Generalized convexities and generalized gradients based on algebraic operations

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• 作者：Dehui Yuan ; Altannar Chinchuluun ; Xiaoling Liu and Panos M. Pardalos
• 关键词：Generalized convexity ; Generalized monotonicity ; ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4H5N28B-4&_mathId=mml4&_user=10&_cdi=6894&_rdoc=16&_handle=V-WA-A-W-BE-MsSAYZW-UUA-U-AA
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2006
• 出版时间：15 September 2006
• 年：2006
• 卷：321
• 期：2
• 页码：675-690
• 全文大小：156 K

文摘

In this paper, we investigate properties of generalized convexities based on algebraic operations introduced by Ben Tal [A. Ben Tal, On generalized means and generalized convex functions, J. Optim. Theory Appl. 21 (1977) 1–13] and relations between these generalized convexities and generalized monotonicities. We also discuss the (h,φ)-generalized directional derivative and gradient, and explore the relation between this gradient and the Clarke generalized gradient. Definitions of some generalized averages of the values of a generalized convex function at n equally spaced points based on the algebraic operations are also presented and corresponding results are obtained. Finally, the e52bbd0"" title=""Click to view the MathML source"">(φ,γ)-convexity is defined and some properties of (φ,γ)-convex functions are derived.