  In this paper we introduce a novel abstract descent scheme suited for the
minimization of proper and lower semicontinuous functions. The proposed
abstract scheme generalizes a set of properties that are crucial for the
convergence of several first-order methods designed for nonsmooth nonconvex
optimization problems. Such properties guarantee the convergence of the full
sequence of iterates to a stationary point, if the objective function satisfies
the Kurdyka-Lojasiewicz property. The abstract framework allows for the design
of new algorithms. We propose two inertial-type algorithms with implementable
inexactness criteria for the main iteration update step. The first algorithm,
i$^2$Piano, exploits large steps by adjusting a local Lipschitz constant. The
second algorithm, iPila, overcomes the main drawback of line-search based
methods by enforcing a descent only on a merit function instead of the
objective function. Both algorithms have the potential to escape local
minimizers (or stationary points) by leveraging the inertial feature. Moreover,
they are proved to enjoy the full convergence guarantees of the abstract
descent scheme, which is the best we can expect in such a general nonsmooth
nonconvex optimization setup using first-order methods. The efficiency of the
proposed algorithms is demonstrated on two exemplary image deblurring problems,
where we can appreciate the benefits of performing a linesearch along the
descent direction inside an inertial scheme.
