Clarke Subdifferentials of Moreau Envelopes and Hypoconvex Optimization

Moreau envelopes play an important role in variational analysis and optimization. In this paper, we show that the Clarke subdifferential of Moreau envelopes of prox-bounded functions can characterize hypoconvex functions, while this nice feature fails for the Mordukhovich limiting subdifferential. Moreover, we establish that the Mordukhovich symmetric subdifferential of the Moreau envelope of a prox-bounded function coincides with its Clarke subdifferential. Thus, an equivalent relation also holds for the Mordukhovich symmetric subdifferential. Applications and algorithmic consequences in hypoconvex optimization are given.