On generalized convex functions and generalized subdifferential

摘要

We introduce and study the notion of sigmadocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sigma $$end{document}-subdifferential of a proper function f which contains the Clarke-Rockafellar subdifferential of f under some mild assumptions on f and sigmadocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sigma $$end{document}. We show that some well known properties of the convex function, namely Lipschitz property on the interior of its domain, remain valid for the large class of sigmadocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sigma $$end{document}-convex functions.