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, i2Piano, 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, which even allows for the escape of local minimizers. Both algorithms are proved to enjoy the full convergence guarantees of the abstract descent scheme. The efficiency of the proposed algorithms is demonstrated on an exemplary image deblurring problem in presence of data corrupted by impulse noise, where we can appreciate the benefits of performing a linesearch along the descent direction inside an inertial scheme.
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View An abstract convergence framework with application to inertial inexact forward-backward methods