Iterative Algorithms For Constrained Convex Optimization Problems In Hilbert Spaces

Nonexpansive mapping is one of the most important nonlinear operators, whose iterative algorithms for fixed point problems are widely applied to optimization problems,equation problems and even economic problems. In this paper, two iterative algorithms are proposed for constrained convex optimization problems in Hilbert spaces. One combines Yamada’s hybrid steepest descent method and the regularized gradient-projection algorithm, it is strictly proved that the strong convergence of this iterative algorithm is obtained, and some variational inequality is also solved. The other uses iterative methods of equilibrium problems, and strong convergence theorems for solving convex optimization problems are obtained. The results of this paper extend and improve some existed results.This paper is mostly composed of three parts.The first part: Firstly, we introduce the background of the convex optimization problems and its applications; Secondly, we introduce the relationship between convex optimization problems and fixed point problems, and the research situations of their iterative algotithms, at the same time, we sketch out the relationship between convex optimization problems and equilibrium problems, and the research situations of iterative algorithms for equilibrium problems; Finally, we introduce the main research work of the author in this paper.The second part: Based on Yamada’s hybrid steepest descent method, we construct an iterative algorithm for a constrained convex optimization problem in Hilbert spaces,under proper conditions, the strong convergence of this algorithm is proved, and the unique solution of some variational inequality is established.The third part: We use iterative methods of equilibrium problems to construct a new iterative method for finding solutions of a type of convex optimization problems in Hilbert spaces, under proper conditions, the strong convergence of this iterative algorithm is proved.