Journal of Mathematical Analysis and ApplicationsVolume 321, Issue 2, 15 September 2006, Pages 675–690Cover imageGeneralized convexities and generalized gradients based on algebraic operations

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 (φ,γ)-convexity is defined and some properties of (φ,γ)-convex functions are derived