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

Abstract

The forward-backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and a cocoercive 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.

Convergence of sequences: a survey

“…Let us consider in this section the more general setting with variable metric, i.e., cases in which the metric is allowed to change at each iteration. Applications of these results involve theoretical problems as monotone inclusions [102,52], as well as inverse problems [31], convex feasibility problems [71,31] and constrained convex minimization [103]. All the results in this section concern Féjer properties and we consider mostly the deterministic case.…”

Distributed Forward-Backward algorithms for stochastic generalized Nash equilibrium seeking

“…Therefore, a variable metric, induced by Φ k should be used for the convergence analysis. Such analysis is possible considering the results in [5] and [38,39]. In particular, the FB algorithm in this case reads as…”

On a variable metric iterative method for solving the elastic net problem

“…The quasi‐Fejér monotonicity is a fundamental monotonicity, which satisfies the following condition ‖ z n − z ‖ + ρ n ≥ ‖ z n + 1 − z ‖, with respect to the sequence , where is a summable nonnegative real number sequence in [0, + ∞ ) ( is a set of nonnegative integer numbers); see Ermol'ev and Tuniev 12 . The quasi‐Fejér monotonicity has been widely used as an efficient and fundamental tool to unify the convergence analysis of a large collection of algorithms in many works; see literature 13‐17 . In order to better understand the convergence properties of splitting algorithms, Combettes and Vũ 18,19 extended the notion of the quasi‐Fejér monotonicity to the context of variable metric iterations in general Hilbert spaces and investigated its properties recently.…”