The variable metric forward-backward splitting algorithm under mild differentiability assumptions

TL;DR

This paper introduces a variable metric forward-backward splitting algorithm for convex minimization that converges under mild differentiability assumptions without requiring Lipschitz continuity of the gradient, expanding applicability to Banach spaces.

Contribution

It proves convergence of the algorithm under weaker assumptions than standard, including line search procedures and applications to divergence-type functions.

Findings

Weak convergence of iterates established

Convergence in objective function values proven

O(1/k) convergence rate achieved with stronger assumptions

Abstract

We study the variable metric forward-backward splitting algorithm for convex minimization problems without the standard assumption of the Lipschitz continuity of the gradient. In this setting, we prove that, by requiring only mild assumptions on the smooth part of the objective function and using several types of line search procedures for determining either the gradient descent stepsizes, or the relaxation parameters, one still obtains weak convergence of the iterates and convergence in the objective function values. Moreover, the $o(1/k)$ convergence rate in the function values is obtained if slightly stronger differentiability assumptions are added. We also illustrate several applications including problems that involve Banach spaces and functions of divergence type.