A Proximal Average for Nonconvex Functions: A Proximal Stability Perspective

Bauschke, Lucet, and Trienis [SIAM Rev., 50 (2008), pp. 115–132] developed the concept of the proximal average of two convex functions. In this work we show the relationship between the proximal average and the Moreau envelope and exploit this relationship to develop stability theory for a generalized proximal average function. This approach allows us to extend the concept of the proximal average to include many nonconvex functions. The most basic theory requires only that the functions of interest be prox-bounded, while the most powerful results hold for prox-regular functions.