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 gene
ral 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.
The purpose of the research is to study the Bergman function and Bergman distance to generalize Moreau – Yosida Approximation.
To do that we replace the quadratic additive terms in Moreau – Yosida Approximates by more general Bergman distance and s
tudy properties of generalized approximation and prove equivalence between epigraph – convergence and pointwise convergence of the generalized Moreau – Yosida Approximation.
The supporting functions are powerful tools for studying several problems in mathematics and engineering sciences, since they have useful advantages.
In this paper, we prove that the following conditions and statements are equivalent:
1. is convex.
2. is supporting
3. is subadditive on the unit sphere.
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 Br
egman 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.
