On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces. (1st February 2016)

• Record Type: Journal Article
• Authors: Bello Cruz, J. Y.; De Oliveira, W.
• Is Part Of: Numerical functional analysis and optimization. Volume 37:Number 2(2016)
• Page Start: 129
• Page End: 144
• Publication Date: 2016-02-01
• Subjects: Armijo line search -- convex minimization -- projection method -- strong and weak convergence; 90C25 -- 90C30; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2015.1080271
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.