  We consider structured minimization problems subject to smooth inequality
constraints and present a flexible algorithm that combines interior point (IP)
and proximal gradient schemes. While traditional IP methods cannot cope with
nonsmooth objective functions and proximal algorithms cannot handle complicated
constraints, their combined usage is shown to successfully compensate the
respective shortcomings. We provide a theoretical characterization of the
algorithm and its asymptotic properties, deriving convergence results for fully
nonconvex problems, thus bridging the gap with previous works that successfully
addressed the convex case. Our interior proximal gradient algorithm benefits
from warm starting, generates strictly feasible iterates with decreasing
objective value, and returns after finitely many iterations a primal-dual pair
approximately satisfying suitable optimality conditions. As a byproduct of our
analysis of proximal gradient iterations we demonstrate that a slight
refinement of traditional backtracking techniques waives the need for upper
bounding the stepsize sequence, as required in existing results for the
nonconvex setting.
