# Optimality and Duality for  Multiobjective Fractional Programming Involving Nonsmooth Generalized(ℱ,b,ϕ,ρ,θ)-Univex  Functions

Authors: Jen-Chwan Liu; Chun-Yu Liu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

We establish properly efficient necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth generalized(ℱ,b,ϕ,ρ,θ)-univex functions. Utilizing the necessary optimality conditions, we formulate the parametric dual model and establish some duality results in the framework of generalized (ℱ,b,ϕ,ρ,θ)-univex functions.


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## Body

## 1. Introduction

In this paper, we consider the following nondifferentiable nonconvex multiobjective fractional programming problem:(1)(MFP)Minimize(f1(x)g1(x),f2(x)g2(x),…,fm(x)gm(x)),subjecttox∈S={x∈ℝn∣hj(x)≤0,000000000000000000for1≤j≤pℝnhj},
where  (a1)
fi:X0↦ℝ, gi:X0↦ℝ, i=1,2,…,m and hj:X0↦ℝ, j=1,2,…,p are Lipschitz on X0, and X0 is an open subset of ℝn;(a2)
fi(x)≥0, gi(x)>0, i=1,2,…,m.Minimize means obtaining  efficient solution in the following sense. A point x¯∈S is said to be an  efficient solution for (MFP) if there is no x∈S such that
(2)fi(x)gi(x)≤fi(x¯)gi(x¯),1≤i≤m,
with at least one strict inequality.A pointx¯∈S is said to be a  properly efficient solution for (MFP) which was introduced by Geoffrion [1] if and only if  (a)
x¯ is an efficient solution for (MFP),(b)
there exists a scalarM>0 such that for each i, we have
(3)fi(x¯)/gi(x¯)-fi(x)/gi(x)fj(x)/gj(x)-fj(x¯)/gj(x¯)≤M,for somej such that fj(x¯)/gj(x¯)<fj(x)/gj(x), whenever x∈S and fi(x¯)/gi(x¯)>fi(x)/gi(x).An efficient point for (MFP) that is not properly efficient is said to beimproperly efficient. Thus, for x¯ to be improperly efficient for (MFP) means that to every scalar M>0, there is a point x∈S and an i such that fi(x¯)/gi(x¯)>fi(x)/gi(x) and
(4)fi(x¯)/gi(x¯)-fi(x)/gi(x)fj(x)/gj(x)-fj(x¯)/gj(x¯)>M,
for all j such that fj(x¯)/gj(x¯)<fj(x)/gj(x).Many papers have been devoted to the multiobjective fractional programming problem in recent decades; see for example [1–13].In [8], Preda introduced generalized (ℱ,ρ)-convexity, an extension of ℱ-convexity and generalized ρ-convexity defined by Vial [14, 15]. Bhatia and Jain [2] defined generalized (ℱ,ρ)-convexity for nonsmooth functions, an extension of generalized (ℱ,ρ)-convexity defined by Preda [8], and they derived some duality theorems for nonsmooth multiobjective programs. In [5, 6], Liu also established the Kuhn-Tucker type necessary and sufficient optimality conditions for multiobjective fractional programming involving nonsmooth pseudoinvex functions in [5] or (ℱ,ρ)-convex functions in [6] and considered the parameter dual problem in the framework of generalized convex functions.Recently, Zalmai [13, 16, 17] introduced generalized (ℱ,b,ϕ,ρ,θ)-univex n-set functions and he also established sufficient efficiency conditions in multiobjective fractional subset programming [13] and sufficient optimality conditions in minimax fractional subset programming [16, 17] under various generalized (ℱ,b,ϕ,ρ,θ)-univexity assumptions. In [10], Preda  et al. obtained duality results for a dual model of Zalmai [13] replacing the assumptions of sublinearity or convexity by that of quasiconvexity in the third argument.In this paper, we are inspired to consider the optimality and duality of properly efficient for (MFP) containing nonsmooth generalized(ℱ,b,ϕ,ρ,θ)-univex functions. We organize this paper as follows. Some definitions and notations are given in Section 2. In Section 3, we establish some properly efficient sufficient optimality conditions for (MFP) involving nonsmooth generalized (ℱ,b,ϕ,ρ,θ)-univex functions. We also derive some duality results in Section 4.


## 2. Notations and Preliminary

Throughout the paper, letℝn be the n-dimensional Euclidean space and let ℝ+n be its nonnegative orthant.Definition 1.
The functionf:X0↦ℝ is said to be Lipschitz on X0 if there exists c>0 such that for all y,x∈X0,
(5)|f(y)-f(x)|≤c∥y-x∥,
where ∥·∥ denotes any norm in ℝn.For eachd in ℝn, f∘(x;d) is the generalized directional derivative of Clarke [18] defined by
(6)f∘(x;d)=limsupy→xt↓0f(y+td)-f(y)t.
It then follows that
(7)f∘(x;d)=max{ξ⊤d∣ξ∈∂f(x)}foranyxandd.
where ∂f(·) denotes Clarke’s generalized gradient [18].Definition 2.
A functionalℱ:ℝn↦ℝ is said to be sublinear (superlinear) if ℱ(a1+a2)≤(≥)ℱ(a1)+ℱ(a2) for all a1,a2∈ℝn, and ℱ(αa)=αℱ(a) for all α∈ℝ+ and a∈ℝn.LetX be an open subset of ℝn. We suppose that the function f:X↦ℝ is Lipschitz on X and d(·,·):X×X↦ℝ such that d(x,x)=0 for any x.Definition 3 (cf. [13, 16, 17, 19, 20]).
(1) Letℱ(x,x*;·):ℝn↦ℝ be a sublinear function, b:X×X↦ℝ a function with positive values, ϕ:ℝ↦ℝ a function, ρ a real number, and θ:X×X↦X×X a function such that x≠x*⇒θ(x,x*)≠(0,0). The function f is said to be (ℱ,b,ϕ,ρ,θ)-univex at x* if
(8)ϕ(f(x)-f(x*))≥ℱ(x,x*;b(x,x*)ξ)+ρd2(θ(x,x*))
for all x∈X and ξ∈∂f(x*).
(2) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-pseudounivex at x* if
(9)ℱ(x,x*;b(x,x*)ξ)≥-ρd2(θ(x,x*))⟹ϕ(f(x)-f(x*))≥0
for all x∈X and ξ∈∂f(x*).
(3) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-strictly-pseudounivex at x* if
(10)ℱ(x,x*;b(x,x*)ξ)≥-ρd2(θ(x,x*))⟹ϕ(f(x)-f(x*))>0
for all x∈X and ξ∈∂f(x*).
(4) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-prestrictly-pseudounivex at x* if
(11)ℱ(x,x*;b(x,x*)ξ)>-ρd2(θ(x,x*))⟹ϕ(f(x)-f(x*))≥0
for all x∈X and ξ∈∂f(x*).
(5) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-quasiunivex at x* if
(12)ϕ(f(x)-f(x*))≤0⟹ℱ(x,x*;b(x,x*)ξ)≤-ρd2(θ(x,x*))
for all x∈X and ξ∈∂f(x*).
(6) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-strictly-quasiunivex at x* if
(13)ϕ(f(x)-f(x*))≤0⟹ℱ(x,x*;b(x,x*)ξ)<-ρd2(θ(x,x*))
for all x∈X and ξ∈∂f(x*).
(7) The functionf is said to be (ℱ,b,ϕ,ρ,θ)-prestrictly-quasiunivex at x* if
(14)ϕ(f(x)-f(x*))<0⟹ℱ(x,x*;b(x,x*)ξ)≤-ρd2(θ(x,x*))
for all x∈X and ξ∈∂f(x*).In order to establish some Kuhn-Tucker conditions for efficient optimality of (MFP), we give the scalar minimization problem as follows:(15)(SFP)iMinimizefi(x)gi(x),subjecttofk(x)gk(x)≤fk(x¯)gk(x¯),k≠i,subjecttohj(x)≤0,j=1,2,…,p.In [6], we established the following necessary efficient optimality conditions for (MFP).Lemma 4.
Letx¯∈S. If   x¯ is an efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (SFP)i, i=1,2,…,m. Then there exist vi¯=fi(x¯)/gi(x¯)≥0, yi¯>0, 1≤i≤m, and β¯∈ℝ+p such that
(16)0∈∑i=1m‍yi¯Ti(x¯)+∑j=1p‍βj¯∂hj(x¯),βj¯hj(x¯)=0,1≤j≤p,
where
(17)Ti(x¯)=∂fi(x¯)+vi¯∂(-gi)(x¯)gi(x¯),1≤i≤m.If we chooseλi¯=yi¯/gi(x¯)>0, the result of Lemma 4 can be restated as follows.Theorem 5 (necessary optimality conditions).
Letx¯∈S. If x¯ is an (properly) efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (SFP)i, i=1,2,…,m. Then there exist λi¯>0, 1≤i≤m, v¯∈ℝ+m, and β¯∈ℝ+p such that
(18)0∈∑i=1m‍λi¯[∂fi(x¯)+vi¯∂(-gi)(x¯)]+∑j=1p‍βj¯∂hj(x¯),(19)fi(x¯)-vi¯gi(x¯)=0,1≤i≤m,(20)βj¯hj(x¯)=0,1≤j≤p.In order to simplify the complication of (MFP), we consider the following nonfractional multiobjective programming problem with a parameterv∈ℝ+m:
(21)(MP)vminimize(f1(x)-v1g1(x),Minimize0000f2(x)-v2g2(x),…,Minimize0000fm(x)-vmgm(x))subjecttox∈S.
We obtained the equivalence between (MFP) and (MP)v as follows.Lemma 6.
Letx¯∈S and vi¯=fi(x¯)/gi(x¯),1≤i≤m. Then x¯ is a properly efficient solution for (MFP) if and only of x¯ is a properly efficient solution for (MP)v¯.Proof.
(a) Letx¯ be an efficient solution for (MP)v¯. Assume that x¯ is not an efficient solution for (MFP). Then there exists x∈S such that
(22)fi(x)gi(x)≤fi(x¯)gi(x¯),1≤i≤m,
with at least one strict inequality. Thus, we have
(23)fi(x)-fi(x¯)gi(x)gi(x¯)≤0=fi(x¯)-fi(x¯)gi(x¯)gi(x¯),1≤i≤m,
or equivalently,
(24)fi(x)-vi¯gi(x)≤fi(x¯)-vi¯gi(x¯),1≤i≤m,
with at least one strict inequality which contradicts the efficient solution of x¯ for  (MP)v¯. Therefore, we have x¯ as an efficient solution for (MFP). Conversely, if x¯ is an efficient solution for   (MFP), along with the same lines of above, we can obtain that x¯ is an efficient solution for    (MP)v¯.
(b) Letx¯ be a properly efficient solution for   (MFP). Assume that x¯ is not a properly efficient solution for   (MP)v¯. Then every scalar M>0, there is a point x∈S and an i such that fi(x¯)-vi¯gi(x¯)>fi(x)-vi¯gi(x) and
(25)(fi(x¯)-vi¯gi(x¯))-(fi(x)-vi¯gi(x))(fj(x)-vj¯gj(x))-(fj(x¯)-vj¯gj(x¯))>Mgi(x)gj(x),
for all j such that fj(x¯)-vj¯gj(x¯)<fj(x)-vj¯gj(x).
It follows that for somei such that 0>fi(x)-fi(x¯)/gi(x¯)gi(x) and
(26)(fi(x¯)/gi(x¯))gi(x)-fi(x)fj(x)-(fj(x¯)/gj(x¯))gj(x)>Mgi(x)gj(x),
for all j such that 0<fj(x)-fj(x¯)/gj(x¯)gj(x), or equivalently, for some i such that fi(x¯)/gi(x¯)>fi(x)/gi(x) and
(27)fi(x¯)/gi(x¯)-fi(x)/gi(x)fj(x)/gj(x)-fj(x¯)/gj(x¯)>M,
for all j such that fj(x¯)/gj(x¯)<fj(x)/gj(x), which contradicts the properly efficient solution of x¯ for (MFP). Therefore, we have x¯ is a properly efficient solution for   (MP)v¯. Conversely, if x¯ is a properly efficient solution for   (MP)v¯, along with the same lines of above, we can obtain that x¯ is a properly efficient solution for (MFP). Thus, the proof is complete.Lemma 7.
Letx¯∈S and vi¯=fi(x¯)/gi(x¯), 1≤i≤m. If there exists λ¯∈ℝ+m∖{0} such that
(28)∑i=1m‍λi¯(fi(x)-vi¯gi(x))≥0,foranyx∈S,
then x¯ is a properly efficient solution to (MFP).Proof.
Assume thatx¯ is not an efficient for (MFP). Then there exists a point x∈S such that
(29)fi(x)gi(x)≤fi(x¯)gi(x¯),1≤i≤m,
with at least one strict inequality. Thus, we have
(30)fi(x)-fi(x¯)gi(x)gi(x¯)≤0,1≤i≤m,
or equivalently,
(31)fi(x)-vi¯gi(x)≤0,1≤i≤m,
with at least one strict inequality.
Thus, for anyλ∈ℝ+m∖{0}, we have
(32)∑i=1m‍λi(fi(x)-vi¯gi(x))<0,
which contradicts the inequality (28).
It remains to show thatx¯ is properly efficient. Assume that it is an improperly efficient point for the problem (MFP). Then, by Lemma 6, x¯ is an improperly efficient solution for  (MP)v¯. Then every scalar M>0, there is a point x∈S and an i such that fi(x¯)-vi¯gi(x¯)>fi(x)-vi¯gi(x) and
(33)(fi(x¯)-vi¯gi(x¯))-(fi(x)-vi¯gi(x))(fj(x)-vj¯gj(x))-(fj(x¯)-vj¯gj(x¯))>M,
for all j such that fj(x¯)-vj¯gj(x¯)<fj(x)-vj¯gj(x).
For anyλ∈ℝ+m∖{0}, we choose M=(m-1)maxi,j{λj/λi}>0, (we may assume that m≥2) such that
(34)[fi(x¯)-vi¯gi(x¯)]-[fi(x)-vi¯gi(x)]>(m-1)λjλi[(fj(x)-vj¯gj(x))-(fj(x¯)-vj¯gj(x¯))],
for all j≠i. These imply that
(35)λim-1[vi¯gi(x)-fi(x)]>λj[fj(x)-vj¯gj(x)],∀j≠i.
Summing over j≠i yield
(36)λi[vi¯gi(x)-fi(x)]>∑j≠i‍λj[fj(x)-vj¯gj(x)],
or
(37)∑i=1m‍λi[fi(x)-vi¯gi(x)]<0,
which contradicts the inequality (28). Thus, the proof is complete.


## 3. Sufficient Optimality Conditions

Applying Lemmas6 and 7, we can establish the following sufficient optimality conditions for properly efficient solution of (MFP).Let(38)P={1,2,…,p},P(x)={j∈P∣hj(x)=0}.Theorem 8 (sufficient optimality conditions).
Letx¯∈S, and let there exist λi¯>0, 1≤i≤m, v¯∈ℝ+m, and β¯∈ℝ+p such that (18), (19), and (20). Let
(39)A(x)=∑i=1m‍λi¯[fi(x)+vi¯(-gi)(x)]+∑j=1p‍βj¯hj(x),B(x)=∑i=1m‍λi¯[fi(x)+vi¯(-gi)(x)],C(x)=∑j=1p‍βj¯hj(x).
If any one of the following conditions holds:  (a)
(i)
fi is (ℱ,b,ϕ1,ρ1i,θ)-univex at x¯, -gi is (ℱ,b,ϕ1,ρ2i,θ)-univex at x¯, 1≤i≤m,ϕ1 is superlinear, and ϕ1(a)≥0⇒a≥0;(ii)
for eachj∈P(x¯),   hj is (ℱ,b,ϕ2j,ρ3j,θ)-quasiunivex at x¯,   ϕ2j is increasing, and ϕ2j(0)=0;(iii)
∑i=1m‍λi¯ρ1i+∑i=1m‍λi¯vi¯ρ2i+∑j∈P(x¯)βj¯ρ3j≥0;(b)
(i)
fi is (ℱ,b,ϕ1,ρ1i,θ)-univex at x¯, -gi is (ℱ,b,ϕ1,ρ2i,θ)-univex at x¯, 1≤i≤m, ϕ1 is superlinear, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ3,θ)-quasiunivex at x¯,    ϕ2 is increasing, and ϕ2(0)=0;(iii)
∑i=1m‍λi¯ρ1i+∑i=1m‍λi¯vi¯ρ2i+ρ3≥0;(c)
(i)
A is (ℱ,b,ϕ,0,θ)-pseudounivex at x¯ and ϕ(a)≥0⇒a≥0;(d)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at x¯, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-quasiunivex at x¯, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2≥0;(e)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-prestrictly-quasiunivex at x¯, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-quasiunivex at x¯, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2>0;(f)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-prestrictly-quasiunivex at x¯, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-strictly-pseudounivex at x¯, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2≥0;(g)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at x¯, and ϕ1(a)≥0⇒a≥0;(ii)
for eachj∈P(x¯), hj is (ℱ,b,ϕ2j,ρ2j,θ)-quasiunivex at x¯, ϕ2j is increasing, and ϕ2j(0)=0;(iii)
ρ1+∑j∈P(x¯)βj¯ρ2j≥0.Thenx¯ is a properly efficient solution for (MFP).Proof.
Letx be any feasible solution of (MFP). By (18), there exist ξi∈∂fi(x¯), ςi∈∂(-gi)(x¯), for 1≤i≤m, and ζj∈∂hj(x¯), for 1≤j≤p such that
(40)∑i=1m‍λi¯(ξi+vi¯ςi)+∑j=1p‍βj¯ζj=0.
From here it results
(41)ℱ(x,x¯;∑i=1m‍λi¯(ξi+vi¯ςi)+∑j=1p‍βj¯ζj)≥0.
If hypothesis (a) holds,
(42)ϕ1(fi(x)-fi(x¯))≥ℱ(x,x¯;b(x,x¯)ξi)+ρ1id2(θ(x,x¯)),1≤i≤m,(43)ϕ1(-gi(x)+gi(x¯))≥ℱ(x,x¯;b(x,x¯)ςi)+ρ2id2(θ(x,x¯)),1≤i≤m.
Now, multiplying (42) by λi¯, (43) by λi¯vi¯, and adding the resulting inequalities, and then using the superlinearity of ϕ1 and sublinearity of ℱ(x,x¯;·), we obtain
(44)ϕ1(∑i=1m‍λi¯[fi(x)-vi¯gi(x)]-∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)])≥ℱ(x,x¯;b(x,x¯)∑i=1m‍λi¯(ξi+vi¯ςi))+(∑i=1m‍λi¯ρ1i+∑i=1m‍λi¯vi¯ρ2i)d2(θ(x,x¯)).
Since x∈S, it is clear that
(45)hj(x)-hj(x¯)≤0,∀j∈P(x¯).
Using the increasing property of ϕ2j and ϕ2j(0)=0, we have
(46)ϕ2j(hj(x)-hj(x¯))≤0,∀j∈P(x¯).
Using the (ℱ,b,ϕ2j,ρ3j,θ)-quasiunivexity of hj at x¯, we get from (46)
(47)ℱ(x,x¯;b(x,x¯)ζj)≤-ρ3jd2(θ(x,x¯)),∀j∈P(x¯).
Because β¯∈ℝ+p and (20) holds, β¯j=0 for each j∈P∖P(x¯), and ℱ(x,x¯;·) is sublinear, the above inequality yields
(48)ℱ(x,x¯;b(x,x¯)∑j=1p‍βj¯ζj)≤-(∑j∈P(x¯)‍βj¯ρ3j)d2(θ(x,x¯)).
From the nonnegativity of b(x,x¯), sublinearity of ℱ(x,x¯;·), and (41), it follows that
(49)ℱ(x,x¯;b(x,x¯)∑i=1m‍λi¯(ξi+vi¯ςi))0000+ℱ(x,x¯;b(x,x¯)∑j=1p‍βj¯ζj)≥0.
Now adding (44) and (48), and then using (49) and ∑i=1m‍λi¯ρ1i+∑i=1m‍λi¯vi¯ρ2i+∑j∈P(x¯)βj¯ρ3j≥0, we obtain
(50)ϕ1(∑i=1m‍λi¯[fi(x)-vi¯gi(x)]-∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)])≥0.
But ϕ1(a)≥0⇒a≥0, and so (50) yields
(51)∑i=1m‍λi¯[fi(x)-vi¯gi(x)]-∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)]≥0.
Thus, we have
(52)∑i=1m‍λi¯[fi(x)+vi¯(-gi(x))]≥∑i=1m‍λi¯[fi(x¯)+vi¯(-gi(x¯))]=0.
Therefore, by Lemma 7, x¯ is a properly efficient solution for (MFP).
If hypothesis (b) holds, from (20), x∈S, and β¯∈ℝ+p, we have
(53)∑j=1p‍βj¯hj(x)≤0=∑j=1p‍βj¯hj(x¯).
and so using the properties of ϕ2, we obtain
(54)ϕ2(∑j=1p‍β¯j[hj(x)-hj(x¯)])≤0.
Using the (ℱ,b,ϕ2,ρ3,θ)-quasiunivexity of C at x¯, we get from (54)
(55)ℱ(x,x¯;b(x,x¯)∑j=1p‍βj¯ζj)≤-ρ3d2(θ(x,x¯)).
Now combining (44), (49), and (55), and using (iii), we obtain (52). Therefore, by Lemma 7, x¯ is a properly efficient solution for (MFP).
If hypothesis (c) holds, using the(ℱ,b,ϕ,0,θ)-pseudounivexity of A, we get from (41)
(56)ϕ(∑i=1m‍λi¯[fi(x)-vi¯gi(x)]+∑j=1p‍β¯jhj(x)0000-(∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)]+∑j=1p‍β¯jhj(x¯)))≥0.
In view of the properties of ϕ, we deduce from this inequality that
(57)∑i=1m‍λi¯[fi(x)-vi¯gi(x)]+∑j=1p‍β¯jhj(x)0000-(∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)]+∑j=1p‍β¯jhj(x¯))≥0.
Equation (53) along with (57) yields
(58)∑i=1m‍λi¯[fi(x)+vi¯(-gi(x))]≥∑i=1m‍λi¯[fi(x¯)+vi¯(-gi(x¯))]=0.
Therefore, by Lemma 7, x¯ is a properly efficient solution for (MFP).
If hypothesis (d) holds, using the properties ofϕ2 and (53), we obtain
(59)ϕ2(∑j=1p‍β¯j[hj(x)-hj(x¯)])≤0,
which in view of (ii) implies that
(60)ℱ(x,x¯;b(x,x¯)∑j=1p‍βj¯ζj)≤-ρ2d2(θ(x,x¯)).
Now combining (49), (60), and ρ1+ρ2≥0, we obtain
(61)ℱ(x,x¯;b(x,x¯)∑i=1m‍λi¯(ξi+vi¯ςi))00≥-ℱ(x,x¯;b(x,x¯)∑j=1p‍βj¯ζj)00≥ρ2d2(θ(x,x¯))≥-ρ1d2(θ(x,x¯)).
Along with the fact that the B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at x¯, we get from (61) that
(62)ϕ1(∑i=1m‍λi¯[fi(x)-vi¯gi(x)]-∑i=1m‍λi¯[fi(x¯)-vi¯gi(x¯)])≥0.
In view of the properties of ϕ1, we deduce from this inequality that
(63)∑i=1m‍λi¯[fi(x)+vi¯(-gi(x))]≥∑i=1m‍λi¯[fi(x¯)+vi¯(-gi(x¯))]=0.
Therefore, by Lemma 7, x¯ is a properly efficient solution for (MFP). (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.


## 4. Duality Theorem

From the optimality of properly efficient for the problem (MFP), we can formulate the following parametric dual problem:(64)(D)maximizev=(v1,v2,…,vm)subjectto0∈∑i=1m‍λi[∂fi(u)+vi∂(-gi)(u)]subjectto0∈+∑j=1p‍βj∂hj(u),(65)fi(u)-vigi(u)≥0,1≤i≤m,(66)βjhj(u)≥0,1≤j≤p,(67)u∈ℝn,v∈ℝ+m,β∈ℝ+p,λi>0,1≤i≤m.
We denote by FD the set of all feasible solutions (u,v,λ,β) of problem (D). The point (u¯,v¯,λ¯,β¯)∈FD is called an efficient for (D) if there is no (u,v,λ,β)∈FD such that
(68)vi≥vi¯,1≤i≤m,
with at least one strict inequality.The point(u¯,v¯,λ¯,β¯)∈FD is said to be a properly efficient for (D) if and only if  (a)
(u¯,v¯,λ¯,β¯) is an efficient solution for (D),(b)
there exists a scalarM>0 such that for each i, we have
(69)vi-vi¯vj¯-vj≤M,for somej such that vj<vj¯, whenever (u,v,λ,β)∈FD and vi>vi¯.If(u¯,v¯,λ¯,β¯) was not properly efficient solution for (D), for each fixed real number M>0, there would exist a feasible solution (u,v,λ,β)∈FD and an index i such that vi>vi¯ and
(70)vi-vi¯vj¯-vj>M,
for all j such that vj<vj¯.Theorem 9 (duality theorem).
Letx¯∈S, (u,v,λ,β)∈FD, and let
(71)A(·)=∑i=1m‍λi[fi(·)+vi(-gi)(·)]+∑j=1p‍βjhj(·),B(·)=∑i=1m‍λi[fi(·)+vi(-gi)(·)],C(·)=∑j=1p‍βjhj(·).If any one of the following conditions holds:(a)
(i)
fi is (ℱ,b,ϕ1,ρ1i,θ)-univex at u, -gi is (ℱ,b,ϕ1,ρ2i,θ)-univex at u, 1≤i≤m, ϕ1 is superlinear, and ϕ1(a)≥0⇒a≥0;(ii)
βjhj is (ℱ,b,ϕ2j,ρ3j,θ)-quasiunivex at u,1≤j≤p, ϕ2j is increasing, and ϕ2j(0)=0;(iii)
∑i=1m‍λiρ1i+∑i=1m‍λiviρ2i+∑j=1p‍ρ3j≥0;(b)
(i)
fi is (ℱ,b,ϕ1,ρ1i,θ)-univex at u, -gi is (ℱ,b,ϕ1,ρ2i,θ)-univex at u, 1≤i≤m, ϕ1 is superlinear, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ3,θ)-quasiunivex at u,ϕ2 is increasing, and ϕ2(0)=0;(iii)
∑i=1m‍λiρ1i+∑i=1m‍λiviρ2i+ρ3≥0;(c)
(i)
A is (ℱ,b,ϕ,0,θ)-pseudounivex at u and ϕ(a)≥0⇒a≥0.(d)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at u, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-quasiunivex at u, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2≥0;(e)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-prestrictly-quasiunivex at u, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-quasiunivex at u, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2>0;(f)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-prestrictly-quasiunivex at u, and ϕ1(a)≥0⇒a≥0;(ii)
C is (ℱ,b,ϕ2,ρ2,θ)-strictly-pseudounivex at u, ϕ2 is increasing, and ϕ2(0)=0;(iii)
ρ1+ρ2≥0;(g)
(i)
B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at u, and ϕ1(a)≥0⇒a≥0;(ii)
βjhj is (ℱ,b,ϕ2j,ρ2j,θ)-quasiunivex at u,1≤j≤p, ϕ2j is increasing, and ϕ2j(0)=0;(iii)
ρ1+∑j=1p‍ρ2j≥0.Assume thatx¯ is a properly efficient solution for (MFP) and satisfies an appropriate constraint qualification [21] for (SFP)i, i=1,2,…,m. Then there exist λi¯>0, 1≤i≤m, v¯∈ℝ+m, and β¯∈ℝ+p such that (x¯,v¯,λ¯,β¯) is a properly efficient solution for (D).Proof.
With Theorem5, there exist λi¯>0, 1≤i≤m, v¯∈ℝ+m, and β¯∈ℝ+p such that (x¯,v¯,λ¯,β¯) is a feasible solution of (D). By (64), there exist ξi∈∂fi(u), ςi∈∂(-gi)(u), for 1≤i≤m, and ζj∈∂hj(u), for 1≤j≤p such that
(72)∑i=1m‍λi(ξi+viςi)+∑j=1p‍βjζj=0.
From here it results
(73)ℱ(x¯,u;∑i=1m‍λi(ξi+viςi)+∑j=1p‍βjζj)≥0.
If hypothesis (a) holds, then
(74)ϕ1(fi(x¯)-fi(u))≥ℱ(x¯,u;b(x¯,u)ξi)+ρ1id2(θ(x¯,u)),1≤i≤m,(75)ϕ1(-gi(x¯)+gi(u))≥ℱ(x¯,u;b(x¯,u)ςi)+ρ2id2(θ(x¯,u)),1≤i≤m.
Now, multiplying (74) by λi, (75) by λivi, and adding the resulting inequalities, and then using the superlinearity of ϕ1 and sublinearity of ℱ(x¯,u;·), we obtain
(76)ϕ1(∑i=1m‍λi[fi(x¯)-vigi(x¯)]-∑i=1m‍λi[fi(u)-vigi(u)])≥ℱ(x¯,u;b(x¯,u)∑i=1m‍λi(ξi+viςi))+(∑i=1m‍λiρ1i+∑i=1m‍λiviρ2i)d2(θ(x¯,u)).
Since x¯∈S, β∈ℝ+p, and (66), it is clear that
(77)βjhj(x¯)-βjhj(u)≤0,1≤j≤p.
Using the increasing property of ϕ2j and ϕ2j(0)=0, we have
(78)ϕ2j(βjhj(x¯)-βjhj(u))≤0,1≤j≤p.
Using the (ℱ,b,ϕ2j,ρ3j,θ)-quasiunivexity of βjhj at u, we get from (78)
(79)ℱ(x¯,u;b(x¯,u)βjζj)≤-ρ3jd2(θ(x¯,u)),1≤j≤p.
Because β∈ℝ+p the nonnegativity of b(x¯,u), and ℱ(x¯,u;·) is sublinear, the above inequality yields
(80)ℱ(x¯,u;b(x¯,u)∑j=1p‍βjζj)≤-(∑j=1p‍ρ3j)d2(θ(x¯,u)).
From the nonnegativity of b(x¯,u), sublinearity of ℱ(x¯,u;·), and (73), it follows that
(81)ℱ(x¯,u;b(x¯,u)∑i=1m‍λi(ξi+viςi))0000+ℱ(x¯,u;b(x¯,u)∑j=1p‍βjζj)≥0.
Now adding (76) and (80), and then using (81) and ∑i=1mλiρ1i+∑i=1m‍λiviρ2i+∑j=1p‍ρ3j≥0, we obtain
(82)ϕ1(∑i=1m‍λi[fi(x¯)-vigi(x¯)]-∑i=1m‍λi[fi(u)-vigi(u)])≥0.
But ϕ1(a)≥0⇒a≥0, and so (82) yields
(83)∑i=1m‍λi[fi(x¯)-vigi(x¯)]-∑i=1m‍λi[fi(u)-vigi(u)]≥0.
Equation (65) along with (83) yields
(84)∑i=1m‍λi[fi(x¯)+vi(-gi(x¯))]≥∑i=1m‍λi[fi(u)+vi(-gi(u))]≥0.
Assume that (x¯,v¯,λ¯,β¯) is not an efficient for (D). Then there exists a point (u,v,λ,β)∈ℱD such that
(85)vi≥vi¯=fi(x¯)gi(x¯),1≤i≤m,
with at least one strict inequality.
Thus, for anyλ∈ℝ+m∖{0}, we have
(86)∑i=1m‍λi[fi(x¯)-vigi(x¯)]<0,
which contradicts inequality (84).
It remains to show that(x¯,v¯,λ¯,β¯) is properly efficient. Assume that it is an improperly efficient point for the problem (D) and choose M=(m-1)(λj/λi)(gj(x¯)/gi(x¯))>0 (we may assume that m≥2), there would exist a feasible solution (u,v,λ,β)∈FD and an index i such that vi>vi¯ and
(87)vi-vi¯>M[vj¯-vj]=(m-1)λjλigj(x¯)gi(x¯)[vj¯-vj],
for all j such that vj<vj¯.
These imply that(88)gi(x¯)[vi-fi(x¯)gi(x¯)]>(m-1)λjλigj(x¯)[fj(x¯)gj(x¯)-vj],∀j≠i.
Thus, we have
(89)λim-1[vigi(x¯)-fi(x¯)]>λj[fj(x¯)-vjgj(x¯)],∀j≠i.
Summing over j≠i yield
(90)λi[vigi(x¯)-fi(x¯)]>∑j≠i‍λj[fj(x¯)-vjgj(x¯)],
or
(91)∑i=1m‍λi[fi(x¯)-vigi(x¯)]<0,
which contradicts inequality (84). Thus, the proof of (a) is complete.
If hypothesis (b) holds, from (66), x¯∈S, and β∈ℝ+p, then we have
(92)∑j=1p‍βjhj(x¯)≤0≤∑j=1p‍βjhj(u).
and so using the properties of ϕ2, we obtain
(93)ϕ2(∑j=1p‍βj[hj(x¯)-hj(u)])≤0.
Using the (ℱ,b,ϕ2,ρ3,θ)-quasiunivexity of C at u, then we get from (93)
(94)ℱ(x¯,u;b(x¯,u)∑j=1p‍βjζj)≤-ρ3d2(θ(x¯,u)).
Now combining (76), (81), and (94), and using (iii), we obtain (84). Therefore, along with the same lines of (a), we obtain (x¯,v¯,λ¯,β¯) is a properly efficient solution for (D).
If hypothesis (c) holds, using the(ℱ,b,ϕ,0,θ)-pseudounivexity of A, we get from (73)
(95)ϕ(∑i=1m‍λi[fi(x¯)-vigi(x¯)]+∑j=1p‍βjhj(x¯)000-(∑i=1m‍λi[fi(u)-vigi(u)]+∑j=1p‍βjhj(u)))≥0.
In view of the properties of ϕ, we deduce from this inequality that
(96)∑i=1m‍λi[fi(x¯)-vigi(x¯)]+∑j=1p‍βjhj(x¯)0000-(∑i=1m‍λi[fi(u)-vigi(u)]+∑j=1p‍βjhj(u))≥0.
Equation (92) and (65) along with (96) yields
(97)∑i=1m‍λi[fi(x¯)+vi(-gi(x¯))]≥∑i=1m‍λi[fi(u)+vi(-gi(u))]≥0.
Thus, we obtain (x¯,v¯,λ¯,β¯) is a properly efficient solution for (D).
If hypothesis (d) holds, using the properties ofϕ2 and (92), we obtain
(98)ϕ2(∑j=1p‍βj[hj(x¯)-hj(u)])≤0,
which in view of (ii) implies that
(99)ℱ(x¯,u;b(x¯,u)∑j=1p‍βjζj)≤-ρ2d2(θ(x¯,u)).
Now combining (81), (99), and ρ1+ρ2≥0, we obtain
(100)ℱ(x¯,u;b(x¯,u)∑i=1m‍λi(ξi+viςi))≥-ℱ(x¯,u;b(x¯,u)∑j=1p‍βjζj)≥ρ2d2(θ(x¯,u))≥-ρ1d2(θ(x¯,u)).
Along with the fact that the B is (ℱ,b,ϕ1,ρ1,θ)-pseudounivex at u, we get from (100)
(101)ϕ1(∑i=1m‍λi[fi(x¯)-vigi(x¯)]-∑i=1m‍λi[fi(u)-vigi(u)])≥0.
In view of the properties of ϕ1, we deduce from this inequality that
(102)∑i=1m‍λi[fi(x¯)+vi(-gi(x¯))]≥∑i=1m‍λi[fi(u)+vi(-gi(u))]≥0.
Therefore, (x¯,v¯,λ¯,β¯) is a properly efficient solution for (D). (e), (f), and (g) follows along with the same lines of (d) and (a). Thus, the proof is complete.



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*Source: 284046-2013-11-20.xml*