Journal of Applied Mathematics
Volume 2012 (2012), Article ID 629194, 19 pages
http://dx.doi.org/10.1155/2012/629194
Research Article

A Class of Semilocal 𝐸 -Preinvex Functions and Its Applications in Nonlinear Programming

Hehua Jiao,1,2 Sanyang Liu,1 and Xinying Pai1,3

1College of Science, Xidian University, Xi'an, Shanxi 710071, China
2College of Mathematics and Computer Science, Yangtze Normal University, Fuling, Chongqing 408100, China
3Department of Mathematics, China University of Petroleum, Qingdao, Shandong 266555, China

Received 20 April 2011; Accepted 5 December 2011

Academic Editor: Ch Tsitouras

Copyright © 2012 Hehua Jiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A kind of generalized convex set, called as local star-shaped 𝐸 -invex set with respect to 𝜂 , is presented, and some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocal 𝐸 -preinvex functions, which is a generalization of semi- 𝐸 -preinvex functions and semilocal 𝐸 -convex functions, is introduced. Simultaneously, some of its basic properties are discussed. Furthermore, as its applications, some optimality conditions and duality results are established for a nonlinear programming.