How to regularize a difference of convex functions

Abstract Given a nonconvex function f defined as the difference of two convex functions g and h ( f is a so-called d.c. function), we study the regularized (or smoothed) version f r = g □ r 2 ∥ · ∥ 2 − h □ r 2 ∥ · ∥ 2 of f obtained by performing the infimal convolution of both component functions g and h by the same kernel function r 2 ∥ · ∥ 2 . Critical points of f r and f are compared and the behavior of critical points of f r as r → + ∞ is considered. To a great extent the nice properties of the regularization process ϑ → ϑ □ r 2 ∥ · ∥ 2 when applied to convex functions ϑ are preserved for the process f → f r when performed on d.c. functions f .

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