The forward-backward splitting algorithm is a popular operator splitting method for finding a zero of the sum of two maximal monotone operators, with one of which is cocoercive operator. In this paper, we present a new convergence analysis of a variable metric forward-backward splitting algorithm with relaxation in real Hilbert spaces. We prove the weak convergence of this algorithm under some weak conditions on the relaxation parameters. Moreover, we allow the relaxation parameters larger than one. Consequently, we recover a variable metric proximal point algorithm. 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. 查看全文>>