The Moreau Envelope Function And Proximal Mapping In The Sense Of The Φ-Divergence

Proximal point algorithm is an important method of solving the constrainedoptimization problem. Moreau envelope function and proximal point mapping areat the heart of the proximal point algorithm, they are also natural and efcienttools for regularizing functions and approximating optimization problems.In this paper, we consider replacing· by-divergence d (·,·) to defineMoreau envelope function and its proximal mapping. Moreover we research theanalytical properties of the Moreau envelope function and its proximal operator inthe sense of divergence, such as continuity and diferentiability. In the same time,we discuss the case of taking the special function. Before this we consider some ofthe properties of which influence the divergence function. Meanwhile we describethe directional derivative of Moreau envelope function. At last, we construct ageneralized augmented Lagrangian by using the entropy-like approximate of thequadratic Lagrangian, i.e., augmented Lagrangian function with divergence.