How to regularize a difference of convex functions
Given a nonconvex function \(f\) defined as the difference of two convex functions \(g\) and \(h\) (\(f\) is a so-called d.c. function), the author studies the regularized (or smoothed) version \(f_ r= g\square r/2\| \cdot\|^ 2- h\square r/2\| \cdot\|^ 2\) of \(f\) obtained by performing the infimal convolution of both component functions \(g\) and \(h\) by the same kernel function \(r/2\|\cdot \|^ 2\). He compares critical points of \(f_ r\) and \(f\) and considers the behavior of critical points of \(f_ r\) as \(r\to +\infty\). To a great extent the nice properties of the regularization process \(\varphi\to\varphi \square r/2\| \cdot\|^ 2\) when applied to convex functions \(\varphi\) are preserved for the process \(f\to f_ r\) when performed on d.c. functions \(f\).
scientific article; zbMATH DE number 61589
scientific article