  We introduce a unified algorithmic framework, called proximal-like
incremental aggregated gradient (PLIAG) method, for minimizing the sum of a
convex function that consists of additive relatively smooth convex components
and a proper lower semi-continuous convex regularization function, over an
abstract feasible set whose geometry can be captured by using the domain of a
Legendre function. The PLIAG method includes many existing algorithms in the
literature as special cases such as the proximal gradient method, the Bregman
proximal gradient method (also called NoLips algorithm), the incremental
aggregated gradient method, the incremental aggregated proximal method, and the
proximal incremental aggregated gradient method. It also includes some novel
interesting iteration schemes. First we show the PLIAG method is globally
sublinearly convergent without requiring a growth condition, which extends the
sublinear convergence result for the proximal gradient algorithm to incremental
aggregated type first order methods. Then by embedding a so-called Bregman
distance growth condition into a descent-type lemma to construct a special
Lyapunov function, we show that the PLIAG method is globally linearly
convergent in terms of both function values and Bregman distances to the
optimal solution set, provided that the step size is not greater than some
positive constant. These convergence results derived in this paper are all
established beyond the standard assumptions in the literature (i.e., without
requiring the strong convexity and the Lipschitz gradient continuity of the
smooth part of the objective). When specialized to many existing algorithms,
our results recover or supplement their convergence results under strictly
weaker conditions.
