Annals of Operations Research 133, 47–61, 2005
2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.
Parameter-Free Dual Models for Fractional
Programming with Generalized Invexity
JIN-CHIRNG LEE and HANG-CHIN LAI
Department of Applied Mathematics, Chung-Yuan Christian University, Chung Li 320, Taiwan
Abstract. By parameter-free approach, we establish sufﬁcient optimality conditions for nondifferentiable
fractional variational programming under certain speciﬁc structure of generalized invexity. Employing the
sufﬁcient optimality conditions, two parameter-free dual models are formulated. The weak duality, strong
duality and strict converse duality theorems are proved in the framework of generalized invexity.
Keywords: Euler–Lagrange equation, Kuhn–Tucker condition, (F,ρ,θ)-invex, -pseudoinvex, -quasiinvex,
fractional variational programming, weak-, strong-, strict
The term invex means invariant convex which was coined by Craven (1981), and certain
generalized invexity is studied by Lai and Liu (2003). Recently, Chen and Lai (2003)
investigated the fractional variational programming problem involving a kind of gener-
alized invexity, namely (F ,ρ,θ)-invexity, and established one parametric dual problem.
In a programming problem the sufﬁcient (existence) theorem for optimal solution fol-
lows from the converse of necessary optimality conditions with some extra assumptions.
Such as convexity, invexity, generalize convexity/invexity etc. It then several sufﬁcient
optimality theorems are established. By using such optimality theorems, several duality
moduls to a primary programming problem are constituted by many authors (cf. Bha-
tia and Jain, 1994; Lai and Lee, 2002; Lai and Liu, 1999, 2002a, 2002b, 2003; Lai,
Liu, and Tanaka, 1999; Liu, 1998; Mond, Chandra, and Husain, 1988; Zalmai, 1994,
1995; etc.). In (Chen and Lai, 2003), Chen and Lai have established one parameter dual
model and proved some duality theorems. This paper is a continuous work of (Chen
and Lai, 2003) in which we consider two parameter-free dual models for fractional dy-
namic variational problem with generalized invex functions. In (Chen and Lai, 2003), the
(F,ρ,θ)-invexity as well as generalized (F,ρ,θ)-invexity are discussed, the authors in-
troduce one parameter in the sufﬁcient optimality theorems (see theorems 4.1 and 4.2 of
(Chen and Lai, 2003)) which involves (F,ρ,θ)-invexity for nondifferentiable fractional
variational programming. They constructed naturally the one parameter dual problem
for the given minimization problem, and proved the weak, strong and strict converse
This research was partly supported by NSC, Taiwan.