IJPAM: Volume 41, No. 4 (2007)

OPTIMALITY CONDITIONS AND DUALITY FOR
NONLINEAR PROGRAMMING PROBLEMS INVOLVING
LOCALLY $(H_p,r,\alpha)$-PRE-INVEX FUNCTIONS
AND $H_p$-INVEX SETS

Dehui Yuan$^1$, Xiaoling Liu$^2$, Shengyun Yang$^3$,
Nyamsuren Damdin$^4$, Altannar Chinchuluun$^5$
$^{1,2,3}$Department of Mathematics
Hanshan Teacher's College
Chaozhou, Guangdong Province, 521041, P.R. CHINA
$^4$Department of Statistics
School of Mathematics
Mongolian University of Science and Technology
Ulaanbaatar, MONGOLIA
$^5$Department of Industrial and Systems Engineering
College of Engineering
University of Florida
303 Weil Hall, Gainesville, FL 32611, USA
e-mail: altannar@ufl.edu


Abstract.In this paper, we introduce new types of generalized convex functions and sets which are called locally $(H_p,r,\alpha)$-pre-invex functions and locally $H_p$-invex sets, respectively. Some properties of these new classes of functions and sets are established. We also obtain necessary optimality conditions, sufficient optimality conditions and duality theorems for nonlinear programming problems introducing new concepts of generalized differential and gradient sets of functions. Furthermore, we give some examples to illustrate that the new generalized convexity is an extension of some generalized convexities introduced in literature.

Received: November 20, 2007

AMS Subject Classification: 90C26, 90C46

Key Words and Phrases: locally $H_p$ invex set, locally $(H_p,r,\alpha)$-pre-invex function, right super order $\alpha-(p,r)$ differential

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 41
Issue: 4