OPTIMALITY AND DUALITY FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS WITH n-SET FUNCTIONS AND GENERALIZED V -TYPE-I UNIVEXITY

MULTIOBJECTIVE FRACTIONAL PROGRAMMING PROBLEMS WITH n-SET FUNCTIONS

AND GENERALIZED V -TYPE-I UNIVEXITY

ANDREEA M ˘AD ˘ALINA STANCU

We study optimality conditions and generalized Mond-Weir duality for multiob- jective fractional programming involvingn-set functions which satisfy appropriate generalizedV-type-I univexity conditions. Our results generalize to fractional pro- gramming those obtained by Predaet al. [6].

AMS 2010 Subject Classification: 90C29, 90C30, 90C32, 90C46.

Key words: optimality, duality, multiobjective fractional programming, n-set functions, generalizedV-type-I univexity.

1. INTRODUCTION

Problems of multiobjective optimization are widespread in mathematical modelling of real world systems for a very broad range of applications. In par- ticular, several classes of multiobjective problems with set andn-set functions have been the subject of several papers in the last few decades.

For a historical survey of optimality conditions and duality for pro- gramming problems involving set and n-set functions the reader is referred to Stancu-Minasian and Preda’s review paper [8].

General theory for optimizingn-set functions was first developed by Mor- ris [5] who, for fractions of a single set, obtained results that are similar to the standard mathematical programming problem. Corley [1] gave the con- cept of derivative of a real-valued n-set function and generalized the results of Morris [5] to n-set functions and established optimality conditions and La- grangian duality.

Along the lines of Jeyakumar and Mond [3], and Mishraet al. [4], Preda et al. [6] defined new classes of n-set functions, called (ρ, ρ⁰)-V-univex type- I, (ρ, ρ⁰)-quasi-V-univex type-I, (ρ, ρ⁰)-pseudo-V-univex type-I, (ρ, ρ⁰)-quasi pseudo-V-univex type-I and (ρ, ρ⁰)-pseudo quasi-V-univex type-I. They ob- tained necessary and sufficient optimality criteria and some duality results.

REV. ROUMAINE MATH. PURES APPL.,57(2012),4, 401-421

Fractional programming have been the subject of many research papers, survey papers and books. Fractional programming models have been suc- cessfully developed for various branches of human activity and especially to economics.

In this paper we generalize to fractional programming the results ob- tained by Preda et al. [6]. We study optimality conditions and generalized Mond-Weir duality for multiobjective fractional programming involving n-set functions which satisfy appropriate generalizedV-type-I univexity conditions.

In Section 2 of this paper we give some definitions introduced in [6] for multi-objective programming and state necessary optimality condition for a multi-objective optimization problem involving set-functions. In Sections 3 and 4 we study optimality conditions and generalized Mond-Weir duality for multiobjective fractional programming involvingn-set functions which satisfy appropriate generalized V-type-I univexity conditions.

2. DEFINITIONS AND PRELIMINARIES

For any vectorsx ∈ Rⁿ, y ∈Rⁿ, we use the following notations: x < y iff xi < yi,i= 1,2, . . . , n;x5yiffxi5yi,i= 1,2, . . . , n;x≤y iffx5y, but x6=y;x^>y=Pn

i=1xiyi.

For an arbitrary vectorx∈Rⁿand a subsetJof the index set{1,2, . . . , n}, we denote by xJ the vector with the components xj, j ∈ J. Let Rⁿ₊ be the nonnegative orthant of Rⁿ.

Let (X,Γ, µ) be a finite atomless measure space withL₁(X,Γ, µ) a sepa- rable space. For h ∈ L1(X,Γ, µ) and Z ∈ Γ with indicator (characteris- tic) function I_Z ∈ L∞(X,Γ, µ), the general integral R

Zhdµ will be denoted by hh, I_Zi.

Now we shall define the notion of differentiability for n-set functions.

Morris [5] introduced the differentiability for set functions and Corley [1] de- fined this notion for n-set functions.

A set function ϕ: Γ→ Ris said to be differentiable at T if there exists Dϕ_T ∈L1(X,Γ, µ),called the derivative ofϕatT, such that for eachS ∈Γ,

ϕ(S) =ϕ(T) +hDϕ_T, I_S−I_Ti+ψ(S, T),

where ψ: Γ×Γ→R and has the property thatψ(S, T) iso(d(S, T)), that is

d(S,T)→0lim ψ(S, T)/d(S, T) = 0, and dis a pseudometric on Γ.

A function h : Γⁿ → R is said to have a partial derivative at S⁰ = (S₁⁰, . . . , S_n⁰) with respect to its k-th argument (15k5n),if the function

ϕ(Sk) =h(S₁⁰, . . . , S_k−1⁰ , Sk, S_k+1⁰ , . . . , S_n⁰)

has derivativeDϕ_S⁰

k,and we defineDkh(S⁰) =Dϕ_S⁰

k. If there exist allDkh(S⁰), 1 5k5n, then we put Dh(S⁰) = (D₁h(S⁰), . . . , D_nh(S⁰)).If H : Γⁿ →R^m, H = (H1, . . . , Hm),we putDkH(S⁰) = (DkH1(S⁰), . . . , DkHm(S⁰)).

The functionh: Γⁿ→Ris differentiable atS⁰ ∈Γⁿif there existDh(S⁰) and ψ: Γⁿ×Γⁿ→Rsuch that

h(S) =h(S⁰) +

n

X

k=1

hD_kh(S⁰), I_Sk−I_S⁰

ki+ψ(S, S⁰), where ψ(S, S⁰) iso[d(S, S⁰)], and dis a pseudometric on Γⁿ.

A vector valued set function f = (f1, . . . , fp) : Γ → R^p is differentiable on Γ if all component functions f_i,15i5p, are differentiable on Γ.

Consider the following n-set function multiobjective optimization pro- blem

(VP) minimize{f(S) = (f₁(S), . . . , f_p(S))|g(S)50, S ∈Γⁿ},

where f : Γⁿ→R^p and g: Γⁿ→R^m are differentiablen-set functions on Γⁿ. The problem is to find the collection of (properly) efficient sets defined below.

We denote P = {S ∈Γⁿ |g(S)50} the set of all feasible solutions of problem (VP).

Definition 2.1. A set S⁰ ∈ P is anefficient solution (Pareto solution) of problem (VP) if there exists no S∈ P such thatf(S)≤f(S⁰).

Definition 2.2. An efficient solution S⁰ of (VP) is called properly ef- ficient, if there exists a positive number M with the property that, if f_i(S)<

fi(S⁰) for any i and for S ∈ P, then _f^{fi(S0)−fi(S)}

j(S)−f_j(S⁰) ≤ M for some j for which f_j(S)> f_j(S⁰).

With respect to the constraints of the problem (VP), we consider the par- tition{J₀, J1, . . . , J_k}of the index set{1,2, . . . , m},that is,

k

S

s=0

Js={1,2, . . . , m},and for any s6=t, we have Js∩Jt=∅.

According to the partition {J₀, J₁, . . . , J_k} associated to problem (VP), we introduce the notation

ψi(S, λJ0) =fi(S) +λ^>_J0gJ0(S)

for any i, 1 5 i 5 p, where λ ∈ R^m+ is a given vector. Also let the vectors ρ= (ρ1, . . . , ρp)∈R^p, ρ⁰= (ρ⁰₁, . . . , ρ⁰_k)∈R^k,and the real numbersρ0, ρ⁰₀∈R. The following definitions were introduced by Predaet al. [6] and extend similar concepts defined by Jeyakumar and Mond [3] and Mishra et al. [4] .

Definition 2.3. We say that problem (VP) is (ρ, ρ⁰)-V-univex type I at S⁰ ∈ P according to the partition{J₀, J1, . . . , Jk} relative toλ∈R^m₊, if there

exist positive real functions α1, . . . , αp and β1, . . . , βk defined on Γⁿ ×Γⁿ, nonnegative functions b₀ and b₁, also defined on Γⁿ×Γⁿ, ϕ₀ :R → R, ϕ₁ : R→R,such that

(2.1)

b0(S, S⁰)ϕ0[ψi(S, λJ0)−ψi(S⁰, λJ0)]=

=αi(S, S⁰)

n

X

t=1

hD_tψi(S⁰, λJ0), ISt−I_S⁰

ti+ρid²(S, S⁰) and

(2.2)

−b₁(S, S⁰)ϕ₁



 X

j∈J_s

λ_jg_j(S⁰)



=

=βs(S, S⁰)X

j∈Js

λj n

X

t=1

hD_tgj(S⁰), ISt −I_S⁰

ti+ρ⁰_sd²(S, S⁰), for any S ∈ P,i∈ {1, . . . , p}and s∈ {1, . . . , k}.

If (VP) is (ρ, ρ⁰)-V-univex type I at allS⁰ ∈ P, according to the partition {J₀, J1, . . . , J_k} relative to λ∈R^m+,then we say that (VP) is (ρ, ρ⁰)-V-univex type I on P according to the partition{J₀, J₁, . . . , J_k}relative toλ∈R^m+.

If strict inequality holds in (2.1) (whenever S 6= S⁰), then we say that (VP) is (ρ, ρ⁰)-semi strictlyV-univex type I atS⁰ or on P,according to the partition {J₀, J1, . . . , Jk}relative toλ∈R^m₊,depending on the case.

Definition 2.4. We say that problem (VP) is (ρ0, ρ⁰₀)-quasiV-univex type I at S⁰ ∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p₊ and λ ∈ R^m+, if there exist positive real functions α1, . . . , αp and β1, . . . , β_k defined on Γⁿ×Γⁿ, nonnegative functionsb₀ andb₁,also defined on Γⁿ×Γⁿ, ϕ0 :R→R, ϕ1:R→R,such that the implications

(2.3)

b0(S, S⁰)ϕ0

" _p X

i=1

τiαi(S, S⁰)[ψi(S, λJ0)−ψi(S⁰, λJ0)]

# 50

⇒

p

X

i=1

τ_i

n

X

t=1

hD_tψ_i(S⁰, λ_J0), I_St−I_S⁰

ti5−ρ₀d²(S, S⁰), ∀S ∈ P and

(2.4)

b1(S, S⁰)ϕ1





k

X

s=1

βs(S, S⁰)X

j∈Js

λjgj(S⁰)



50

⇒

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

hD_tg_j(S⁰), I_St −I_S0

ti5−ρ⁰₀d²(S, S⁰), ∀S ∈ P.

both hold.

If (VP) is (ρ0, ρ⁰₀)-quasi V-univex type I at all S⁰ ∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p+ and λ ∈ R^m+, then we say that (VP) is (ρ0, ρ⁰₀)-quasi V-univex type I on P according to the partition {J₀, J₁, . . . , J_k} relative toτ ∈R^p+ and λ∈R^m+.

If the second (implied) inequality in (2.3) is strict (S 6=S⁰), then we say that (VP) is (ρ₀, ρ⁰₀)-semi strictly quasiV-univex type I atS⁰ or onP,accord- ing to the partition{J₀, J₁, . . . , J_k}relative toτ ∈R^p₊ andλ∈R^m₊,depending on the case.

Definition 2.5. We say that problem (VP) is (ρ₀, ρ⁰₀)-pseudo V-univex type I atS⁰ ∈ P according to the partition{J₀, J1, . . . , J_k}relative toτ ∈R^p+

and λ ∈ R^m+, if there exist positive real functions α₁, . . . , α_p and β₁, . . . , β_k defined on Γⁿ×Γⁿ, nonnegative functionsb0 andb1,also defined on Γⁿ×Γⁿ, ϕ₀ :R→R, ϕ₁:R→R,such that for allS∈ P,the implications

(2.5)

p

X

i=1

τ_i

n

X

t=1

D

D_tψ_i S⁰, λ_J0

, I_St−I_S⁰

t

E

=−ρ₀d² S, S⁰

⇒b₀ S, S⁰ ϕ₀

" _p X

i=1

τ_iα_i S, S⁰ ψ_i(S, λ_J0)−ψ_i S⁰, λ_J0

#

=0, and

(2.6)

m

X

j=1, j /∈J0

λ_j

n

X

t=1

D

D_tg_j(S⁰), I_St−I_S0 t

E

=−ρ⁰₀d² S, S⁰

⇒b1 S, S⁰ ϕ1





k

X

s=1

βs S, S⁰ X

j∈Js

λjgj S⁰



50.

both hold.

If (VP) is (ρ0, ρ⁰₀)-pseudo V-univex type I at all S⁰ ∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p+ and λ ∈ R^m+, then we say that (VP) is (ρ0, ρ⁰₀)-pseudoV-univex type I on P according to the partition {J₀, J₁, . . . , J_k} relative toτ ∈R^p+ and λ∈R^m+.

If the second (implied) inequality in (2.5) is strict (S 6= S⁰), then we say that (VP) is (ρ₀, ρ⁰₀)-semi strictly pseudo V-univex type I at S⁰ or onP, according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p₊ and λ ∈ R^m₊, depending on the case.

If the second (implied) inequalities in (2.5) and (2.6) are both strict, then we say that (VP) is (ρ0, ρ⁰₀)-strictly pseudo V-univex type I at S⁰ or on P, according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p+ and λ ∈ R^m+, depending on the case.

Definition 2.6. We say that problem (VP) is (ρ0, ρ⁰₀)-quasi pseudo V- univex type I at S⁰ ∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p₊ and λ ∈ R^m₊, if there exist positive real functions α1, . . . , αp and β₁, . . . , β_k defined on Γⁿ×Γⁿ, nonnegative functions b₀ and b₁, also defined on Γⁿ×Γⁿ,ϕ₀ :R→R, ϕ₁:R→R,such that the implications

(2.7)

b₀ S, S⁰ ϕ₀

" _p X

i=1

τ_iα_i S, S⁰ ψ_i(S, λ_J0)−ψ_i S⁰, λ_J0

# 50

⇒

p

X

i=1

τi n

X

t=1

D

Dtψi S⁰, λJ0

, ISt −I_S⁰

t

E

5−ρ₀d² S, S⁰

, ∀S∈ P

and

(2.8)

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

D

D_tg_j(S⁰), I_St −I_S0 t

E

=−ρ⁰₀d² S, S⁰

⇒b1 S, S⁰ ϕ1





k

X

s=1

βs S, S⁰ X

j∈Js

λjgj S⁰



50, ∀S ∈ P.

both hold.

If (VP) is (ρ0, ρ⁰₀)-quasi pseudo V-univex type I at all S⁰∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p+ and λ ∈ R^m+, then we say that (VP) is (ρ0, ρ⁰₀)-quasi pseudo V-univex type I on P according to the partition {J₀, J₁, . . . , J_k}relative toτ ∈R^p+ and λ∈R^m+.

If the second (implied) inequality in (2.8) is strict (S 6= S⁰), then we say that (VP) is (ρ₀, ρ⁰₀)-quasi strictly pseudo V-univex type I at S⁰ or on P, according to the partition {J₀, J1, . . . , Jk} relative to τ ∈ R^p₊ and λ ∈ R^m+,depending on the case.

Definition 2.7. We say that problem (VP) is (ρ₀, ρ⁰₀)-pseudo quasi V- univex type I at S⁰ ∈ P according to the partition {J₀, J1, . . . , J_k} relative to τ ∈ R^p₊ and λ ∈ R^m+, if there exist positive real functions α₁, . . . , α_p and β1, . . . , βk defined on Γⁿ×Γⁿ, nonnegative functionsb0 andb1,also defined on Γⁿ×Γⁿ,ϕ₀:R→R, ϕ₁ :R→R,such that for anyS ∈ P, the implications

(2.9)

p

X

i=1

τ_i

n

X

t=1

D

D_tψ_i S⁰, λ_J0

, I_St−I_S0 t

E

=−ρ₀d² S, S⁰

⇒b₀ S, S⁰ ϕ₀

" _p X

i=1

τ_iα_i S, S⁰ ψ_i(S, λ_J0)−ψ_i S⁰, λ_J0

#

=0,

and

(2.10)

b₁ S, S⁰ ϕ₁





k

X

s=1

β_s S, S⁰ X

j∈Js

λ_jg_j S⁰



=0

⇒

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

D

D_tg_j(S⁰), I_St −I_S⁰

t

E

5−ρ⁰₀d² S, S⁰ . both hold.

If (VP) is (ρ0, ρ⁰₀)-pseudo quasi V-univex type I at all S⁰∈ P according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p+ and λ ∈ R^m+, then we say that (VP) is (ρ0, ρ⁰₀)-pseudo quasi V-univex type I on P according to the partition {J₀, J₁, . . . , J_k}relative toτ ∈R^p+ and λ∈R^m+.

If the second (implied) inequality in (2.9) is strict (S 6= S⁰), then we say that (VP) is (ρ₀, ρ⁰₀)-strictly pseudo quasiV-univex type I at S⁰ or on P, according to the partition {J₀, J₁, . . . , J_k} relative to τ ∈ R^p₊ and λ ∈ R^m₊, depending on the case.

The following necessary conditions are from Zalmai ([9]).

Theorem 2.1(Zalmai [9]). Suppose that:

(a1)S⁰ is a properly efficient solution of (VP);

(a2)there exists anS^∗ ∈ PwithgM0(S^∗)<0whereM0=

j|gj(S⁰) = 0 such that

gj(S⁰) +

n

X

t=1

D

Dtgj(S⁰), IS^∗_t −I_S⁰

t

E

<0, ∀j∈ {1, . . . , m}. Then there exist τ⁰∈R^p, τ⁰ >0,and λ⁰ ∈R^m₊ such that

p

X

i=1

τ_i⁰

n

X

t=1

D

D_tf_i(S⁰), I_St −I_S⁰

t

E +

m

X

j=1

λ⁰_j

n

X

t=1

D

D_tg_j(S⁰), I_St−I_S⁰

t

E

=0,

∀S ∈ P,and

λ⁰_jgj(S⁰) = 0, j ∈ {1, . . . , m}. 3. OPTIMALITY CONDITIONS

Consider the multiobjective fractional programming problem

(VFP) Minimize

f1(S)

g₁(S), . . . ,fp(S) g_p(S)

subject to

hj(S)50, j = 1, . . . , m, S ∈Γⁿ. Assume thatgi(S)>0,i∈P ={1, . . . , p},S∈Γⁿ.

Denote by S₀ ={h_j(S)50,j= 1, . . . , m,S∈Γⁿ} the set of all feasible solutions of problem (VFP).

In order to derive a set of necessary and sufficient conditions for (VFP), we employ a Dinkelbach-type indirect approach via the following auxiliary problem

(VFP_µ) Minimize

S∈S₀ (f1(S)−µ1g1(S), . . . , fp(S)−µpgp(S)),

where µ_i, i ∈ P, are parameters. This problem is equivalent to (VFP) in the sense that for particular choices of µ_i, i ∈P,the two problems have the same set of efficient solutions. This equivalence is stated more precisely in the following lemma whose proof is straightforward, and hence, omitted.

Lemma 3.1 ([7]). An S^∗ ∈ S₀ is an efficient solution of (VFP) if and only if it is an efficient solution of (VFP_µ^∗) with µ^∗_i =f_i(S^∗)/g_i(S^∗), i∈P.

Put ψ_i(S, µ, λ_J0) = f_i(S) −µ_ig_i(S) +λ^T_J

0h_J0(S) where µ_i, i ∈ P, are parameters and λ∈R^m+ is a given vector.

The following theorem gives sufficient conditions for a set to be an effi- cient set solution of problem (VFP) under generalized type I conditions with respect to a partition of the constraints.

Theorem 3.1(Sufficiency). Suppose that:

(a1)S⁰ ∈ S₀;

(a2)there exist τ⁰ ∈R^p+,

p

P

i=1

τ_i⁰ = 1, µ⁰ ∈R^p+ and λ⁰ ∈R^m+ such that (a)for any S ∈ S₀ we have

p

X

i=1

τ_i⁰

n

X

t=1

D

Dtfi(S⁰)−µ⁰_iDtgi S⁰

, ISt−I_S⁰

t

E + +

m

X

j=1

λ⁰_j

n

X

t=1

D

Dthj S⁰

, ISt−I_S⁰

t

E

=0,

(b)with respect to the partition {J₀, J₁, . . . , J_k} we have X

j∈J_s

λ⁰_jh_j(S⁰) = 0 for any s∈ {0,1, . . . , k};

(a3)problem (VFP_µ⁰)is (ρ₀, ρ⁰₀)-quasi strictly pseudo V-univex type I at S⁰ with ρ₀+ρ⁰₀ ≥0, according to the partition {J₀, J₁, . . . , J_k}with respect to τ⁰, λ⁰ and for some positive functionsαi, i∈ {1, . . . , p}and βj, j ∈ {1, . . . , k}.

Further, suppose that, for r∈R, we have r50⇒ϕ0(r)≤0, (3.1)

ϕ₁(r)<0⇒r <0, (3.2)

b₀ S, S⁰

>0, b₁ S, S⁰

>0, ∀S∈ S₀. (3.3)

Then S⁰ is an efficient solution for (VFP).

Proof. Suppose that S⁰ is not an efficient solution of (VFP). According to Lemma 3.1, S⁰ is not an efficient solution of (VFP_µ⁰) with µ⁰_i = ^f_g^i(S0)

i(S⁰),i= 1, . . . , p. Then there exists anS⁰ ∈ S₀ such thatf_i(S⁰)−µ⁰_ig_i(S⁰)≤f_i(S⁰)− µ⁰_igi(S⁰), for any i = 1, . . . , p, with strict inequality for at least one i. Since (λ⁰_J

0)^Th_J0(S)50 for ∀S ∈ S₀ and by hypothesis (a2)(b), (λ⁰_J

0)^Th_J0(S⁰) = 0, it follows that for any i= 1, . . . , p,we have

ψi S⁰, µ⁰, λ⁰_J0

−ψ_i S⁰, µ⁰, λ⁰_J0

5(fi S⁰

−µ⁰_igi(S⁰))−(f_i S⁰

−µ⁰_igi(S⁰))≤0.

Since τ⁰ ∈R^p₊ and α_i S⁰, S⁰

>0, i= 1, . . . , p, it follows

p

X

i=1

τ_i⁰α_i S⁰, S⁰ ψ_i S⁰, µ⁰, λ⁰_J0

−ψ_i S⁰, µ⁰, λ⁰_J0

≤0 and using (3.1) and (3.3) we get

b₀(S⁰, S⁰)ϕ₀

" _p X

i=1

τ_i⁰α_i(S⁰, S⁰) ψ_i S⁰, µ⁰, λ⁰_J0

−ψ_i S⁰, µ⁰, λ⁰_J0

#

≤0.

(3.4)

From (2.7) and (3.4) it follows that

p

X

i=1

τ_i⁰

n

X

t=1

D

D_tψ_i S⁰, µ⁰, λ⁰_J0 , I_S⁰

t−I_S⁰

t

E

5−ρ₀d² S⁰, S⁰ .

Since ψi S⁰, µ⁰, λ⁰_J

0

= fi−µ⁰_igi

(S⁰) + P

j∈J₀

λ⁰_jhj(S⁰) and

p

P

i=1

τ_i⁰ = 1, the last inequality becomes

(3.5)

p

X

i=1

τ_i⁰

n

X

t=1

D

D_t(f_i−µ⁰_ig_j)(S⁰), I_S⁰

t −I_S⁰

t

E +

+X

j∈J0

λ⁰_j

n

X

t=1

D

Dthj(S⁰), I_S⁰

t−I_S⁰

t

E

5−ρ₀d² S⁰, S⁰ .

By inequality (3.5), assumption (a2) (a) and from the assumption that ρ0+ ρ⁰₀ ≥0, we have

(3.6)

m

X

j=1, j /∈J0

λ⁰_j

n

X

t=1

D

Dthj(S⁰), I_S⁰

t−I_S⁰

t

E

=ρ0d²(S⁰, S⁰)=−ρ⁰₀d²(S⁰, S⁰).

From (3.6) and assumption (a3), it follows that (3.7) b₁(S⁰, S⁰)ϕ₁





k

X

s=1

β_s(S⁰, S⁰)X

j∈J_s

λ⁰_jh_j(S⁰)



<0.

By (3.7), (3.2) and (3.3), we have (3.8)

k

X

s=1

βs(S⁰, S⁰)X

j∈Js

λ⁰_jhj(S⁰)<0.

On the other hand, by hypotheses (a2) (b) we have P

j∈Js

λ⁰_jh_j(S⁰) = 0 for any s∈ {1, . . . k},which implies

(3.9)

k

X

s=1

βs(S⁰, S⁰)X

j∈Js

λ⁰_jhj(S⁰) = 0.

Equation (3.9) contradicts inequality (3.8), and therefore the theorem is proved.

The next theorem gives necessary condition for a properly efficient set solution of (VFP) and results from Lemma 3.1 and Theorem 2.1.

Theorem 3.2(Necessity). Suppose that:

(b1)S⁰ is a properly efficient solution of (VFP);

(b2)there exists an S^∗ ∈ S₀ withhM0(S^∗)<0where M0={j|hj(S⁰) = 0}such that

hj(S⁰) +

n

X

t=1

D

Dthj(S⁰), I_St^∗−I_S⁰

t

E

<0, ∀j∈ {1, . . . , m}. Then there exist τ⁰∈R^p, τ⁰ >0,µ⁰ ∈R^p and λ⁰∈R^m+ such that

(3.10)

p

X

i=1

τ_i⁰

n

X

t=1

D

D_tf_i(S⁰)−µ⁰_iD_tg_i(S⁰), I_St−I_S0 t

E + +

m

X

j=1

λ⁰_j

n

X

t=1

D

Dthj(S⁰), I_St −I_S⁰

t

E

=0, for any S∈ S₀

and

(3.11) λ⁰_jhj(S⁰) = 0, j∈ {1, . . . , m}.

The above theorems contain two sets of parametersτ_i⁰ and µ⁰_i, i∈P.It is possible to eliminate one of these two sets of parameters, and thus, obtain a semi-parametric version of Theorems 3.1 and 3.2. This can be accomplished by simply replacingµ⁰_i by fi(S⁰)/gi(S⁰),i∈P, and redefiningτ⁰ andλ⁰. For further reference, we state this in next theorems.

Theorem 3.3 (Sufficiency). Suppose that (a1)S⁰ ∈ S₀;

(a2)there exist τ⁰ ∈R^p+,

p

P

i=1

τ_i⁰ = 1,and λ⁰ ∈R^m+ such that (a)for any S ∈ S₀;we have

p

X

i=1

τ_i⁰

n

X

t=1

D

g_i(S⁰)D_tf_i(S⁰)−f_i S⁰

D_tg_i(S⁰), I_St −I_S⁰

t

E + +

m

X

j=1

λ⁰_j

n

X

t=1

D

D_th_h(S⁰), I_St−I_S⁰

t

E

=0,

(b)with respect to the partition {J₀, J1, . . . , J_k}we have X

j∈Js

λ⁰_jh_h(S⁰) = 0, for any s∈ {0,1, . . . , k};

(a3)problem (VFP_µ) is (ρ0, ρ⁰₀)-quasi strictly pseudo V-univex type I at S⁰ with ρ₀+ρ⁰₀ ≥0, according to the partition {J₀, J₁, . . . , J_k},with respect to τ⁰, λ⁰ and for some positive functionsαi, i∈ {1, . . . , p}and βj, j ∈ {1, . . . , k}. Further, suppose that, for r ∈R,we have

r50⇒ϕ0(r)50, ϕ1(r)<0⇒r <0 and

b₀ S, S⁰

>0, b₁ S, S⁰

>0, ∀S∈ P.

Then S⁰ is an efficient solution for (VFP).

Theorem 3.4(Necessity). Suppose that:

(b1)S⁰ is a properly efficient solution of (VFP);

(b2)there exists an S^∗ ∈ S₀ withhM0(S^∗)<0where M0={j|hj(S⁰) = 0}such that

h_j(S⁰) +

n

X

t=1

D

D_th_j(S⁰), I_S^∗

t −I_S0 t

E

<0, ∀j∈ {1, . . . , m}.

Then there exist τ⁰ ∈R^p, τ⁰ >0,and λ⁰ ∈R^m₊ such that for any S ∈ S₀ we have

p

X

i=1

τ_i⁰

n

X

t=1

D

gi(S⁰)Dtfi(S⁰)−fi S⁰

Dtgi(S⁰), ISt −I_S⁰

t

E + +

m

X

j=1

λ⁰_j

n

X

t=1

D

Dthj(S⁰), ISt −I_S⁰

t

E

=0 and

λ⁰_jhj(S⁰) = 0, j∈ {1, . . . , m}.

4. GENERALIZED MOND-WEIR DUALITY

We associate to problem (VFP), with respect to the partition{J₀, J1, . . . , J_k} of its constraints, the following generalized Mond-Weir dual problem:

(GMWD)

maximize

f1(T)−µ₁g1(T)+λ^T_J0hJ0(T), . . . , fp(T)−µpgp(T) +λ^T_J0hJ0(T) , subject to (T, τ, µ, λ)∈ D,

where Dis the set of feasible solutions of problem (GMWD), that is

D=























(T, τ, µ, λ)

p

X

i=1

τ_i

n

X

t=1

D_t f_i−µ_ig_i+λ^T_J0h_J0

(T), I_St−I_Tt + +

m

X

j=1

λ_j

n

X

t=1

hD_th_j(T), I_St −I_Tti=0, ∀S ∈Γⁿ λ^T_J

sh_Js(T)=0, s= 1, . . . , k,

T ∈Γⁿ, τ ∈R^p+, e^Tτ = 1, λ∈R^m+, µ∈R^p+





















 and e= (1, . . . ,1)^T ∈R^p.

Theorem 4.1(Weak duality). Suppose that:

(i1) S∈ S₀;

(i2) (T, τ, µ, λ)∈ D and τ >0;

(i3) problem (VFP_µ) is (ρ0, ρ⁰₀)-pseudo quasi V-univex type I at T with ρ₀+ρ⁰₀ ≥0,according to the partition {J₀, J₁, . . . , J_k}with respect to τ,λand some positive functions αi, i∈ {1, . . . , p}, βs, s∈ {1, . . . , k}.

Further, assume that for r∈R we have ϕ₀(r)=0⇒r=0, (4.1)

r=0⇒ϕ1(r)=0, (4.2)

b₀(S, T)>0, b₁(S, T)=0.

(4.3)

Then, the following cannot hold

f_i(S)5f_i(T)−µ_ig_i(T) +λ^T_J0h_J0(T), for anyi∈ {1, . . . , p}, (4.4)

f_j(S)< f_j(T)−µ_jg_j(T) +λ^T_J0h_J0(T), for some j∈ {1, . . . , p}. (4.5)

Proof. By hypothesis (i2), we haveλ^>_J

sh_Js(T)=0,for alls∈ {1, . . . , k}. Since βs, s∈ {1, . . . , k},are all positive functions, we have

(4.6)

k

X

s=1

βs(S, T)λ^>_JshJs(T)=0.

By hypothesis (i3), (4.2), (4.3) and (4.6), it follows that (4.7)

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

hD_th_j(T), I_St−I_Tti5−ρ⁰₀d²(S, T).

From (4.7), hypothesis (i2), and by assumption that ρ0+ρ⁰₀≥0, we have

p

X

i=1

τi n

X

t=1

D Dt

fi−µigi+λ^>_J0hJ0

(T), ISt −ITt

E

= (4.8)

=ρ⁰₀d²(S, T)=−ρ₀d²(S, T).

From (4.8) and using again hypothesis (i3), we get b0(S, T)ϕ0

" _p X

i=1

τiαi(S, T)

fi−µigi+λ^>_J0hJ0

(4.9) (S)−

−

fi−µigi+λ^>_J0hJ0

(T)

#

=0.

From (4.9), (4.1) and (4.3), it follows

p

X

i=1

τiαi(S, T)

fi−µigi+λ^>_J0hJ0

(S)−

fi−µigi+λ^>_J0gJ0

(T)

=0.

(4.10)

Assume that (4.1) and (4.5) hold. ForS∈ S₀andλJ0 =0 we haveλ^>_J0gJ0(S)5 0.Sinceα_i>0, τ >0,from (4.2) and (4.3) we get

p

X

i=1

τiαi(S, T) h

fi−µigi+λ^>_J0gJ0

(S)−

fi−µigi−λ^>_J0gJ0

(T)

i

<0, which contradicts (4.10). Therefore, the theorem is proved.

Theorem 4.2 (Weak duality). Suppose that assumptions (i1) and (i2) of Theorem 4.1hold. We also assume

(i3⁰) problem (VFP_µ) is (ρ, ρ⁰)-semi strictly V-univex type I at T with

p

P

i=1 τiρi

αi(S,T)+

k

P

s=1 ρ⁰_s

βs(S,T) ≥0,according to the partition {J₀, J1, . . . , J_k}with res- pect to λand some positive functions α^∗_i, i∈ {1, . . . , p}andβ_s^∗, s∈ {1, . . . , k}. Further, assume that the functions ϕ₀ and ϕ₁ have the properties

ϕ₀(r)=0⇒r=0, (4.11)

r=0⇒ϕ₁(r)=0 (4.12)

with ϕ₀ linear, and

(4.13) b₀(S, T)<0, b₁(S, T)=0.

Then, the following cannot hold

fi(S)5fi(T)−µigi(T) +λ^T_J0hJ0(T), for any i∈ {1, . . . , p}, (4.14)

fj(S)< fj(T)−µjgj(T) +λ^T_J0hJ0(T), for some j ∈ {1, . . . , p}. (4.15)

Proof. By hypothesis (i2) we haveλ^>_J

sh_Js(T)=0,for all s∈ {1, . . . , k}. From (4.12) and sinceβ_s^∗,s={1, . . . k}are all positive functions, it follows that (4.16) β_s^∗(S, T)ϕ₁

X

j∈J_s

λ_jh_j(T)

=0.

If we replace in Definition 2.3 the expressionb1 S, S⁰

/βs S, S⁰

byβ_s^∗(S, T), from (4.16) and (i3⁰) it follows

X

j∈Js

λj n

X

t=1

hD_thj(T), ISt−ITti5 −ρ⁰_sd²(S, T) βs(S, T) . Summing these relations over s∈ {1, . . . , k},we get

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

hD_th_j(T), I_St−I_Tti5

k

X

s=1

−ρ⁰_s

β_s(S, T)d²(S, T).

Since

p

P

i=1 τiρi

αi(S,T) + Pk s=1

ρ⁰_s

βs(S,T) ≥0,we obtain (4.17)

m

X

j=1, j /∈J₀

λ_j

n

X

t=1

hD_th_j(T), I_St −I_Tti5

p

X

i=1

τ_iρ_i

αi(S, T)d²(S, T).

From (4.17) and hypothesis (i2), it follows (4.18)

p

X

i=1

τ_i

n

X

t=1

D D_t

f_i−µ_ig_i+λ^>_J0h_J0

(T), I_St−I_TtE

=−

p

X

i=1

τ_iρ_i

αi(S, T)d²(S, T).

Dividing both sides of (2.1) by αi S, S⁰

and taking S⁰ = T, by hypothesis (i3⁰) we get

b0(S, T) α_i(S, T)ϕ0

h

fi−µigi+λ^>_J0hJ0

(S)−

fi−µigi+λ^>_J0hJ0

(T)

i

>

>D

D

f_i−µ_ig_i+λ^>_J0h_J0

(T), I_S−I_TE

+ ρ_i

αi(S, T)d²(S, T), ∀i.

Multiplying by τ_i and takingα^∗_i(S, T) = 1/α_i(S, T),we get for ∀i, b₀(S, T)τ_iα^∗_i(S, T)ϕ₀h

f_i−µ_ig_i+λ^>_J0h_J0

(S)−

f_i−µ_ig_i+λ^>_J0h_J0 (T)i

>

> τihDψ_i(T, µ, λJ0), IS−ITi+ρiα_i^∗(S, T)d²(S, T).

Applying (4.18), using (4.11), (4.13) and then summing with respect to i, we have

(4.19)

p

X

i=1

τ_iα^∗_i(S, T)h

f_i−µ_ig_i+λ^>_J0h_J0 (S)−

f_i−µ_ig_i+λ^>_J0h_J0 (T)i

>0.

If we assume that (4.14) and (4.15) hold, since α^∗_i >0 andτ >0,we have

p

X

i=1

τiα^∗_i(S, T) h

fi−µigi+λ^>_J0hJ0

(S)−

fi−µigi+λ^>_J0hJ0

(T)

i

<0, which contradicts (4.19).

Theorem 4.3(Strong duality). Suppose that:

(j1)S⁰ is a properly efficient solution of (VFP_µ);

(j2)there exists an S^∗∈ S₀withh_M0(S^∗)<0,where M₀={j|h_j(S⁰) = 0},such that

hj(S⁰) +

n

X

t=1

D

Dthj(S⁰), I_St^∗−I_S⁰

t

E

<0, ∀j∈ {1, . . . , m}.

Then there exist τ⁰ ∈ R^p, τ⁰ > 0, µ⁰ ∈ R^p and λ⁰ ∈ R^m₊ such that (S⁰, τ⁰, µ⁰, λ⁰) ∈ D and the objective functions of (VFP_µ) and (GMWD) have the same values at S⁰ and S⁰, τ⁰, µ⁰, λ⁰

, respectively.

If problem(VFP_µ)is (ρ₀, ρ⁰₀)-pseudo quasi V-univex type I withρ₀+ρ⁰₀≥ 0 at all feasible solutions of (GMWD),according to the partition {J₀, J₁, . . . , J_k}with respect to τ⁰, λ⁰ and conditions (4.1)–(4.3)of Theorem 4.1are satis- fied for all feasible solutions of (GMWD), then S⁰, τ⁰, µ⁰, λ⁰

∈ D is an efficient solution of (GMWD).

Proof. By Theorem 2.1, there exist τ⁰ ∈R^p, τ⁰ >0,and λ⁰ ∈R^m+ such that, for any S ∈ S₀ we have

p

P

i=1

τ_i⁰

n

P

t=1

D

D_tf_i(S⁰)−µ⁰_iD_tg_i(S⁰), I_St−I_S⁰

t

E +