lasso problem, and strong convergence theorems for the mathematical programming with a modified proximal point algorithm and a modified gradient-projection method in the infinite dimensional Hilbert space. We also apply our result on the lasso problem to the image deblurring problem. Some numerical examples are given to demonstrate our results. The main result of this paper entails a unified study of many types of optimization problems. Our algorithms to solve these problems are different from any results in the literature. Some results of this paper are original and some results of this paper improve, extend, and unify comparable results in existence in the literature.

Highlights

Let R be the set of real numbers, H and H be Hilbert spaces with inner product ·, · and norm ·
We give a new iteration to study the (SFP), we give our modified CQ method to study (SFP), we study two strong convergence theorem to the solution of this problem
A special case of one of our iteration is modified gradient-projection algorithm. We use this modified gradient-projection algorithm to establish a strong convergence theorem of this problem (AP ), and our results improve recent results given by Xu in [ ]

Summary

Introduction

Let R be the set of real numbers, H and H be (real) Hilbert spaces with inner product ·, · and norm ·. We establish a modified proximal point algorithm and prove a strongly convergence theorem to study this problem. We use this modified gradient-projection algorithm to establish a strong convergence theorem of this problem (AP ), and our results improve recent results given by Xu in [ ].

Results

Conclusion