Convergence analysis of a variable metric forward–backward splitting algorithm with applications

Journal of Inequalities and Applications

Fuying Cui¹  Chuanxi Zhu²  Yuchao Tang² 

[1] 0000 0001 2182 8825, grid.260463.5, Department of Mathematics, Nanchang University, Nanchang, P.R. China;0000 0001 2182 8825, grid.260463.5, Department of Mathematics, Nanchang University, Nanchang, P.R. China;0000 0001 2182 8825, grid.260463.5, School of Management, Nanchang University, Nanchang, P.R. China;

关键词: Forward–backward splitting algorithm;    Monotone inclusion;    Variable metric;    Split feasibility problem;    90C25;    47H05;    65K05;   

DOI  :  10.1186/s13660-019-2097-4

来源: publisher

【 摘 要 】

The forward–backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and an inverse strongly monotone operator. In this paper, we present a new convergence analysis of a variable metric forward–backward splitting algorithm with extended relaxation parameters in real Hilbert spaces. We prove that this algorithm is weakly convergent when certain weak conditions are imposed upon the relaxation parameters. Consequently, we recover the forward–backward splitting algorithm with variable step sizes. As an application, we obtain a variable metric forward–backward splitting algorithm for solving the minimization problem of the sum of two convex functions, where one of them is differentiable with a Lipschitz continuous gradient. Furthermore, we discuss the applications of this algorithm to the fundamental of the variational inequalities problem, constrained convex minimization problem, and split feasibility problem. Numerical experimental results on LASSO problem in statistical learning demonstrate the effectiveness of the proposed iterative algorithm.

【 授权许可】

CC BY