Applying FISTA to optimization problems (with or) without minimizers

Beck and Teboulle’s FISTA method for finding a minimizer of the sum of two convex functions, one of which has a Lipschitz continuous gradient whereas the other may be nonsmooth, is arguably the most important optimization algorithm of the past decade. While research activity on FISTA has exploded ever since, the mathematically challenging case when the original optimization problem has no minimizer has found only limited attention. In this work, we systematically study FISTA and its variants. We present general results that are applicable, regardless of the existence of minimizers.