ON HIGHER-ORDER DUALITY
FOR MULTIOBJECTIVE PROGRAMMING INVOLVING GENERALIZED (F, ρ, γ, b)-CONVEXITY
ANTON B ˘AT ˘ATORESCU, VASILE PREDA and MIRUNA BELDIMAN
The concepts of higher-order (F, ρ, γ, b)-convexity and generalized higher-order (F, ρ, γ, b)-convexity are defined. We consider a higher-order dual model associ- ated to a nondifferentiable multiobjective programming problem involving support functions and a weak duality result is established under appropriate higher-order (F, ρ, γ, b)-convexity conditions.
AMS 2002 Subject Classification: 90C29, 90C30, 90C46.
Key words: higher-order (F, ρ, γ, b)-(pseudo/quasi)-convexity, nondifferentiable multiobjective programming, higher-order duality, duality theorem.
1. INTRODUCTION
For nonlinear programming problems, a number of duals have been sug- gested among which the Wolfe dual [13] is well known. While studying duality under generalized convexity, Mond and Weir [18] proposed a number of differ- ent duals for nonlinear programming problems with nonnegative variables and proved various duality theorems under appropriate pseudo-convexity/quasi- convexity assumptions.
Taking motivation from Bazaraa and Goode [1] and Hanson and Mond [8], Nanda and Das [20] attempted to extend the results of Mond and Weir [18]
to cone domains with appropriate pseudo-invexity and quasi-invexity assump- tions on objective and constraint functions. However, certain shortcomings were pointed out in the work of Nanda and Das [20] and appropriate modifi- cations were suggested for studying duality under pseudo-invexity assumptions in Chandra and Abha [4].
The study of second order duality is significant due to the computa- tional advantage over first order duality as it provides tighter bounds for the value of the objective function when approximations are used [10, 12, 16, 24].
Mangasarian [12] considered a nonlinear programming problem and discussed second order duality under inclusion condition. Mond [16] was the first who present second order convexity. He also gave in [16] simpler conditions than
MATH. REPORTS9(59),2 (2007), 161–174
Mangasarian using a generalized form of convexity which was later called sec- ond order convexity by Mahajan [11] and bonvexity by Bector and Chandra [3].
Further, Jeyakumar [10] and Yang [24] discussed also second order Mangasar- ian type dual formulation under ρ-convexity and generalized representation conditions respectively. In [28] Zhang and Mond established some duality the- orems for second-order duality in nonlinear programming under generalized second-order B-invexity, defined in their paper. In [2, 18] it was shown that second order duality can be useful from computational point of view, since one may obtain better lower bounds for the primal problem than otherwise. The case of some optimization problems that involve n-set functions was studied by Preda [22].
Recently, Yang et al. [27] proposed four second-order dual models for nonlinear programming problems and established some duality results under generalized second-orderF-convexity assumptions. In [15] Mishra and Rueda generalized Zhang’s Mangasarian type and Mond-Weir type higher-order dual- ity [28] to higher-order type I functions. Yang et al. [26] extended this results to a class of nondifferentiable multiobjective programming problems. They also presented an unified higher-order dual model for nondifferentiable multi- objective programs, where every component of the objective function contains a support function of a compact convex set.
In Section 2 we define the higher-order (F, ρ, γ, b)-convexity and gener- alized higher-order (F, ρ, γ, b)-convexity. In Section 3 we consider a class of nondifferentiable programming problems and for the dual model defined by Yang et al. [26], we prove a weak duality result. Finally, we consider a special case, which emphasizes the generalizations introduced in this paper.
2. DEFINITIONS AND SOME PRELIMINARY RESULTS
We denote by Rn the n-dimensional Euclidean space, and by Rn+ the nonnegative orthant ofRn.Further,R∗+={x∈R|x >0}.
For any vectors x∈Rn, y ∈Rn we define: xy=ni=1xiyi.
Let C ⊂ Rn be a compact convex set. The support function of C is defined by
s(x |C) = max
xy |y∈C
and being convex and everywhere finite, it has a subdifferential [23], that is, there exists z∈Rn such that
s(y |C)≥s(x|C) +z(y−x) for all y∈C.
The subdifferential ofs(x|C) is given by
∂s(x |C) =
z∈C |zx=s(x |C) .
For any setD⊂Rn,the normal cone toDat a pointx∈Dis defined by ND(x) =
y∈Rn |y(z−x)≤0, for all z∈D .
For a compact convex set C we obviously have: y ∈ NC(x) if and only if s(y |C) =xy, or equivalently, ifx∈∂s(y |C).
Let us considerH :Rn →Rp, G :Rn → Rq, X ⊂ Rn and the multiob- jective programming problem
(P)
minimize H(x)
subject to: G(x)≥0, x∈X
Let us denote the set of feasible solutions of (P) byP,that is, P ={x∈X |G(x)≥0}.
Definition2.1. A vector ¯x∈ Pis an efficient solution of (P) if there exists no otherx∈ P such thatH(¯x)−H(x)∈Rp+\ {0},that is, Hi(x)≤Hi(¯x) for alli∈ {1, . . . , p},and for at least onej∈ {1, . . . , p} we haveHj(x)< Hj(¯x).
¯
x ∈ P is said to be a weak efficient solution of (P) if there exists no x ∈ P such that for alli∈ {1, . . . , p}, Hi(x)< Hi(¯x).
Definition 2.2. An efficient solution ¯x∈ P of (P) is properly efficient, if there exists a positive numberM with the property that, whenever Hi(x) <
Hi(¯x) forx∈ P and i∈ {1, . . . , p},there exists somej∈ {1, . . . , p}such that Hj(x)> Hj(¯x) and HHi(¯x)−Hi(x)
j(x)−Hj(¯x) ≤M.
We denote by∇f(¯x) the gradient vector at ¯xof a differentiable function f :Rp → R,and by ∇2f(¯x) the Hessian matrix of f at ¯x. For a real-valued twice differentiable function ψ(x, y) defined on an open set in Rp ×Rq, we denote by∇xψ(¯x,y) the gradient vector of¯ ψ with respect tox at (¯x,y)¯ ,and by ∇xxψ(¯x,y) the Hessian matrix with respect to¯ x at (¯x,y)¯ . Similarly we may define∇yψ(¯x,y)¯ ,∇xyψ(¯x,y) and¯ ∇yyψ(¯x,y)¯ .
The following result will be used in the next section.
Lemma 2.1 ([6]). If x¯ ∈ P is a properly efficient solution of (P), there exist α∈Rp+\ {0} and β ∈Rq+\ {0} such that
p i=1
αi∇Hi(x)− q j=1
βj∇Gj(x) = 0.
Let us consider a function F :X×X×Rn→ R (where X ⊆Rn) with the property that for all (x, y)∈X×X, we have
F(x, y;·) is a convex function, (i)
F(x, y; 0)≥0.
(ii)
IfF satisfies (i) and (ii), we obviously have
F(x, y;−a)≥ −F(x, y;a), for any a∈Rn.
Example 2.1. Let us considerF(x, y;a) = 2a+a2,whereadepends onx andy. This function satisfies (i) and (ii), but is neither sub-additive nor positive homogeneous, that is, the relations
(i) F(x, y;a+b)≤F(x, y;a) +F(x, y;b) (ii) F(x, y;λa) =λF(x, y;a) are not fulfilled for any a, b∈Rn and λ∈R, λ≥0.
We may conclude that the class of functions that verify (i) and (ii) is more general than the class of sub-linear functions with respect the third argument, i.e. those which satisfy (i) and (ii). We notice that till now, most results in optimization theory were stated under generalized convexity assumptions involving functions F which are sub-linear. The results of this paper are obtained by using weaker assumptions with respect to the functionF.
Let us consider functionsb:X×X →R+, d:X×X→R+, γ:X×X → R∗+,and a numberρ∈R.Further, suppose that F :X×X×Rn→Rsatisfies (i) and (ii) and that ϕ : X → R and h : X ×Rn → R are differentiable functions. We introduce in the subsequent definition the class of higher-order (F, ρ, γ, b)-convexity.
Definition 2.3. • We say that ϕ is higher-order (F, ρ, γ, b)-convex at u∈X with respect toh, if for all (x, y)∈X×Rn we have
(2.1) b(x, u) (ϕ(x)−ϕ(u))≥F(x, u;γ(x, u) [∇ϕ(u) +∇yh(u, y)]) + +b(x, u)
h(u, y)−y[∇yh(u, y)]
+ρd(x, u).
•We say thatϕis higher-order (F, ρ, γ, b)-pseudo-convex atu∈X with respect toh, if for all (x, y)∈X×Rnwe have
F(x, u;γ(x, u) [∇ϕ(u) +∇yh(u, y)])≥ −ρd(x, u)
=⇒b(x, u) (ϕ(x)−ϕ(u))≥b(x, u)
h(u, y)−y[∇yh(u, y)]
.
• We say that ϕ is higher-order (F, ρ, γ, b)-quasi-convex at u ∈X with respect toh, if for all (x, y)∈X×Rnwe have
b(x, u) (ϕ(x)−ϕ(u))≤b(x, u)
h(u, y)−y[∇yh(u, y)]
=⇒F(x, u;γ(x, u) [∇ϕ(u) +∇yh(u, y)])≤ −ρd(x, u).
• If ϕ is higher-order (F, ρ, γ, b)-convex (pseudo/quasi-convex) at each u∈X with respect to the same function h, thenϕ is said to be higher-order (F, ρ, γ, b)-convex (pseudo/quasi-convex) on X with respect to h.
• If −ϕ is higher-order (F, ρ, γ, b)-convex (pseudo/quasi-convex) atu ∈ X with respect to h, then ϕ is said to be higher-order (F, ρ, γ, b)-concave (pseudo/quasi-concave) atu∈X with respect toh.
Remark 2.1. The elementsρ, γandbgive more felexibility for the defined properties to hold for all (x, y)∈X×Rn.In particular, if we take ρ = 0 and b≡1, γ ≡1,then
(1) The above definition reduces to Definition 4 of Chen [5].
(2) When h(u, y) = y∇uuϕ(u)y/2 and F(x, u;a) = η(x, u)a, where η:X×X→Rn,the higher-order (F, ρ, γ, b)-(pseudo/quasi)-convexity reduces to η-(pseudo/quasi)-bonvexity in [7, 21].
(3) When h(u, y) = y∇uuϕ(u)y/2 and F(x, u;a) = η(x, u)a, where η:X×X→Rn,the higher-order (F, ρ, γ, b)-(pseudo/quasi)-convexity reduces to the second-orderF (pseudo/quasi) invexity in [9].
(4) When h(u, y) = −y∇uϕ(u) + ψ(u, y) and F(x, u;a) = α(x, u)aη(x, u), where α : X ×X → R+\ {0}, η : X × X → Rn are positive functions, and ψ : X ×Rn → R is a differentiable function, the higher-order (F, ρ, γ, b)-(pseudo/quasi)-convex function becomes the higher- order (pseudo/quasi) type I function in [14, 19].
Example2.2. We consider here a function which is higher-order (F, ρ, γ, b)- convex. Similarly, we can proceed for the other classes of functions introduced in Definition 2.3. Let us consider X=
0,π2 and ϕ:X→R, ϕ(x) =x+ sinx, h:X×R→R, h(u, y) =−ycosu.
Obviously,ϕis not convex. We have
∇xϕ(x) = 1 + cosx, ∇xxϕ(x) =−sinx,
∇yh(u, y) =−cosu, ∇uh(u, y) =ysinu.
Let us consider F :X×X×R→R defined byF(x, y;a) = 2|a|+|a|2, which satisfies (i) and (ii) and is not sublinear.
Before proceeding, it is worth to notice that in this example we have h(u, y) =−ycosu= 1
2y∇uuϕ(u)y=−y2 2 sinu,
i.e., h(u, y) has not the form used in [7, 21], as we mentioned in Remark 2.1 (item 2). Further, if we consider the case presented in Remark 2.1 (item 4), we see that
−y∇uϕ(u) +ψ(u, y) =−y(1 + cosu) +ψ(u, y),
and if we take ψ(u, y) =y, then h(u, y) = −ycosu has the same form as in [14, 19], but now our mappingF is not sublinear as required there.
Let us define the functions
b(x, u) =
2xsinu+x2sin2u
x−u+ sinx−sinu if (x−u+ sinx−sinu)>0, 0 if (x−u+ sinx−sinu)≤0, γ(x, u) = xsinu,
d(x, u) = 2xsinu+x2sin2u.
Since
∇ϕ(u) +∇yh(u, y) = 1 + cosu−cosu= 1, h(u, y)−y∇yh(u, y) =−ycosu−y(−cosu) = 0, we have
F(x, u;γ(x, u) [∇ϕ(u) +∇yh(u, y)]) = 2xsinu+x2sin2u, and relation (2.1) becomes
1≥1 +ρ, if (x−u+ sinx−sinu)>0, 0≥1 +ρ, if (x−u+ sinx−sinu)≤0.
It now follows that forρ ≤ −1 the function ϕ(x) =x+ sinx is higher-order (F, ρ, γ, b)-convex at u∈X with respect toh(u, y) =−ycosu.
3. HIGHER-ORDER DUALITY INVOLVING NONDIFFERENTIABLE FUNCTIONS
We consider in this section differentiable functions
f = (f1, . . . , fp) :Rn→Rp, h= (h1, . . . , hp) :Rn×Rn→Rp, g= (g1, . . . , gm) :Rn→Rm, k= (k1, . . . , km) :Rn×Rn→Rm, compact convex sets Ci ⊂ Rn, i ∈ P = {1,2, . . . , p}, and an open subset D⊂Rn. Let M ={1,2, . . . , m} and consider a partition of the index set M defined byIα ⊂M, α = 0,1, . . . , µ,with
µ
α=0Iα =M and Iα∩Iβ =∅ for any α=β.
Consider the nondifferentiable multiobjective programming problem:
(NMP) minimize: {f1(x) +s(x|C1), . . . , fp(x) +s(x|Cp)}
subject to: g(x)≥0, x∈D.
With this problem, we associate the dual problem
(NMD)
maximize: {f1(u) +h1(u, π)−π∇πh1(u, π) +uw1−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) ,
· · ·
fp(u) +hp(u, π)−π∇πhp(u, π) +uwp−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) },
subject to (3.1)
p i=1
λi(∇πhi(u, π) +wi) = m τ=1
yτ[∇gτ(u) +∇πkτ(u, π)],
(3.2)
τ∈Iα
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π)
≤0, α= 1, . . . , µ,
(3.3) y ≥0,
(3.4) wi ∈Ci, i∈P, λ∈Λ+, where Λ+=
λ∈Rp |λ >0, p
i=1λi= 1
.We putw= (w1, . . . , wp).
Remark 3.1. The higher-order convexity assumptions, which will be con- sidered in the sequel, connect the functions h and k with the objective and the constraint functions of the problem. In this way, the objective function is indirectly involved in the constraints of the dual problem.
3.1. WEAK DUALITY
In this subsection we prove a weak duality result between the problems (NMP) and (NMD) under higher-order (F, ρ, γ, b)-quasi/-pseudo-convexity as- sumptions. Ifx is a feasible solution of (NMP) and (u, λ, w, y, π) is a feasible solution of (NMD), then we suppose that the following properties are satisfied.
(i1) For all α = 1, . . . , µ, the function
−
τ∈Iα
yτgτ( ·)
is higher- order (F, βα, γ, bα)-quasi-convex with respect toδα(u, π) =−
τ∈Iαyτkτ(u, π),
that is, (3.5)
bα(x, u)
τ∈Iα
(−yτ)
gτ(x)−gτ(u)−kτ(u, π) +π∇πkτ(u, π) ≤0
=⇒F
x, u,−γ(x, u)
τ∈Iα
yτ[∇gτ(u) +∇πkτ(u, π)]
≤ −βαd(x, u).
(i2) Fori∈P,the functionfi(·) + ( ·)wi is higher-order (F, ρi, γ, bi)- quasi-convex with respect to
χi(u, π) =hi(u, π)−
τ∈I0
yτ[gτ(u) +kτ(u, π)],
that is, bi(x, u)
fi(x) +xwi−(fi(u) +uwi) ≤
hi(u, π)−π∇πhi(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) bi(x, u)
⇒F
x, u, γ(x, u)[∇fi(x) +wi+∇πhi(u, π)−
τ∈I0
yτ∇πkτ(u, π)]
≤
≤ −ρid(x, u)
and there existsj∈P such that the functionfj( ·) + ( ·)wj is higher-order (F, ρj, γ, bj)-pseudo-convex with respect toχj(u, π).
Theorem3.1. Let xbe a feasible solution of (NMP)and let (u, λ, w, y, π) be a feasible solution of (NMD).Suppose that properties (i1), (i2)and the con- ditions
(i3) F
x, u,−γ(x, u)[λ∇f(u) +
τ∈I0
yτ∇gτ(u)]
≤0;
(i4) µ α=1βα+
p
i=1λiρi ≥0
do hold. Then, the relations below cannot hold simultaneously:
(3.6)
for alli∈P, bi(x, u) [fi(x) +s(x |Ci)]≤
≤bi(x, u){fi(u) +uwi+hi(u, π)−π∇πhi(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) },
and
(3.7)
for some j∈P, bj(x, u) [fj(x) +s(x|Cj)]<
< bj(x, u)
fj(u) +uwj+hj(u, π)−π∇πhj(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) .
Proof. Since x is feasible for (NMP) and (u, λ, w, y, π) is feasible for (NMD), it follows from (3.5) and the fact thatF satisfies (i) and (ii) that (3.8) F
x, u,−γ(x, u)
τ∈M\I0
yτ[∇gτ(u) +∇πkτ(u, π)]
≤ − µ α=1
βαd(x, u).
From (3.1), (3.8) and the the fact thatF satisfies (i) and (ii), we obtain
(3.9)
F
x, u, γ(x, u) p
i=1
λi(∇πhi(u, π) +wi)−
−
τ∈I0
yτ[∇gτ(u) +∇πkτ(u, π)]
≥ µ α=1
βαd(x, u).
Now, suppose on the contrary that (3.6) and (3.7) hold. Since bi(x, u) ≥ 0, xwi ≤s(x |Ci), i∈P,we have
(3.10)
bi(x, u)
fi(x) +xwi
≤bi(x, u) [fi(x) +s(x |Ci)]≤
≤bi(x, u)
fi(u) +uwi+hi(u, π)−π∇πhi(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) , ∀i∈P, and
(3.11)
bj(x, u)
fj(x) +xwj
≤bi(x, u) [fj(x) +s(x|Cj)]<
< bj(x, u)
fj(u) +uwj+hj(u, π)−π∇πhj(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) , for somej∈P.
By (i2), we obtain (3.12)
F
x, u, γ(x, u)
∇fi(x) +wi+∇πhi(u, π)−
τ∈I0
yτ∇πkτ(u, π) ≤
≤ −ρid(x, u), for anyi∈P,
and (3.13) F
x, u, γ(x, u)
∇fj(x) +wj+∇πhj(u, π)−
τ∈I0
yτ∇πkτ(u, π)
<
<−ρjd(x, u), for somej∈P.
Since λ >0 and p
i=1λi = 1, it follows from (3.12), (3.13) and the fact that F satisfies (i) and (ii), that
(3.14)
F
x, u, γ(x, u) p
i=1
λi(∇fi(x) +wi+∇πhi(u, π))−
−
τ∈I0
yτ∇πkτ(u, π)
<−p
i=1λiρid(x, u). Using now (i4), i.e. −p
i=1λiρi ≤ µ
α=1βα,from (3.14) we get
(3.15)
F
x, u, γ(x, u) p
i=1
λi(∇fi(x) +wi+∇πhi(u, π))−
−
τ∈I0
yτ∇πkτ(u, π)
<
µ α=1
βαd(x, u).
Combining (3.9) and (3.15) we have F
x, u, γ(x, u) p
i=1
λi[∇fi(x) +wi+∇πhi(u, π)]−
τ∈I0
yτ∇πkτ(u, π)
<
< F
x, u, γ(x, u) p
i=1
λi(∇πhi(u, π)+wi)−
τ∈I0
yτ[∇gτ(u)+∇πkτ(u, π)]
, and using again the fact thatF satisfies (i) and (ii) we obtain
F
x, u,−γ(x, u)[λ∇f(u) +
τ∈I0
yτ∇gτ(u)]
>0,
which contradicts assumption (i3). Hence (3.6) and (3.7) cannot hold, and the theorem is proved.
We remark that in Theorem 3.1 not all terms that appear in connection with the higher-order (F, ρ, γ, b)-quasi/-pseudo-convexity are necessary to es- tablish the weak duality property. If we replace restriction (3.1) in (NMD) by (3.16)
p i=1
λi(∇πhi(u, π) +wi) = m τ=1
yτ∇πkτ(u, π)
then we obtain a new dual problem (NMD1). Now, for (NMP) and (NMD1) we can state the weak duality theorem, below,which can be proved similarly to the previous one.
Theorem 3.2. Let x be feasible for (NMP) and let (u, λ, w, y, π) be feasible for (NMD1). Suppose that for all feasible (x, u, y, w, π) there exists a function F :Rn×Rn×Rn→Rthat satisfies (i)and (ii), and the implication
(3.17)
bα(x, u)
τ∈Iα
(−yτ)
gτ(u)−kτ(u, π) +π∇πkτ(u, π) ≤0
⇒ F
x, u,−γ(x, u)
τ∈Iα
yτ∇πkτ(u, π)
≤−βαd(x, u), α= 1, . . . , µ,
holds. Furthermore, assume that (a)for i∈P,
bi(x, u)
fi(x) +xwi−
fi(u) +uwi+hi(u, π)−π∇πhi(u, π)
+
+
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) ≤0
⇒ F
x, u, γ(x, u)
∇πhi(u, π) +wi−
τ∈I0
yτ∇πkτ(u, π)
≤ −ρid(x, u);
(b)for some j∈P, F
x, u, γ(x, u)
∇πhj(u, π) +wj−
τ∈I0
yτ∇πkτ(u, π)
≥ −ρjd(x, u)
⇒bj(x, u)
fj(x) +xwj−
fj(u) +uwj +hj(u, π)−π∇πhj(u, π)
+
+
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) ≥0;
(c) µ
α=1βα+p
i=1λiρi ≥0.
Then, the relations below cannot hold simultaneously
(3.18)
for all i∈P, bi(x, u) [fi(x) +s(x |Ci)]≤
≤bi(x, u)
fi(u) +uwi+hi(u, π)−π∇πhi(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) ,
and
(3.19)
for some j∈P, bj(x, u) [fj(x) +s(x |Cj)]<
< bj(x, u)
fj(u) +uwj+hj(u, π)−π∇πhj(u, π)−
−
τ∈I0
yτ
gτ(u) +kτ(u, π)−π∇πkτ(u, π) .
3.2. A SPECIAL CASE
Let us consider the compact convex setsCi defined as
(3.20) Ci =
Biv |vBiv≤1 . It is easily shown that
xBix =s(x |Ci). In this case, problems (NMP) and (NMD) can be restated as follows:
(NMP*) minimize
f1(x) +
xB1x, . . . , fp(x) + xBpx
subject to g(x)≥0, x∈D.
and
(NMD*)
maximize { f1(u) +h1(u, π)−π∇πh1(u, π) +uB1v−
−
τ∈I0
yτgτ(u) +yτkτ(u, π)−π∇πyτkτ(u, π) ,
· · ·
fp(u) +hp(u, π)−π∇πhp(u, π) +uBpv−
−
τ∈I0
yτgτ(u) +yτkτ(u, π)−π∇πyτkτ(u, π) },
subject to
p i=1
λi(∇πhi(u, π) +Biv) = m τ=1
yτ∇πkτ(u, π),
τ∈Iα
yτgτ(u) +yτkτ(u, π)−π∇π(yτkτ(u, π)) ≤0,
α= 1, . . . , µ, y≥0, vBiv≤1, i∈P, λ∈Λ+.
Remark 3.2. (1) If p = 1 the problems (NMP*) and (NMD*) become (NDP) and (NDHGD) considered by Mishra et al. [15]. If, in addition,π= 0, we obtain the Mond-Weir dual associated to the optimization problem that involve a square root term, which was studied for the first time by Mond [17].
(2) If p = 1, I0 = M and Iα = ∅ for α = 1, . . . , µ, then (NMP*) and (NMD*) become respectively the problems (NDP) and (NDHMD) considered in [15].
(3) If p= 1, I0 =∅, I1 =M,and Iα=∅ forα= 2, . . . , µ,then (NMP*) and (NMD*) become respectively (NDP) and (NDHD) considered in [15].
(4) Ifh(u, π) =π∇f(u) andki(u, π) =π∇gi(u), i∈M,then (NMD*) reduces to (VD) considered by Yang, Teo and Yang [25].
(5) WhenF(x, u,∇φ(u)) =∇φ(u)η(x, u),whereη :Rn×Rn→Rnis a vector valued function, then conditions (a), (b) and (c) of Theorem 3.1 reduce to higher-order generalized invexity considered in [15].
Acknowledgements. This research was partially supported by Grant CNCSIS no.
27694/2005.
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Received 5 October 2006 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania
