GENERALIZED (�,b,φ,ρ,θ)-UNIVEX n-SET FUNCTIONS AND SEMIPARAMETRIC DUALITY MODELS IN MULTIOBJECTIVE FRACTIONAL SUBSET PROGRAMMING

We construct a number of semiparametric duality models and 
establish appropriate duality results under various generalized 
(ℱ,b,ϕ,ρ,θ)-univexity assumptions for a multiobjective fractional subset programming problem.


Introduction
In this paper, we will present a number of semiparametric duality results under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the following multiobjective fractional subset programming problem: (P) Minimize F 1 (S) G 1 (S) , F 2 (S) G 2 (S) ,..., F p (S) G p (S) subject to H j (S) 0, j ∈ q, S ∈ A n , ( where A n is the n-fold product of the σ-algebra A of subsets of a given set X,F i ,G i ,i ∈ p ≡ {1, 2,..., p}, and H j , j ∈ q, are real-valued functions defined on A n , and for each i ∈ p, G i (S) > 0 for all S ∈ A n such that H j (S) 0, j ∈ q.This paper is essentially a continuation of the investigation that was initiated in the companion paper [6] where some information about multiobjective fractional programming problems involving point-functions as well as n-set functions was presented, a fairly comprehensive list of references for multiobjective fractional subset programming problems was provided, a brief overview of the available results pertaining to multiobjective fractional subset programming problems was given, and numerous sets of semiparametric sufficient efficiency conditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions were established.These and some other related material that were discussed in [6] will not be repeated in the present paper.Making use of the semiparametric sufficient efficiency criteria developed in [6] in conjunction with a certain necessary efficiency result that will be recalled in the next section, here we will construct several semiparametric duality models for (P) with varying degrees of generality and, in each case, prove appropriate weak, strong, and strict converse duality theorems under a number of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions.The rest of this paper is organized as follows.In Section 3 we consider a simple dual problem and prove weak, strong, and strict converse duality theorems.In Section 4 we formulate another dual problem with a relatively more flexible structure that allows for a greater variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions under which duality can be established.In Sections 5 and 6 we state and discuss two general duality models which are, in fact, two families of dual problems for (P), whose members can easily be identified by appropriate choices of certain sets and functions.
Evidently, all of these duality results are also applicable, when appropriately specialized, to the following three classes of problems with multiple, fractional, and conventional objective functions, which are particular cases of (P): (P1) Minimize S∈F F 1 (S),F 2 (S),...,F p (S) ; (1.2) where F (assumed to be nonempty) is the feasible set of (P), that is, F = S ∈ A n : H j (S) 0, j ∈ q . (1.5) Since in most cases the duality results established for (P) can easily be modified and restated for each one of the above problems, we will not explicitly state these results.

Preliminaries
In this section, we gather, for convenience of reference, a few basic definitions and auxiliary results which will be used frequently throughout the sequel.
Let (X,A,µ) be a finite atomless measure space with L 1 (X,A,µ) separable, and let d be the pseudometric on A n defined by where denotes symmetric difference; thus (A n ,d) is a pseudometric space.For h ∈ L 1 (X,A,µ) and T ∈ A with characteristic function χ T ∈ L ∞ (X,A,µ), the integral T hdµ will be denoted by h,χ T .
We next define the notion of differentiability for n-set functions.It was originally introduced by Morris [3] for a set function, and subsequently extended by Corley [1] for n-set functions.

G. J. Zalmai 1111
Definition 2.1.A function F : A → R is said to be differentiable at S * if there exists DF(S * ) ∈ L 1 (X,A,µ), called the derivative of F at S * , such that for each S ∈ A, where where W G (S,S * ) is o(d(S,S * )) for all S ∈ A n .We next recall the definitions of the generalized (Ᏺ,b,φ,ρ,θ)-univex n-set functions which will be used in the statements of our duality theorems.For more information about these and a number of other related classes of n-set functions, the reader is referred to [6].We begin by defining a sublinear function which is an integral part of all the subsequent definitions.
In the proofs of the duality theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions.These are obtained by considering the contrapositive statements.For example, (Ᏺ,b,φ,ρ,θ)-quasiunivexity can be defined in the following equivalent way: F is said to be (Ᏺ,b,φ,ρ,θ)-quasiunivex at S * if for each (2.7) Needless to say, the new classes of generalized convex n-set functions specified in Definitions 2.5, 2.6, and 2.7 contain a variety of special cases; in particular, they subsume all the previously defined types of generalized n-set functions.This can easily be seen by appropriate choices of Ᏺ, b, φ, ρ, and θ.
In the sequel we will also need a consistent notation for vector inequalities.For all a,b ∈ R m , the following order notation will be used: a b if and only if a i b i for all i ∈ m; a b if and only if a i b i for all i ∈ m, but a = b; a > b if and only if a i > b i for all i ∈ m; a b is the negation of a b.
Throughout the sequel we will deal exclusively with the efficient solutions of (P).An x * ∈ ᐄ is said to be an efficient solution of (P) if there is no other x ∈ ᐄ such that ϕ(x) ϕ(x * ), where ϕ is the objective function of (P).
Next, we recall a set of parametric necessary efficiency conditions for (P).
Theorem 2.8 [5].Assume that F i ,G i , i ∈ p, and H j , j ∈ q, are differentiable at S * ∈ A n , and that for each i ∈ p, there exist S i ∈ A n such that ) If S * is an efficient solution of (P) and The above theorem contains two sets of parameters u * i and λ * i ,i ∈ p, which were introduced as a consequence of our indirect approach in [5] requiring two intermediate auxiliary problems.It is possible to eliminate one of these two sets of parameters and thus obtain a semiparametric version of Theorem 2.8.Indeed, this can be accomplished by simply replacing λ * i by F i (S * )/G i (S * ), i ∈ p, and redefining u * and v * .For future reference, we state this in the next theorem.
Theorem 2.9.Assume that F i ,G i ,i ∈ p, and H j , j ∈ q, are differentiable at S * ∈ A n , and that for each i ∈ p, there exist S i ∈ A n such that (2.11) If S * is an efficient solution of (P), then there exist u * ∈ U and For simplicity, we will henceforth refer to an efficient solution S * of (P) satisfying (2.11) and (2.12) for some S i ,i ∈ p, as a normal efficient solution.
The form and contents of the necessary efficiency conditions given in Theorem 2.9 in conjunction with the sufficient efficiency results established in [6] provide clear guidelines for constructing various types of semiparametric duality models for (P).

Duality model I
In this section, we discuss a duality model for (P) with a somewhat restricted constraint structure that allows only certain types of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions for establishing duality.More general duality models will be presented in subsequent sections.
In the remainder of this paper, we assume that the functions F i ,G i ,i ∈ p, and H j , j ∈ q, are differentiable on A n and that F i (T) 0 and G i (T) > 0 for each i ∈ p and for all T such that (T,u,v) is a feasible solution of the dual problem under consideration.

Consider the following problem: (DI)
Minimize ) where Ᏺ(S,T;•) : L n 1 (X,A,µ) → R is a sublinear function.The following two theorems show that (DI) is a dual problem for (P).
Proof.(a) From (i) and (ii) it follows that Multiplying (3.6) by u i G i (T) and (3.7) by u i F i (T), i ∈ p, adding the resulting inequalities, and then using the superlinearity of φ and sublinearity of Ᏺ(S,T;•), we obtain (3.9) Likewise, from (3.8) we deduce that (3.10) Since v 0, S ∈ F, and (3.3) holds, it is clear that which implies, in view of the properties of φ, that the left-hand side of (3.10) is less than or equal to zero, that is, 0 Ᏺ S,T;b(S,T) q j=1 v j DH j (T) + q j=1 ρ j d 2 θ(S,T) . (3.12) From the sublinearity of Ᏺ(S,T;•) and (3.2) it follows that (3.13) Now adding (3.9) and (3.12), and then using (3.13) and (iii), we obtain But φ(a) 0 ⇒ a 0, and so (3.14) yields which in turn implies that (3.17) (b) Since for each j ∈ q, v j H j (S) 0, it follows from (3.3) that q j=1 v j H j (S) 0 and so using the properties of φ, we obtain which in view of (ii) implies that Now combining (3.9), (3.13), and (3.20), and using (iii), we obtain (3.15).Therefore, the rest of the proof is identical to that of part (a).(c) From the (Ᏺ,b, φ,0,θ)-pseudounivexity assumption and (3.2) it follows that v j H j (T) 0. (3.21) In view of the properties of φ, this inequality becomes which because of (3.3), primal feasibility of S, and nonnegativity of v, reduces to (3.15), and so the rest of the proof is identical to that of part (a).for any differentiable function F : A n → R and S ∈ A n , and assume that any one of the three sets of hypotheses specified in Theorem 3.1 holds for all feasible solutions of (DI).Then there exist u * ∈ U and v * ∈ R q + such that (S * ,u * ,v * ) is an efficient solution of (DI) and ϕ(S * ) = ξ(S * ,u * ,v * ).
We also have the following converse duality result for (P) and (DI).
Theorem 3.3 (strict converse duality).Let S * and Ᏺ(S,S * ;•) be as in Theorem 3.2, let ( S, u, v) be a feasible solution of (DI) such that Furthermore, assume that any one of the following three sets of hypotheses is satisfied: (a) the assumptions specified in part (a) of Theorem 3.1 are satisfied for the feasible solution ( S, u, v) of (DI); F i is strictly (Ᏺ,b, φ, ρi ,θ)-univex at S for at least one index i ∈ p with the corresponding component u i of u positive, and φ(a) > 0 ⇒ a > 0, or −G i is strictly (Ᏺ,b, φ, ρ i ,θ)-univex at S for at least one index i ∈ p with u i positive, and φ(a) > 0 ⇒ a > 0, or H j is strictly (Ᏺ,b, φ, ρ j ,θ)-univex at S for at least one index j ∈ q with v j positive, and φ(a) > 0 ⇒ a > 0, or is strictly (Ᏺ,b, φ,0,θ)-pseudounivex at S, and φ(a) > 0 ⇒ a > 0. Then S = S * , that is, S is an efficient solution of (P).
Proof.(a) Suppose to the contrary that S = S * .Proceeding as in the proof of part (a) of Theorem 5.1, we arrive at the strict inequality
Proof.(a) From the primal feasibility of S and (4.3) it is clear that for each j ∈ J + , H j (S) H j (T) and so using the properties of φ j , we obtain φ j (H j (S) − H j (T)) 0, which by virtue of (ii) implies that for each j ∈ J + ,

Ᏺ S,T;b(S,T)DH j (T)
− ρ j d 2 θ(S,T) .( Since v 0, v j = 0 for each j ∈ q \ J + , and Ᏺ(S,T;•) is sublinear, these inequalities can be combined as follows: Ᏺ S,T;b(S,T) q j=1 v j DH j (T) From (3.13) and (4.7) we see that where the second inequality follows from (iii).In view of (i), (4.8) implies that φ( f (S,T, u) − f (T,T,u)) 0, which because of the properties of φ, reduces to f (S,T,u) − f (T,T, u) 0. But f (T,T,u) = 0 and hence f (S,T,u) 0, which is precisely (3.15).Therefore, the rest of the proof is identical to that of part (a) of Theorem  (5.6) Since S ∈ F and v 0, it is clear from (5.4) that for each t ∈ m, and so using the properties of φ t , we get φ t Λ t (S,v) − Λ t (T,v) 0, which in view of (ii) implies that for each t ∈ m, Ᏺ S,T;b(S,T (5.9) From (5.6) and (5.9) we deduce that where the second inequality follows from (iii).Because of (i), this inequality implies that φ(Π(S,T,u,v) − Π(T,T,u,v)) 0. But φ(a) ≥ 0 ⇒ a ≥ 0, and so we get Π(S,T,u,v) Π(T,T,u,v) = 0. Inasmuch as S ∈ F, u > 0, v 0, and G i (T) > 0, i ∈ p, this inequality yields (5.11) Since u > 0, this inequality implies that G. J. Zalmai 1125 which in turn implies that (5.13) (b)-(d) The proofs are similar to that of part (a).
Theorem 5.2 (weak duality).Let S and (T,u,v) be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any one of the following six sets of hypotheses is satisfied: ) is strictly (Ᏺ,b, φi , ρi ,θ)-pseudounivex at T, φi is increasing, and φi (0 ) is prestrictly (Ᏺ,b, φi , ρi ,θ)-quasiunivex at T, and for each i ∈ I + , φi is increasing and φi (0) = 0, where {I 1+ ,I 2+ } is a partition of increasing, and φ t (0) = 0, and for each t ∈ m 2 , Λ t (•,v) is (Ᏺ,b, φ t , ρ t ,θ)-quasiunivex at T, and for each t ∈ m, φ t is increasing and φ t (0 Proof.(a) Suppose to the contrary that ϕ(S) ω (T,u,v).This implies that for each i ∈ p, G i (T)F i (S) − [F i (T) + Λ 0 (T,v)]G i (S) 0, with strict inequality holding for at least one index ∈ p.Using these inequalities along with the primal feasibility of S and nonnegativity of v, we see that where the second inequality follows from (iii).Obviously, this inequality contradicts (5.9), which is valid for the present case because of (ii).Hence we must have ϕ(S) ω(T,u,v).(b)-(f) The proofs are similar to that of part (a).for any differentiable function F : A n → R and S∈A n , and assume that any one of the ten sets of hypotheses specified in Theorems 5.1 and 5.2 holds for all feasible solutions of (DIII).Then there exist u * ∈ U and v * ∈ R q + such that (S * ,u * ,v * ) is an efficient solution of (DIII) and ϕ(S * ) = ω(S * ,u * ,v * ).
Proof.By Theorem 2.9, there exist u * ∈ U and v ∈ R q + such that (5.18) vj H j (S * ) = 0, j ∈ q. (5.19) Now if we let v * j = vj /G i (S * ) for each j ∈ J 0 , v * j = vj for each j ∈ q \ J 0 , and observe that Λ 0 (S * ,v * ) = 0, then (5.18) and (5.19) can be rewritten as follows: (5.21) From (5.20) and (5.21) it is clear that (S * ,u * ,v * ) is a feasible solution of (DIII).Proceeding as in the proof of Theorem 3.3, it can easily be verified that it is an efficient solution of (DIII).
We next show that certain modifications in Theorem 5.1 lead to a number of strict converse duality results for (P) and (DIII).
Theorem 5.4 (strict converse duality).Let S * and Ᏺ be as in Theorem 5.3, let ( S, u, v) be a feasible solution of (DIII) such that Proof.The proof is similar to that of Theorem 3.3.
Evidently, (DIII) contains a number of important special cases which can easily be identified by appropriate choices of the partitioning sets J 0 ,J 1 ,...,J m , and the sublinear function Ᏺ(S,T;•).We conclude this section by briefly looking at a few of these special cases.In each case, we specify the required conditions for duality by specializing part (a) of Theorem 5.2.
If we set m = q, J t = {t}, t ∈ q, then (DIII) reduces to the following problem: v j DH j (T) 0 ∀S ∈ A n , v j H j (T) 0, j ∈ q.
In a similar manner, one can obtain a vast number of duality theorems for (P) by specializing the other nine sets of conditions for (DIIIa)-(DIIIe) and other special cases of (DIII).

Duality model IV
In this section, we present another general duality model for (P) that is different from (DIII) in that here in constructing the constraints we not only use a partition of the index set q, but also a partition of the set p. A parametric point-function version of this dual problem was considered earlier in [4].

Theorem 3 . 2 (
strong duality).Let S * be a regular efficient solution of (P), let Ᏺ(S,S * ; DF(S * )) = n k=1 D k F(S * ),χ Sk − χ S * k 0; (b) the assumptions specified in part (b) of Theorem 3.1 are satisfied for the feasible solution ( S, u, v) of (DI), F i and φ or −G i and φ satisfy the requirements described in part (a), or the function R → q j=1 v j H j (R) is strictly (Ᏺ,b, φ, ρ,θ)-pseudounivex at S, or p i=1 u i [G i ( S) ρi + F i ( S) ρ i ] + ρ > 0; (c) the assumptions specified in part (c) of Theorem 3.1 are satisfied for the feasible solution ( S, u, v) of (DI), and the function .25) in contradiction to (3.23).Hence we conclude that S = S * .(b) and (c) The proofs are similar to that of part (a).
3.1.(b)-(f) The proofs are similar to that of part (a).
i (T)DF i (T) − F i (T)DG i (T) + q j=1 v j DH j (T) 0 ∀S ∈ A n , j∈Jt v j H j (T) 0, t ∈ m.
i (T)DF i (T) − F i (T)DG i (T) + q j=1