Optimality and duality for the multiobjective fractional programming with the generalized (F,ρ) convexity

A class of multiobjective fractional programmings (MFP) are first formulated, where the involved functions are local Lipschitz and Clarke subdifferentiable. In order to deduce our main results, we give the definitions of the generalized (F,ρ) convex class about the Clarke subgradient. Under the above generalized convexity assumption, the alternative theorem is obtained, and some sufficient and necessary conditions for optimality are also given related to the properly efficient solution for the problems. Finally, we formulate the two dual problems (MD) and (MD1) corresponding to (MFP), and discuss the week, strong and reverse duality.  2002 Elsevier Science (USA). All rights reserved. 1. Preliminaries and formulations In recent, optimality and duality for the multiobjective programming have been studied. Under kinds of generalized convexities, some results had been obtained. Jeyakumar [1] gave the optimality and duality for nondifferentiable nonconvex program under the ρ-invexity assumption. When the involved functions are continuous differentiable, Lin [2] obtained the sufficient conditions for a class of multiobjective program with F -convexity. Kanniappan [3] got the necessary E-mail address: [email protected]. 0022-247X/02/$ – see front matter  2002 Elsevier Science (USA). All rights reserved. PII: S0022-247X(02)0 02 48 -2 X. Chen / J. Math. Anal. Appl. 273 (2002) 190–205 191 conditions under the subgradient assumptions. In this paper, we first define a kind of generalized convexity about the Clarke subgradient. Then, the alternative theorem is given. Finally, optimality and duality are obtained for a class of nondifferential and nonconvex multiobjective fractional programming. Suppose that h :X → R (X ⊆ R) is local Lipschitz function; that is, for a given x ∈ X, h is Lipschitz in some neighborhood of x . Denote by ∂0h(x) the Clarke subgradient of h at x . Lemma 1 [4]. If φi :X → R is local Lipschitz function, and is proper at x ∈ X, si is a real number, i = 1,2, . . . , t , then ∂0 ( t ∑ i=1 siφi ) (t)= t ∑ i=1 si∂ φi(x). Further more, if φ1(x) 0, φ2(x) 0, then ∂(φ1 · φ2)(x)= φ2(x)∂φ1(x)+ φ1(x)∂φ2(x). Suppose that F :X×X×Rn →R (X ⊆Rn) is sublinear about third variable, d(· , ·) is a pseudometric on R,ρ ∈ R. Definition 1. If for all ξ ∈ ∂0h(x0) and for all x ∈X, we have h(x)− h(x0) F(x, x0; ξ)+ ρ · d2(x, x0), then the function h is said to be G− (F,ρ) convex at x0. Definition 2. If for all ξ ∈ ∂0h(x0) and for all x ∈X, we have h(x) h(x0) ⇒ F(x, x0; ξ) −ρ · d2(x, x0), then the function h is said to be G− (F,ρ) quasiconvex at x0. Definition 3. If for all ξ ∈ ∂0h(x0) and for all x ∈X, we have F(x, x0; ξ) −ρ · d2(x, x0) ⇒ h(x) h(x0), then the function h is said to be G− (F,ρ) pseudoconvex at x0. Remark. (1) If F(x, x0; ξ) = ξ η(x, x0) in above definitions, then we get ρinvexity. (2) If h is continuous differentiable at x0 and ρ = 0, then we obtain F -class generalized convexity. In this paper, we consider the following multiobjective fractional problem: 192 X. Chen / J. Math. Anal. Appl. 273 (2002) 190–205 (MFP) minimize G(x)= ( f1(x) g(x) , f2(x) g(x) , . . . , fp(x) g(x) ) subject to h(x) 0, x ∈ C, where C ⊆ R is close convex set; fi :C → R (i = 1,2, . . . , p), g :C → R; h = (h1, h2, . . . , hm) , hj :C → R (j = 1,2, . . . ,m) are local Lipschitz, and Clarke subdifferentiable at x ∈ C. And, suppose that g(x) > 0. If g(x) is not linear function, then fi(x) 0, x ∈ C, i = 1,2, . . . , p. Define X = {x ∈ C | h(x) 0}, Λ+ = { λ= (λ1, λ2, . . . , λp) | λi 0, i = 1,2, . . . , p, p ∑ i=1 λi = 1 }