A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces. (17th April 2014) - Tian, Ming; Huang, Li-Hua
Abstract : The constrained convex minimization problem is to find a point x ∗ with the property that x ∗ ∈ C, and h ( x ∗ ) = min h ( x ), ∀ x ∈ C, where C is a nonempty, closed, and convex subset of a real Hilbert space H, h ( x ) is a real-valued convex function, and h ( x ) is not Fréchet differentiable, but lower semicontinuous. In this paper, we discuss an iterative algorithm which is different from traditional gradient-projection algorithms. We firstly construct a bifunction F 1 ( x, y ) defined as F 1 ( x, y ) = h ( y ) − h ( x ) . And we ensure the equilibrium problem for F 1 ( x, y ) equivalent to the above optimization problem. Then we use iterative methods for equilibrium problems to study the above optimization problem. Based on Jung's method (2011), we propose a general approximation method and prove the strong convergence of our algorithm to a solution of the above optimization problem. In addition, we apply the proposed iterative algorithm for finding a solution of the split feasibility problem and establish the strong convergence theorem. The results of this paper extend and improve some existing results.