Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

(1)

www.elsevier.com/locate/jmaa

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized

type I functions

Mohamed Hachimi

^a

, Brahim Aghezzaf

^b,^∗

aFaculté des Sciences Juriduques Economiques et Sociales, Université Ibn Zohr, B.P. 32/S, Agadir, Morocco bDépartement de Mathématiques et d’Informatique Faculté des Sciences Aïn chock, Université Hassan II,

B.P. 5366, Maârif Casablanca, Morocco Received 27 August 2004 Available online 24 February 2006

Submitted by J.P. Dauer

Abstract

Recently, Hachimi and Aghezzaf defined generalized(F, α, ρ, d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F, α, ρ, d)-type I functions are extended to nondifferentiable functions. By utilizing the new con- cepts, we obtain several sufficient optimality conditions and prove mixed type and Mond–Weir type duality results for the nondifferentiable multiobjective programming problem.

2005 Published by Elsevier Inc.

Keywords:Nondifferentiable multiobjective programming problem; Efficient solution; Sufficiency; Type I functions; Generalized convexity; Duality

1. Introduction

Convexity plays an important role in deriving sufficient conditions and duality for multiobjective programming problem. Various generalizations of convexity have been made

* Corresponding author.

E-mail address:[email protected] (B. Aghezzaf).

0022-247X/$ – see front matter2005 Published by Elsevier Inc.

doi:10.1016/j.jmaa.2005.02.064

(2)

in the literature [1–3,5,6,8–17,19,20]. Recently, Hachimi and Aghezzaf [7] considered a class of functions called generalized(F, α, ρ, d)-type I functions for a multiobjective programming problem with inequality constraints in the differentiable case, and they have derived some sufficient conditions and duality results. However, the corresponding conclu- sions cannot be obtained for nondifferentiable programming with the help of generalized (F, α, ρ, d)-type I functions because the derivative is required in the definitions of such functions.

In this paper, we are concerned with the following nondifferentiable multiobjective programming problem with inequality constraints:

(MOP) minimizef (x)=

f₁(x), . . . , f_p(x) , subject to x∈A=

x∈X|g(x)0 ,

whereXis an open subset of a Banach spaceEandf:X→R^p,g:X→R^q are locally Lipschitz functions onX.

We extend the generalized(F, α, ρ, d)-type I functions to nondifferentiable functions.

The sufficient optimality conditions are obtained for problem (MOP) involving generalized (F, α, ρ, d)-type I. Furthermore, duality results are also obtained by associating a mixed type and Mond–Weir type duals problems with the problem (MOP).

Notations.Throughout this paper we use the following convention. For locally Lipschitz functionϕ:X→R, the Clarke directional derivative, denotedϕ^◦(x;d), is given by [4]

ϕ^◦(x¯;d)=lim sup

x→ ¯x t→0⁺

1 2

ϕ(x+t d)−ϕ(x)

. (1a)

The Clarke subdifferential ofϕatx¯is given by

∂ϕ(x)¯ =

x^∗∈E^∗: ϕ^◦(x¯;d)x^∗, d

, (1b)

whereE^∗ denotes the topological dual of E and·,·is the duality pairing. The index setP = {1,2, . . . , p}andQ= {1,2, . . . , q}. Forx¯∈A, the index setI= {j |gj(x)¯ =0} andg_I denotes the vector for active constraints. Ifx andy∈Rⁿ, thenxy⇔x_iy_i, i=1, . . . , n;xy⇔xy andx=y;x < y⇔x_i< y_i,i=1, . . . , n;xy orx^tydenote the inner product. The symboledenotes the vector whose components are all ones, it’s dimension depends on the context. For vector function f =(f₁, . . . , f_p):X→R^p and u∈R^p, we denote

∂f (x)¯ = µ^∗=

µ^∗₁, . . . , µ^∗_pt

:µ^∗_i ∈∂f_i(x), i¯ =1, . . . , p

, (2a)

u∂f (x)¯ = p i=1

u_i∂f_i(x).¯ (2b)

For the multiobjective programming problem (MOP), the solution is defined in terms of a (weak) efficient solution in the following sense:

Definition 1.We say thatx¯∈Ais an efficient solution for problem (MOP) if and only if there exists nox∈Asuch thatf (x)f (x).¯

(3)

Definition 2.We say thatx¯∈Ais a weak efficient solution for problem (MOP) if and only if there exists nox∈Asuch thatf (x) < f (x).¯

Weak efficient solutions are often useful, since they are completely characterized by scalarization [18].

The following results will be needed:

Theorem 3.Letf:E→R^pbe locally Lipschitz at a pointy. Then, foru∈R^pwe have

∂(uf )(y)⊂u∂f (y). (3a)

Proof. See, for example, Miettinen [14] or Clarke [4]. 2

Theorem 4.Let f:E→R^p be locally Lipschitz at a pointy andu∈R^p. Ifu0, we have

∂(Uf )(y)=U ∂f (y), (3b)

whereU is the diagonal matrix such thatU_ii=u_i,i=1, . . . , p.

Proof. From the definition of Clarke subdifferential, it is easily checked that for eachiwe have

∀u_i0, u_i∂f_i(y)=∂(u_if_i)(y).

Hence, foru=(u₁, . . . , u_p)^t0, we obtain

∂(Uf )(y)=∂







u₁f₁(y) u2f2(y)

... upfp(y)





=







∂(u₁f₁)(y)

∂(u2f2)(y) ...

∂(upfp)(y)





=







u₁∂f₁(y) u2∂f2(y)

... up∂fp(y)





=U ∂f (y). 2

2. Generalized(F, α, ρ, d)-type I functions

In the following, we give some definitions which generalize the concepts of(F, α, ρ, d)- type I functions given by Hachimi and Aghezzaf [7] to nondifferentiable cases.

Definition 5.A functionalF:X×X×E^∗→Ris sublinear if for anyx,x¯∈X, F

x,x¯;a^∗₁+a₂^∗

F

x,x¯;a₁^∗ +F

x,x¯;a₂^∗

, ∀a₁^∗, a₂^∗∈E^∗, (4a) F (x,x¯;αa^∗)=αF (x,x¯;a^∗), ∀α∈R, α0, ∀a^∗∈E^∗. (4b) LetF be a sublinear functional and the functionsf =(f₁, . . . , f_p):X→R^pandh= (h1, . . . , hr):X→R^r are locally Lipschitz functions atx¯∈X. Letρ=(ρ¹, ρ²), where ρ¹=(ρ1, . . . , ρp)∈R^p,ρ²=(ρ1+p, . . . , ρr+p)∈R^r. Letα=(α¹, α²), whereα¹:X× X→R+\ {0},α²:X×X→R+\ {0}, and letd(.,.):X×X→R.

For vector-valued function f:X→R^p andµ^∗=(µ^∗₁, . . . , µ^∗_p)∈∂f (x), the symbol¯ F (x,x¯;µ^∗)denotes the vector of componentsF (x,x¯;µ^∗₁), . . . , F (x,x¯;µ^∗_p).

(4)

Definition 6.(f, h)is said(F, α, ρ, d)-type I atx, if for all¯ x∈Awe have f (x)−f (x)¯ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x),¯ ∀µ^∗∈∂f (x),¯ (5a)

−h(x)¯ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x),¯ ∀ν^∗∈∂h(x).¯ (5b) Definition 7.(f, h)is said pseudoquasi(F, α, ρ, d)-type I atx¯, if for allx∈Awe have

f (x) < f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0, ∀µ^∗∈∂f (x),¯ (6a)

−h(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂h(x).¯ (6b) If in the above definition, inequality (6a) is satisfied as

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0, ∀µ^∗∈∂f (x),¯ (6c) then we say that(f, h)is strictly-pseudoquasi(F, α, ρ, d)-type I atx.¯

Definition 8. (f, h)is said weak strictly-pseudoquasi (F, α, ρ, d)-type I at x, if for all¯ x∈Awe have

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0, ∀µ^∗∈∂f (x),¯ (7a)

−h(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂h(x).¯ (7b) Definition 9.(f, h)is said strong pseudoquasi(F, α, ρ, d)-type I atx¯, if for allx∈Awe have

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ (8a)

−h(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂h(x).¯ (8b) If in the above definition, inequality (8a) is satisfied as

f (x) < f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ (8c) then we say that(f, h)is weak pseudoquasi(F, α, ρ, d)-type I atx.¯

Remark 10. Note that for the scalar objective functions the class of pseudoquasi (F, α, ρ, d)-type I, the class of weak strictly-pseudoquasi(F, α, ρ, d)-type I, and the class of strong pseudoquasi(F, α, ρ, d)-type I functions coincide.

Definition 11.(f, h)is said weak quasistrictly-pseudo(F, α, ρ, d)-type I atx¯, if for all x∈Awe have

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ (9a)

−h(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂h(x).¯ (9b) Remark 12.Note that when functionsf andhare differentiable, the concepts of the above generalized(F, α, ρ, d)-type I functions are reduced to those of generalized(F, α, ρ, d)- type I functions defined in [7].

(5)

Definition 13.(f, h)is said weak strictly-pseudosubquasi(F, α, ρ, d)-type I atx, if for all¯ x∈Awe have

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0,

∀µ^∗∈∂f (x),¯ (10a)

−e^th(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0,

∀ν^∗∈∂h(x).¯ (10b)

Definition 14.(f, h)is said strong pseudosubquasi(F, α, ρ, d)-type I atx¯, if for allx∈A we have

f (x)f (x)¯ ⇒ F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0,

∀µ^∗∈∂f (x),¯ (11a)

−e^th(x)¯ 0 ⇒ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0,

∀ν^∗∈∂h(x).¯ (11b)

3. Sufficient optimality conditions

In recent paper, Hachimi and Aghezzaf [7] considered a number of sufficient optimality conditions which depend on generalized(F, α, ρ, d)-type I functions. In this section, we adapt their results to the classes of nondifferentiable functions.

Theorem 15.Suppose that there exists a feasible solutionx¯for(MOP)and vectorsu¯∈R^p andv¯∈R^qsuch that

0∈ ¯u∂f (x)¯ + ¯v∂g(x),¯ (12a)

¯

vg(x)¯ =0, (12b)

¯

u >0, v¯0. (12c)

If(f, g_I)is strong pseudoquasi(F, α, ρ, d)-type I atx¯withuρ¯ ¹α¹(·,x)¯ ⁻¹+ ¯v_Iρ²α²(·,x)¯ ⁻¹ 0, thenx¯is an efficient solution for(MOP).

Proof. Suppose thatx¯ is not an efficient solution for (MOP). Then, there exist anx∈A such thatf (x)f (x). Since¯ gI(x)¯ =0 and from the hypotheses on(f, gI), we have

F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ (13a) F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂g_I(x).¯ (13b) and by the sublinearity ofF we obtain

α¹(x,x)F (x,¯ x¯;µ^∗)−ρ¹d²(x,x),¯ ∀µ^∗∈∂f (x),¯ (14a) α²(x,x)F (x,¯ x¯;ν^∗)−ρ²d²(x,x),¯ ∀ν^∗∈∂g_I(x).¯ (14b)

(6)

Multiplying (14a) and (14b) withuα¯ ¹(x,x)¯ ⁻¹andv¯_Iα²(x,x)¯ ⁻¹, respectively, we get

¯

uF (x,x¯;µ^∗) <− ¯uρ¹α¹(x,x)¯ ⁻¹d²(x,x),¯ ∀µ^∗∈∂f (x),¯ (15)

¯

v_IF (x,x¯;ν^∗)−¯v_Iρ²α²(x,x)¯ ⁻¹d²(x,x),¯ ∀ν^∗∈∂g_I(x).¯ (16) By the sublinearity ofF, we summarize to get

F (x,x¯; ¯uµ^∗+ ¯vν^∗)uF (x,¯ x¯;µ^∗)+ ¯v_IF (x,x¯;ν^∗)

<−

¯

uρ¹α¹(x,x)¯ ⁻¹+ ¯v_Iρ²α²(x,x)¯ ⁻¹

d²(x,x).¯ Sinceuρ¯ ¹α¹(x,x)¯ ⁻¹+ ¯v_Iρ²α²(x,x)¯ ⁻¹0, the above inequalities give

F (x,x¯; ¯uµ^∗+ ¯vν^∗) <0, ∀µ^∗∈∂f (x),¯ ∀ν^∗∈∂g_I(x),¯

we obtain a contradiction to (12a) becauseF (x,x¯;0)=0. Hence,x¯is an efficient solution for (MOP). 2

An interesting case not covered by Theorem 15 above is the case where(x,¯ u,¯ v)¯ is a solution of (12) but the requirement thatu >¯ 0 is not made. This is given by the following two theorems, where instead of requiring thatu >¯ 0, we enforced other convexity conditions on(f, g_I).

0∈ ¯u∂f (x)¯ + ¯v∂g(x),¯ (17a)

¯

vg(x)¯ =0, (17b)

¯

u0, v¯0. (17c)

If (f, g_I) is weak strictly-pseudoquasi (F, α, ρ, d)-type I at x¯ with uρ¯ ¹α¹(·,x)¯ ⁻¹+

¯

v_Iρ²α²(·,x)¯ ⁻¹0, thenx¯is an efficient solution for(MOP).

Proof. Assume that x¯ is not an efficient solution for (MOP). Then, there exists an x∈Asuch thatf (x)f (x). Since¯ g_I(x)¯ =0 and(f, g_I)is weak strictly-pseudoquasi (F, α, ρ, d)-type I atx, we have¯

F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0, ∀µ^∗∈∂f (x),¯ F

x,x¯;α²(x,x)ν¯ ^∗

+ρ²d²(x,x)¯ 0, ∀ν^∗∈∂g_I(x),¯ and now the proof is similar to that of Theorem 15. 2

0∈ ¯u∂f (x)¯ + ¯v∂g(x),¯ (18a)

¯

vg(x)¯ =0, (18b)

(u,¯ v)¯ 0, v¯_I>0. (18c)

(7)

If (f, g_I) is weak quasistrictly-pseudo (F, α, ρ, d)-type I at x¯ with uρ¯ ¹α¹(·,x)¯ ⁻¹+

¯

v_Iρ²α²(·,x)¯ ⁻¹0, thenx¯is an efficient solution for(MOP).

Proof. Assume that x¯ is not an efficient solution for (MOP). Then, there exists an x∈Asuch thatf (x)f (x). Since¯ g_I(x)¯ =0 and (f, g_I)is weak quasistrictly-pseudo (F, α, ρ, d)-type I atx¯, we have

F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ F

x,x¯;α²(x,x)ν¯ ^∗

Remark 18.Similarly, we can prove more results like Theorems 15–17 by varying the convexity condition on(f, g_I)and by changing the sign ofu¯andv.¯

It is obvious that Theorems 15 and 16 hold for weak efficient solutions too. However, it is important to know that the convexity assumptions of Theorems 15 and 16 can be weakened for weak efficient solutions.

Theorem 19.Suppose that there exists a feasible solutionx¯for(MOP)and vectorsu¯∈R^p andv¯∈R^qsuch that the triplet(x,¯ u,¯ v)¯ satisfies system(12)of Theorem15. If(f, g_I)is weak pseudoquasi(F, α, ρ, d)-type I atx¯withuρ¯ ¹α¹(·,x)¯ ⁻¹+ ¯v_Iρ²α²(·,x)¯ ⁻¹0, then

¯

xis an weak efficient solution for(MOP).

Proof. Assume that x¯ is not a weak efficient solution for (MOP). Then, there exists an x∈A such that f (x) < f (x). Since¯ gI(x)¯ =0 and (f, gI) is weak pseudoquasi (F, α, ρ, d)-type I atx¯, we have

F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x)¯ 0, ∀µ^∗∈∂f (x),¯ F

x,x¯;α²(x,x)ν¯ ^∗

Theorem 20.Letx¯ be a feasible solution for(MOP). If there existu¯∈R^p,v¯∈R^q such that the triplet(x,¯ u,¯ v)¯ satisfies system (17)of Theorem16and (f, g_I)is pseudoquasi (F, α, ρ, d)-type I atx¯ withuρ¯ ¹α¹(·,x)¯ ⁻¹+ ¯vIρ²α²(·,x)¯ ⁻¹0, thenx¯is an weak effi- cient solution for(MOP).

Proof. Suppose thatx¯ is not a weak efficient solution for (MOP). Then, there exists an x∈Asuch that f (x) < f (x). Since¯ gI(x)¯ =0 and (f, gI)is pseudoquasi(F, α, ρ, d)- type I atx¯, we have

F

x,x¯;α¹(x,x)µ¯ ^∗

+ρ¹d²(x,x) <¯ 0, ∀µ^∗∈∂f (x),¯ F

x,x¯;α²(x,x)ν¯ ^∗

(8)

Remark 21.The importance of Theorems 19 and 20 lies in the fact that a similar result does not necessarily hold for efficient solutions.

4. Mixed type duality

LetJ₁be a subset ofQandJ₂=Q/J₁. We consider the following mixed type dual of (MOP) defined by:

(XMOP) maximizef (y)+vJ₁gJ₁(y)e, subject to 0∈∂

uf (y)+v_J₁g_J₁(y) +∂

v_J₂g_J₂(y)

, (19a)

v_J₂g_J₂(y)0, (19b)

v0, (19c)

u0, u^te=1. (19d)

Note that we get a sort of Mond–Weir dual forJ₁= ∅and a sort of Wolfe dual forJ₂= ∅ in (XMOP), respectively.

Theorem 22(Weak duality). Assume that for all feasible x for (MOP)and all feasible (y, u, v)for(XMOP), any of the following holds:

(a) u >0, and(f (·)+v_J₁g_J₁(·)e, v_J₂g_J₂(·))is strong pseudoquasi(F, α, ρ, d)-type I at y withuρ¹α¹(·, y)⁻¹+ρ²α²(·, y)⁻¹0;

(b) u >0, and(uf (·)+vJ₁gJ₁(·), vJ₂gJ₂(·))is pseudoquasi(F, α, ρ, d)-type I aty with ρ¹α¹(·, y)⁻¹+ρ²α²(·, y)⁻¹0.

Then the following cannot hold:

f (x)f (y)+v_J₁g_J₁(y)e. (20)

Proof. Suppose contrary to the result of the theorem that (20) holds. Since x is feasible for (MOP) andv0, (20) implies

f (x)+v_J₁g_J₁(x)ef (y)+v_J₁g_J₁(y)e (21a) hold. Since(y, u, v)is feasible for (XMOP), it follows that

−vJ₂gJ₂(y)0. (21b)

By hypothesis (a) and (21), we have F

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y)0, ∀µ^∗∈∂

f (y)+v_J₁g_J₁(y)e

, (22a)

F

x, y;α²(x, y)ξ^∗

+ρ²d²(x, y)0, ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (22b)

Sinceα¹(x, y) >0,α²(x, y) >0 andu >0, the inequalities (22) give F (x, y;uµ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀µ^∗∈∂

f (y)+v_J₁g_J₁(y)e

, (23a) F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (23b)

(9)

Lettingζ^∗=uµ^∗, inequalities (23) give

F (x, y;ζ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀ζ^∗∈u∂

f (y)+v_J₁g_J₁(y)e

, (24a) F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (24b)

Since∂(uf (y)+vJ₁gJ₁(y))⊆u∂(f (y)+vJ₁gJ₁(y)e), we obtain F (x, y;ζ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, (25a)

F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (25b)

By sublinearity ofF, we obtain F (x, y;ζ^∗+ξ^∗) <−

uρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹

d²(x, y).

Sinceuρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹0, we have F (x, y;ζ^∗+ξ^∗) <0, ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, ∀ξ^∗∈∂

v_J₂g_J₂(y)

, (26) which contradicts the duality constraint (19a) becauseF (x, y;0)=0. Hence, (20) cannot hold.

On the other hand, multiplying (21a) withu, we get

uf (x)+v_J₁g_J₁(x) < uf (y)+v_J₁g_J₁(y). (27) When hypothesis (b) holds, inequalities (21b) and (27) implies

F

x, y;α¹(x, y)ζ^∗

+ρ¹d²(x, y) <0, ∀ζ^∗∈∂

uf (y)+vJ₁gJ₁(y)

, (28a)

F

x, y;α²(x, y)ξ^∗

+ρ²d²(x, y)0, ∀ξ^∗∈∂

vJ₂gJ₂(y)

. (28b)

Sinceα¹(x, y) >0 andα²(x, y) >0, the inequalities (28) give F (x, y;ζ^∗) <−α¹(x, y)⁻¹ρ¹d²(x, y), ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, (29a)

F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (29b)

ρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹

d²(x, y).

So we also have (26) which contradicts the duality constraint (19a). 2

We need the condition u >0 in Theorem 22. In order to get the results without the conditionu >0, then other convexity assumption should be enforced, which leads to the following theorem.

Theorem 23(Weak duality). Assume that for all feasiblex for (MOP)and all feasible (y, u, v)for(XMOP), any of the following holds:

(a) (f (·)+v_J₁g_J₁(·)e, v_J₂g_J₂(·))is weak strictly-pseudoquasi(F, α, ρ, d)-type I atywith uρ¹α¹(·, y)⁻¹+ρ²α²(·, y)⁻¹0;

(b) (uf (·)+v_J₁g_J₁(·), v_J₂g_J₂(·)) is strictly-pseudoquasi (F, α, ρ, d)-type I at y with ρ¹α¹(·, y)⁻¹+ρ²α²(·, y)⁻¹0.

(10)

f (x)f (y)+v_J₁g_J₁(y)e. (30)

Proof. Suppose contrary to the result of the theorem that (30) holds. Since x is feasible for (MOP) andv0, (30) implies

f (x)+vJ₁gJ₁(x)ef (y)+vJ₁gJ₁(y)e (31a) hold. Since(y, u, v)is feasible for (DMOP), it follows that

−v_J₂g_J₂(y)0. (31b)

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y) <0, ∀µ^∗∈∂

f (y)+v_J₁g_J₁(y)e

, (32a)

F

x, y;α²(x, y)ξ^∗

+ρ²d²(x, y)0, ∀ξ^∗∈∂

vJ₂gJ₂(y)

. (32b)

Sinceα¹(x, y) >0,α²(x, y) >0 andu0, the inequalities (32) give F (x, y;uµ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀µ^∗∈∂

f (y)+v_J₁g_J₁(y)e

, (33a) F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (33b)

Lettingζ^∗=uµ^∗, inequalities (33) give

F (x, y;ζ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀ζ^∗∈u∂

f (y)+v_J₁g_J₁(y)e

, (34a) F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

vJ₂gJ₂(y)

. (34b)

Since∂(uf (y)+v_J₁g_J₁(y))⊆u∂(f (y)+v_J₁g_J₁(y)e), we obtain F (x, y;ζ^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, (35a)

F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (35b)

uρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹

d²(x, y).

Sinceuρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹0, we have F (x, y;ζ^∗+ξ^∗) <0, ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, ∀ξ^∗∈∂

v_J₂g_J₂(y)

, (36) which contradicts the duality constraint (19a) becauseF (x, y;0)=0. Hence, (30) cannot hold.

On the other hand, multiplying (31a) withu0, we get

uf (x)+vJ₁gJ₁(x)uf (y)+vJ₁gJ₁(y). (37) When hypothesis (b) holds, inequalities (37) and (31b) implies

F

x, y;α¹(x, y)ζ^∗

+ρ¹d²(x, y) <0, ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, (38a)

F

x, y;α²(x, y)ξ^∗

+ρ²d²(x, y)0, ∀ξ^∗∈∂

v_J₂g_J₂(y)

. (38b)

(11)

Sinceα¹(x;, y) >0 andα²(x, y) >0, the inequalities (38) give F (x, y;ζ^∗) <−α¹(x, y)⁻¹ρ¹d²(x, y), ∀ζ^∗∈∂

uf (y)+v_J₁g_J₁(y)

, (39a)

F (x, y;ξ^∗)−α²(x, y)⁻¹ρ²d²(x, y), ∀ξ^∗∈∂

vJ₂gJ₂(y)

. (39b)

ρ¹α¹(x, y)⁻¹+ρ²α²(x, y)⁻¹

d²(x, y).

So we also have (36) which contradicts the duality constraint (19a). 2

Corollary 24. Let (y,¯ u,¯ v)¯ be feasible solution for problem (XMOP) such that

¯

v_J₁g_J₁(y)¯ =0 and assume thaty¯ is feasible for (MOP). If weak duality(any of Theo- rem22 or 23)holds between (MOP)and (XMOP), then y¯ is efficient for(MOP)and (y,¯ u,¯ v)¯ is efficient for(XMOP).

Proof. The proof is similar to the one of Egudo [5, Corollary 1]. 2

5. Mond–Weir type duality

We consider the following Mond–Weir type dual of (MOP) defined in [6]:

(WMOP) maximizef (y),

subject to 0∈u∂f (y)+v∂g(y), (40a)

vg(y)0, (40b)

v0, (40c)

u0, u^te=1. (40d)

We note that, whenJ₁= ∅, problems (XMOP) and (WMOP) are two sort of Mond–Weir duals. But, the feasible region of (WMOP) is bigger than the feasible region of (XMOP).

For any u∈R^p and v∈R^q, let us consider two diagonal matrices U, V such that U_ii=u_ifor alli=1, . . . , pandV_ii=v_i for allj=1, . . . , q.

Theorem 25(Weak duality). Assume that for all feasiblex for (MOP)and all feasible (y, u, v)for(WMOP), any of the following holds:

(a) u >0, and (f (·), V g(·)) is strong pseudosubquasi (F, α, ρ, d)-type I at y with uρ¹α¹(·, y)⁻¹+eρ²α²(·, y)⁻¹0;

(b) u >0, and (Uf (·), V g(·)) is strong pseudosubquasi (F, α, ρ, d)-type I at y with eρ¹α¹(·, y)⁻¹+eρ²α²(·, y)⁻¹0.

f (x)f (y). (41)

(12)

Proof. From the feasibility conditions we have

0=uµ¯^∗+vν¯^∗, (42)

whereµ¯^∗∈∂f (y)andν¯^∗∈∂g(y). Also from inequality (40b), we get

−e^tV g(y)0. (43a)

Now if absurdly we assume that it is

f (x)f (y). (43b)

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y)0, ∀µ^∗∈∂f (y), (44a) F

x, y;α²(x, y)ν^∗

+ρ²d²(x, y)0, ∀ν^∗∈∂(V g)(y). (44b) Taking into account Theorem 4 we obtain

F

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y)0, ∀µ^∗∈∂f (y), (45a) F

x, y;α²(x, y)ν^∗

+ρ²d²(x, y)0, ∀ν^∗∈V ∂g(y). (45b) Lettingµ^∗= ¯µ^∗andν^∗=Vν¯^∗, from inequalities (45) we obtain

F

x, y;α¹(x, y)µ¯^∗

+ρ¹d²(x, y)0, (46a)

F

x, y;α²(x, y)Vν¯^∗

+ρ²d²(x, y)0. (46b)

Sinceα¹(x, y) >0,α²(x, y) >0 andu >0, the inequalities (46) give

F (x, y;uµ¯^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), (47a) F (x, y;Vν¯^∗)−α²(x, y)⁻¹ρ²d²(x, y). (47b) Premultiplying (47b) bye, we have

F (x, y;uµ¯^∗) <−α¹(x, y)⁻¹uρ¹d²(x, y), (48a) F (x, y;vν¯^∗)−α²(x, y)⁻¹eρ²d²(x, y). (48b) By sublinearity ofF, we obtain

F (x, y;uµ¯^∗+vν¯^∗) <−

uρ¹α¹(x, y)⁻¹+eρ²α²(x, y)⁻¹

d²(x, y).

We obtain a contradiction to (42) becauseF (x, y;0)=0. Hence, (41) cannot hold.

On the other hand, premultiplying (43b) withU, sinceu >0 we get

Uf (x)Uf (y). (49)

When hypothesis (b) holds, inequalities (49) and (43a) implies F

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y)0, ∀µ^∗∈∂(Uf )(y), (50a) F

x, y;α²(x, y)ν^∗

+ρ²d²(x, y)0, ∀ν^∗∈∂(V g)(y). (50b)

(13)

Taking into account Theorem 4 we obtain F

x, y;α¹(x, y)µ^∗

+ρ¹d²(x, y)0, ∀µ^∗∈U ∂f (y), (51a) F

x, y;α²(x, y)ν^∗

+ρ²d²(x, y)0, ∀ν^∗∈V ∂g(y). (51b) Lettingµ^∗=Uµ¯^∗andν^∗=Vν¯^∗, from inequalities (51) we obtain

F

x, y;α¹(x, y)Uµ¯^∗

+ρ¹d²(x, y)0, (52a)

F

x, y;α²(x, y)Vν¯^∗

+ρ²d²(x, y)0. (52b)

Premultiplying (52a) byeand (52b) bye, we have

F (x, y;uµ¯^∗) <−α¹(x, y)⁻¹eρ¹d²(x, y), (53a) F (x, y;vν¯^∗)−α²(x, y)⁻¹eρ²d²(x, y). (53b) By sublinearity ofF, we obtain

F (x, y;uµ¯^∗+vν¯^∗) <−

eρ¹α¹(x, y)⁻¹+eρ²α²(x, y)⁻¹

d²(x, y).

We obtain again a contradiction to (42). Hence, (41) cannot hold. 2

Theorem 26(Weak duality). Assume that for all feasiblex for (MOP)and all feasible (y, u, v)for(WMOP), any of the following holds:

(a) (f (·), V g(·)) is weak strictly-pseudosubquasi (F, α, ρ, d)-type I at y with uρ¹α¹(·, y)⁻¹+eρ²α²(·, y)⁻¹0;

(b) (Uf (·), V g(·)) is weak strictly-pseudosubquasi (F, α, ρ, d)-type I at y with eρ¹α¹(·, y)⁻¹+eρ²α²(·, y)⁻¹0.

f (x)f (y). (54)

Proof. This can be proved along the lines of Theorem 25 of this paper. 2

Corollary 27.Let(y,¯ u,¯ v)¯ be feasible solution for (WMOP)and assume thaty is fea- sible for(MOP). If weak duality(any of Theorem25or26)holds between(MOP)and (WMOP), theny¯is efficient for(MOP)and(y,¯ u,¯ v)¯ is efficient for(WMOP).

Proof. The proof is similar to the one of Egudo [5, Corollary 2]. 2

Letx¯be an efficient solution for (MOP). In the following we refers to a constraint qualification any condition which ensures the existence of fixed multipliers such system (12) or (17) holds.

Theorem 28(Strong duality). Letx¯ be an efficient solution for(MOP)and assume that

¯

x satisfies a constraint qualification[14]. Then there existu¯∈R^p and v¯∈R^q such that

(14)

(x,¯ u,¯ v)¯ is feasible for(WMOP). If also weak duality(Theorem25or26)holds between (MOP)and(WMOP)then(x,¯ u,¯ v)¯ is efficient for(WMOP).

Proof. This follows on the lines of Egudo [5, Theorem 3]. 2

References

[1] B. Aghezzaf, M. Hachimi, Sufficiency and duality in multiobjective programming involving generalized (F, ρ)-convexity, J. Math. Anal. Appl. 258 (2001) 617–628.

[2] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global Optim. 18 (2000) 91–101.

[3] C.R. Sector, S. Chandra, Abha, On mixed duality in mathematical programming, J. Math. Anal. Appl. 259 (2001) 346–356.

[4] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[5] R. Egudo, Efficiency and generalized convex duality for multiobjective programs, J. Math. Anal. Appl. 138 (1989) 84–94.

[6] G. Giorgi, A. Guerraggio, The notion of invexity in vector optimization: smooth and nonsmooth case, in:

J.-P. Crouzeix, J.-E. Martinez Legaz, M. Voile (Eds.), Proceedings of the Fifth Symposium on Generalized Convexity, Luminy-Marseille (France), June 17–21, 1996, in: Nonconvex Optimization and Its Applications, vol. 27, Kluwer Academic, Dordrecht, 1998, pp. 389–405.

[7] M. Hachimi, B. Aghezzaf, Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions, J. Math. Anal. Appl. 296 (2004) 382–392.

[8] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550.

[9] M.A. Hanson, B. Mond, Further generalizations of convexity in mathematical programming, J. Inform.

Optim. Sci. 3 (1986) 25–32.

[10] M.A. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program- ming 37 (1987) 51–58.

[11] R.N. Kaul, S. Kaur, Optimality criteria in nonlinear programming involving nonconvex functions, J. Math.

Anal. Appl. 105 (1985) 104–112.

[12] R.N. Kaul, S.K. Suneja, M.K. Srivastava, Optimality criteria and duality in multiobjective optimization involving generalized invexity, J. Optim. Theory Appl. 80 (1994) 465–482.

[13] Z.A. Liang, H.X. Huang, P.M. Pardalos, Optimality conditions and duality for a class of nonlinear fractional programming problems, J. Optim. Theory Appl. 110 (2001) 611–619.

[14] K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic, Dordrecht, 1999.

[15] V. Preda, On efficiency and duality for multiobjective programs, J. Math. Anal. Appl. 166 (1992) 365–377.

[16] N.G. Rueda, M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 (1988) 375–385.

[17] N.G. Rueda, M.A. Hanson, C. Singh, Optimality and duality with generalized convexity, J. Optim. Theory Appl. 86 (1995) 491–500.

[18] Y. Sawaragi, H. Nakayama, T. Tanino, Theory of Multiobjective Optimization, Academic Press, Orlando, FL, 1985.

[19] J.P. Vial, Strong and weak convexity set and functions, Math. Oper. Res. 8 (1983) 231–259.

[20] Z. Xu, Mixed type duality in multiobjective programming problems, J. Math. Anal. Appl. 198 (1996) 621–

635.