The Behavior of Error Bounds via Moreau Envelopes

In this paper, we first establish the equivalence of three types of error bounds: uniformized Kurdyka-Łojasiewicz (u-KL) property, uniformized level-set subdifferential error bound (u-LSEB) and uniformized Hölder error bound (u-HEB) for prox-regular functions. Then we study the behavior of the level-set subdifferential error bound (LSEB) and the local Hölder error bound (LHEB) which is expressed respectively by Moreau envelopes, under suitable assumptions. Finally, in order to illustrate our main results and to compare them with those of recent references, some examples are also given.