An inexact proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth optimization problems

Abstract

In this paper, we propose an inexact version of proximal gradient algorithm with extrapolation for solving a class of nonconvex nonsmooth optimization problems. Specifically, the subproblem in proximal gradient algorithm with extrapolation is allowed to be solved inexactly by certain relative error criterion, in the sense that the criterion can be updated adaptively in each iteration. Under the assumption that an auxiliary function satisfies the Kurdyka-Łojasiewicz (KL) inequality, we prove that the iterative sequence generated by the inexact proximal gradient algorithm with extrapolation converges to a stationary point of the considered problem. Furthermore, the convergence rate of the proposed algorithm can be established when the KL exponent is known. Moreover, we illustrate the advantage by applying the algorithm to solve a nonconvex optimization problem.