Semi-E-Preinvex Functions

Semi-E-Preinvex Functions

Yu-Ru Syau (National Formosa University, Taiwan) and E. Stanley Lee (Kansas State University, USA)
Copyright: © 2010 |Pages: 9
DOI: 10.4018/jalr.2010070103

Abstract

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

Convexity plays an important role in optimization theory. Various generalizations of convex functions have appeared in the literature. A significant generalization of convex functions is the introduction of preinvex functions, originated by (Hanson and Mond, 1987) but so named by (Jeyakumar, 1985). Later, (Weir and Mond, 1988) studied how and where preinvex functions can replace convex functions in multiple objective optimization.

Recently, (Youness, 1999) introduced the concepts of jalr.2010070103.m10-convex sets and jalr.2010070103.m11-convex functions which generalize the definitions of convex sets and convex functions based on the effecting of an operator jalr.2010070103.m12 on the domain on which functions are defined. Youness's initial results inspired a great deal of subsequent work, see (Duca et al., 2000; Duca & Lupsa, 2006; Fulga & Preda, 2009; Syau & Lee, 2005; Yang,2001). In an earlier paper (Syau & Lee, 2005), we extended the class of jalr.2010070103.m13-convex functions to jalr.2010070103.m14-quasiconvex functions. More recently, (Fulga and Preda, 2009) introduced a class of sets, called jalr.2010070103.m15-invex sets, which generalize the concept of jalr.2010070103.m16-convex sets. Based on the proposed jalr.2010070103.m17-invex sets, they introduced the class of jalr.2010070103.m18-preinvex functions as a generalization of jalr.2010070103.m19-convex and preinvex functions, and extended the class of jalr.2010070103.m20-quasiconvex functions to jalr.2010070103.m21-prequasiinvex functions.

Recall that the lower level sets of convex functions are convex sets, and that the lower level sets of preinvex functions are invex sets. Due to these properties, a convex (resp. preinvex) programming with inequality constraints may be easily formulated. However, an jalr.2010070103.m22-convex function does not guarantee jalr.2010070103.m23-convexity of its lower level sets. Based on the concept of jalr.2010070103.m24-convex sets, (Chen, 2002) introduced and studied a new concept of semi-jalr.2010070103.m25-convex functions which guarantee jalr.2010070103.m26-convexity of their lower level sets.

Motivated both by earlier research works (Chen, 2002; Fulga & Preda, 2009) and by the importance of preinvexity, we introduce a class of functions called semi-jalr.2010070103.m27-preinvex functions which are generalizations of semi-jalr.2010070103.m28-convex functions. Similarly, the concept of semi-jalr.2010070103.m29-quasiconvex functions is also generalized to semi-jalr.2010070103.m30-prequasiinvex functions. Some properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the jalr.2010070103.m31-dimensional Euclidean space to be an jalr.2010070103.m32-convex or jalr.2010070103.m33-invex set are given. The relationship between semi-jalr.2010070103.m34-preinvex and jalr.2010070103.m35-preinvex functions is discussed. In addition, some results for the corresponding nonlinear programming problems are considered.

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