The Moreau Envelope Function And Proximal Mapping In The Sense Of The Bregman Distance In Banach Space

Proximal point algorithm is an important method of solving the convex con-strained optimization problem. Moreau envelope function is a smooth approxi-mation of the objective function f, good properties of the envelope function andthe associated proximal mapping make a good performance in solving optimizationproblems.In this paper, we explore some properties of the Moreau envelope function andthe associated proximal mapping in the sense of the Bregman distance induced bya convex function g in Banach space. Precisely, we study the continuity, locallyLipschitz property and diferentiability of the Moreau envelope function and theupper semicontinuity and single-valuedness of the proximal mapping as well as itsrelation to the convexity of λf+g in Banach spaces, where λ is a positive parameter.