Generalized Convexity and Invexity in Optimization Theory: Some New Results

Convex like properties, without vector space structure are intensively used in the minimax theory Since Ky Fan has proved the first minimax theorem for concave-convexlike functions, several authors have proposed other extensions or generalizations for the convexity. In this paper we propose a survey of the recent studies concerning convexity and invexity in optimization theory. In fact, in the first part we consider the studies of Hanson (1981), Craven (1986), Giorgi and Mititelu (1993), Jeyakumar (1985), Kaul and Kaur (1985), Martin (1985) and Caristi, Ferrara and Stefanescu (1999, 2001, 2005) considering some new examples and remarks. In the second part we consider the consistent notation of (Φ, ρ)-invexity establishing new properties in optimization theory.