Proximal gradient-type method with generalized distance and convergence analysis without global descent lemma

We consider solving nonconvex composite optimization problems in which the sum of a smooth function and a nonsmooth function is minimized. Many of convergence analyses of proximal gradient-type methods rely on global descent property between the smooth term and its proximal term. On the other hand, the ability to efficiently solve the subproblem depends on the compatibility between the nonsmooth term and the proximal term. Selecting an appropriate proximal term by considering both factors simultaneously is generally difficult. We overcome this issue by providing convergence analyses for proximal gradient-type methods with general proximal terms, without requiring global descent property of the smooth term. As a byproduct, new convergence results of the interior gradient methods for conic optimization are also provided.