Semi-E-Preinvex Functions

Semi-E-Preinvex Functions

Yu-Ru Syau (National Formosa University, Taiwan) and E. Stanley Lee (Kansas State University, USA)
Copyright: © 2012 |Pages: 7
DOI: 10.4018/978-1-61350-456-7.ch311

Abstract

A class of functions called semi-E-preinvex functions is defined as a generalization of semi-E-convex functions. Similarly, the concept of semi-E-quasiconvex functions is also generalized to semi-E-prequasiinvex functions. Properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the n-dimensional Euclidean space to be an E-convex or E-invex set are given. The relationship between semi-E-preinvex and E-preinvex functions are discussed along with results for the corresponding nonlinear programming problems.
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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

λE(x) + (1 – λ) E(y) ∈M, ∀x, y∈M, ∀λ∈[0, 1].

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

E(S) = {E(x): x∈S}.

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

x, y∈K, λ∈[0, 1] ⟹ y + λη(x, y)∈K.

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],

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