Regularized gradient-projection methods for the constrained convex minimization problem and the zero points of maximal monotone operator

Ming Tian,Si-Wen Jiao

doi:10.1186/s13663-015-0258-9

Abstract

In this paper, based on the viscosity approximation method and the regularized gradient-projection algorithm, we find a common element of the solution set of a constrained convex minimization problem and the set of zero points of the maximal monotone operator problem. In particular, the set of zero points of the maximal monotone operator problem can be transformed into the equilibrium problem. Under suitable conditions, new strong convergence theorems are obtained, which are useful in nonlinear analysis and optimization. As an application, we apply our algorithm to solving the split feasibility problem and the constrained convex minimization problem in Hilbert spaces.

Highlights

Throughout this paper, let N and R be the sets of positive integers and real numbers, respectively
In, Zeng et al [ ] proved a strong convergence theorem for finding a common element of the solution set EP of a generalized equilibrium problem and the set T– ∩ T– for two maximal monotone operators T and T defined on a Banach space X: x ∈ X and xn+ =
Let B be a maximal monotone operator on H such that the domain of B is included in C, and define the set of zero points of B as follows: We always denote Fix(T) as the fixed point set of the nonexpansive mapping T, denote U as the solution set of the constrained convex minimization problem ( . ), and denote EP(F) as the solution set of the equilibrium problem ( . )

Summary

Introduction

Throughout this paper, let N and R be the sets of positive integers and real numbers, respectively. Let B be a maximal monotone operator on H and define the set of zero points of B as follows: B– = {x ∈ H : ∈ Bx}.

Results

Conclusion