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Forward-backward algorithms with different inertial terms for structured non-convex minimization problems

We investigate two inertial forward-backward algorithms in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithms are formulated in the spirit of the famous FISTA method, however the setting is non-convex and we allow different inertial terms. Moreover, the inertial parameters in our algorithms can take negative values too. We also treat the case when the non-smooth function is convex and we show that in this case a better step size can be allowed. We prove some abstract convergence results which applied to our numerical schemes allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka-Lojasiewicz property. Further, we obtain a general result that applied to our numerical schemes ensures convergence rates for the generated sequences and for the objective function values formulated in terms of the KL exponent of a regularization of the objective function. Finally, we apply our results to image restoration.