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 this research is toextendsome results introduced by Rockafellar[19] in finite-dimensioal spaces to general Banach space using the Housdoroff distance convergent instead of epigraphical convergent .These results are aplicationsto study
the second-order epi-derivatives of function to classeand to study the second-order epi-derivatives of sum two convex functionand to studythe second-order epi-derivatives of Moreau-Yosida approximate function alsoto study ofthe second-order epi-derivatives of composition convex function with linear operator .
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.
