Semi-E-Preinvex Functions

2. Preliminaries

Let Rⁿ denote the n-dimensional Euclidean space. Recall (Youness, 1999) that, by definition, a set M⊆Rⁿ is said to be E -convex if there is a mapping E: Rⁿ → Rⁿ such that

Let E: Rⁿ → Rⁿ be a given mapping. For a nonempty set S⊆Rⁿ, let

Lemma 2.1 (Youness, 1999, Proposition 2.2). Let M⊆Rⁿ be a nonempty E -convex set, then E(M)⊆M.

A set K⊆Rⁿ is said to be an invex set with respect to (w.r.t. in short) a given mapping η: Rⁿ × Rⁿ → Rⁿ if

Definition 2.1 (Mohan & Neogy, 1995). Let K⊆Rⁿ be a nonempty invex set w.r.t. a given mapping η: Rⁿ × Rⁿ → Rⁿ. A function f: K → R¹ is said to be preinvex on K w.r.t. η if for all x, y∈K and λ∈[0,1],