On the convergence of the iterates of proximal gradient algorithm with extrapolation for convex nonsmooth minimization problems

In this paper, we consider the proximal gradient algorithm with extrapolation for solving a class of convex nonsmooth minimization problems. We show that for a large class of extrapolation parameters including the extrapolation parameters chosen in FISTA (Beck and Teboulle in SIAM J Imaging Sci 2:183–202, 2009), the successive changes of iterates go to 0. Moreover, based on the Łojasiewicz inequality, we establish the global convergence of iterates generated by the proximal gradient algorithm with extrapolation with an additional assumption on the extrapolation coefficients. The assumption is general enough to allow the threshold of the extrapolation coefficients to be 1. In particular, we prove the length of the iterates is finite. Finally, we perform numerical experiments on the least squares problems with ℓ₁ regularization to illustrate our theoretical results.