Generalized Lasry – Lions regularization function using Bregman distant

The purpose of the research is to study Bergman distance to generalize Lasry – Lions regularization which play important role of theory optimization. To do that we replace the quardatic additive terms in Lasry – Lions regularization by more general Bergman distance (non metric distance), and study properties generalized approximation and proof its continuous as we give a relationship between the solution minimization sets of function and Lions – Lasry Regularization and others properties.

Study the Convergence of Moreau – Bregman Envelope in Reflexive Banach Spaces

It is often useful to replace a function with a sequence of smooth functions approximating the given function to resolve minimizing optimization problems. The most famous one is the Moreau envelope. Recently the function was organized using the Bregman distance h D . It is worth noting that Bregman distance h D is not a distance in the usual sense of the term. In general, it is not symmetric and it does not satisfy the triangle inequality The purpose of the research is to study the convergence of the Moreau envelope function and the related proximal mapping depends on Bregman Distance for a function on Banach space. Proved equivalence between Mosco-epi-convergence of sequence functions and pointwise convergence of Moreau-Bregman envelope We also studied the strong and weak convergence of resolvent operators According to the concept of Bregman distance.