A Multi-step Inertial Forward--Backward Splitting Method for Non-convex Optimization

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In this paper, we propose a multi-step inertial Forward--Backward splitting algorithm for minimizing the sum of two non-necessarily convex functions, one of which is proper lower semi-continuous while the other is differentiable with a Lipschitz continuous gradient.

• We first prove global convergence of the scheme with the help of the Kurdyka-{\L}ojasiewicz property.
• Then, when the non-smooth part is also partly smooth relative to a smooth submanifold, we establish finite identification of the latter and provide sharp local linear convergence analysis.
• The proposed method is illustrated on a few problems arising from statistics and machine learning.