We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka-Łojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of the function, and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg-Marquardt algorithm is detailed.

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Given a Hilbert space H and a closed convex function : H → R ∪ {+∞}, we consider the inertial proximal algorithmwhere (α n ) and (β n ) are nonnegative sequences. The notation ∂ stands for the subdifferential of in the sense of convex analysis. This algorithm can be viewed as the implicit discretization of a continuous gradient system involving a memory term. We give conditions that ensure that a suitable discrete energy decreases to inf as n → +∞. When has a unique minimum, the question of the convergence of (x n ) is solved. In the case of multiple minima, it is proved that if n k=1 α k ∈ l 1 and if a suitable geometric condition on the set argmin is fulfilled, then non stationary sequences of (A) cannot converge.

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