Page 1

Computers and Mathematics with Applications 60 (2010) 3187–3192

Contents lists available at ScienceDirect

Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

Multiobjective higher-order symmetric duality involving generalized

cone-invex functions

S.K. Guptaa, Anurag Jayswalb,∗

aDepartment of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India

bDepartment of Applied Mathematics, Birla Institute of Technology Mesra, Ranchi-835 215, India

a r t i c l ei n f o

Article history:

Received 4 May 2010

Received in revised form 11 October 2010

Accepted 11 October 2010

Keywords:

Multiobjective programming

Cone constraints

Efficiency

Duality

a b s t r a c t

In this paper, a pair of Mond–Weir type multiobjective higher-order symmetric dual

programsoverarbitraryconesisformulatedandusualdualityresultsareestablishedunder

higher-order K-preinvexity/K-pseudoinvexity assumptions. Symmetric minimax mixed

integer primal and dual problems are also discussed.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Mangasarian[1]introducedhigher-orderdualityinnonlinearprogrammingbyintroducingtwicedifferentiablefunctions

h : Rn×Rn→ R and k : Rn×Rn→ Rm. Mond and Zhang [2] obtained duality results for various higher-order dual programs

under higher-order invexity assumptions while Mond and Weir [3] presented two pair of symmetric dual multiobjective

programming problems and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions.

Ahmad et al. [4] considered a general Mond–Weir type higher-order nondifferentiable multiobjective dual programs

and obtained duality relations under higher-order (F,α,ρ,d) type-I functions. Chen [5] studied a pair of Mond–Weir

type symmetric higher-order multiobjective nondifferentiable programs and obtained duality results under higher-order

F-convexity assumptions. Later on, Gulati and Gupta [6] formulated higher-order Wolfe and Mond–Weir type

symmetric dual problems with cone constraints and established usual duality theorems under higher-order η-invexity/η-

pseudoinvexity assumptions.

Khurana [7] formulated a pair of multiobjective symmetric dual programs of Mond–Weir type over arbitrary cones

in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the involved

functions to be cone-pseudoinvex and strongly cone-pseudoinvex. These results were later extended to nondifferentiable

multiobjective symmetric dual programs by Kim and Kim [8].

Second-order multiobjective symmetric duality over arbitrary cones has been discussed by Ahmad and Husain [9] for

Wolfe type programs (assuming the function involved to be second-order invex) and Gulati and Geeta [10] for Mond–Weir

type problems (assuming pseudoinvexity/F-convexity on the kernel function). Recently, Ahmad and Husain [11] studied

mixed symmetric multiobjective dual programs and obtained duality results under K-preinvexity and K-pseudoinvexity

assumptions.

∗Corresponding address: Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India. Tel.: +91 542 2440628.

E-mail addresses: skgiitr@gmail.com (S.K. Gupta), anurag_jais123@yahoo.com (A. Jayswal).

0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.camwa.2010.10.023

Page 2

3188

S.K. Gupta, A. Jayswal / Computers and Mathematics with Applications 60 (2010) 3187–3192

In the present paper, we formulate Mond–Weir type higher-order multiobjective symmetric dual programs over

arbitrary cones and appropriate duality theorems are established under higher-order cone-invexity/cone-pseudoinvexity

assumptions. These duality results have also been used to discuss symmetric minimax mixed integer dual programs. Our

study extends some of the known results in [6,7,12,13].

2. Notations and preliminaries

Let Rndenote the n-dimensional Euclidean space and Rn

Consider the following multiobjective programming problem:

K-Minimize f(x)

subject to x ∈ X0= {x ∈ S : −g(x) ∈ Q},

where S ⊂ Rnis open, f : S → Rk, g : S → Rm, K and Q are closed convex pointed cones with nonempty interiors in Rkand

Rm, respectively.

+be its non-negative orthant.

(P)

Definition 2.1. A point ¯ x ∈ X0is an efficient solution of (P) if there exists no other x ∈ X0such that

f(¯ x) − f(x) ∈ K \ {0}.

Definition 2.2. The positive polar cone K∗of K is defined as

K∗= {z ∈ Rk: xTz ? 0, for all x ∈ K}.

Let h = {h1,h2,...,hk} : S × Rn→ Rkbe a differentiable function and η : S × S → Rn.

Definition 2.3. A differentiable function f : S → Rkis said to be higher-order K-preinvex at u ∈ S with respect to h and η,

if for all x ∈ S,pi∈ Rn,i = 1,2,...,k,

(f1(x) − f1(u) − h1(u,p1) + pT

−fk(u) − hk(u,pk) + pT

1[∇p1h1(u,p1)] − ηT(x,u)(∇xf1(u) + ∇p1h1(u,p1)),...,fk(x)

k[∇pkhk(u,pk)] − ηT(x,u)(∇xfk(u) + ∇pkhk(u,pk))) ∈ K.

Definition 2.4. A differentiable function f : S → Rkis said to be higher-order K-pseudoinvex at u ∈ S with respect to h and

η, if for all x ∈ S,pi∈ Rn,i = 1,2,...,k,

ηT(x,u)(∇xf1(u) + ∇p1h1(u,p1),...,∇xfk(u) + ∇pkhk(u,pk)) ∈ K

⇒ (f1(x) − f1(u) − h1(u,p1) + pT

1[∇p1h1(u,p1)],...,fk(x) − fk(u) − hk(u,pk) + pT

k[∇pkhk(u,pk)]) ∈ K.

Remark 2.1.

Definition 2.3 becomes the higher-order type-I functions given in Mond and Zhang [2].

(ii) For K = R1

respect to η and h given in Gulati and Gupta [6].

(iii) For K = R1

considered in Chandra and Kumar [12].

(iv) If hi(u,pi) =

functions given in Gulati and Mehndiratta [13].

(v) Let K = R1

respectively, introduced by Pandey [14].

(i) If K = R1

+, Definitions 2.3 and 2.4 reduce to that of higher-order invex and higher-order pseudoinvex functions with

+and h(u,p) = 0, Definitions 2.3 and 2.4 reduce to invexity and pseudoinvexity with respect to η as

1

2pT

+and h(u,p) = −pT∇xf(u) + ψ(u,p) where ψ : S × Rn→ R is a differentiable function, then

i∇xxfi(u)pi,i = 1,2,...,k, then Definitions 2.3 and 2.4 become K-η-bonvex and K-η-pseudobonvex

+and h(u,p) =

1

2pT∇xxf(u)p, the above Definitions 2.3 and 2.4 becomeη-bonvexity andη-pseudobonvexity,

3. Mond–Weir type higher-order symmetric duality

In this section, we present the following pair of Mond–Weir type higher-order multiobjective symmetric dual problems

with k-objectives:

Primal Problem (MP).

K-minimize (f1(x,y) + h1(x,y,p1) − pT

subject to

1[∇p1h1(x,y,p1)],...,fk(x,y) + hk(x,y,pk) − pT

k[∇pkhk(x,y,pk)])

−

k −

k −

λ ∈ int K∗,

i=1

λi[∇yfi(x,y) + ∇pihi(x,y,pi)] ∈ C∗

2,

(1)

yT

i=1

λi[∇yfi(x,y) + ∇pihi(x,y,pi)] ? 0,

(2)

x ∈ C1.

(3)

Page 3

S.K. Gupta, A. Jayswal / Computers and Mathematics with Applications 60 (2010) 3187–3192

3189

Dual Problem (MD).

K-maximize (f1(u,v) + g1(u,v,r1) − rT

subject to

1[∇r1g1(u,v,r1)],...,fk(u,v) + gk(u,v,rk) − rT

k[∇rkgk(u,v,rk)])

k −

i=1

λi[∇xfi(u,v) + ∇rigi(u,v,ri)] ∈ C∗

1,

(4)

uT

k −

i=1

λi[∇xfi(u,v) + ∇rigi(u,v,ri)] ? 0,

(5)

λ ∈ int K∗,v ∈ C2,

(6)

where fi: S1× S2→ R, hi: S1× S2× Rm→ R and gi: S1× S2× Rn→ R are differentiable functions for i = 1,2,...,k.

C1and C2are closed convex cones in Rnand Rm, respectively, with nonempty interiors. S1⊆ Rnand S2⊆ Rmare open sets

such that C1× C2⊂ S1× S2. C∗

Rksuch that int K ̸= φ.

We will use p = (p1,p2,...,pk),r = (r1,r2,...,rk) and λ = (λ1,λ2,...,λk) ∈ Rk.

Theorem 3.1 (Weak Duality). Let (x,y,λ,p) and (u,v,λ,r) be feasible solutions to (MP) and (MD), respectively. Suppose that

f(.,v)ishigher-order K-preinvexwithrespecttog(u,v,r)andη1forfixedv and−f(x,.)ishigher-order K-preinvexwithrespect

to −h(x,y,p) and η2for fixed x. Moreover, if η1(x,u) + u ∈ C1and η2(v,y) + y ∈ C2, then

(f1(x,y) + h1(x,y,p1) − pT

−(f1(u,v) + g1(u,v,r1) − rT

1and C∗

2are positive polar cones of C1and C2, respectively and K is a closed convex cone in

1[∇p1h1(x,y,p1)],...,fk(x,y) + hk(x,y,pk) − pT

1[∇r1g1(u,v,r1)],...,fk(u,v) + gk(u,v,rk) − rT

k[∇pkhk(x,y,pk)])

k[∇rkgk(u,v,rk)]) ̸∈ −K \ {0}.

Proof. Suppose contrary to the result that

(f1(x,y) + h1(x,y,p1) − pT

−(f1(u,v) + g1(u,v,r1) − rT

Since λ ∈ int K∗, we have

k −

From (4) and η1(x,u) + u ∈ C1, we have

k −

The above inequality together with (5) yield

1[∇p1h1(x,y,p1)],...,fk(x,y) + hk(x,y,pk) − pT

1[∇r1g1(u,v,r1)],...,fk(u,v) + gk(u,v,rk) − rT

k[∇pkhk(x,y,pk)])

k[∇rkgk(u,v,rk)]) ∈ −K \ {0}.

i=1

λi(fi(x,y) + hi(x,y,pi) − pT

i[∇pihi(x,y,pi)] − fi(u,v) − gi(u,v,ri) + rT

i[∇rigi(u,v,ri)]) < 0.

(7)

[η1(x,u) + u]T

i=1

λi[∇xfi(u,v) + ∇rigi(u,v,ri)] ? 0.

ηT

1(x,u)

k −

i=1

λi[∇xfi(u,v) + ∇rigi(u,v,ri)] ? 0.

(8)

Now, by higher-order K-preinvexity of f(.,v) w.r.t. g(u,v,r) and η1, we get

(f1(x,v) − f1(u,v) − g1(u,v,r1) + rT

+∇r1g1(u,v,r1)),...,fk(x,v) − fk(u,v) − gk(u,v,rk)

+rT

This further using constraint (3) yields

1[∇r1g1(u,v,r1)] − ηT

1(x,u)(∇xf1(u,v)

k[∇rkgk(u,v,rk)] − ηT

1(x,u)(∇xfk(u,v) + ∇rkgk(u,v,rk))) ∈ K.

k −

i=1

λi(fi(x,v) − fi(u,v) − gi(u,v,ri) + rT

i∇rigi(u,v,ri)) ? ηT

1(x,u)

k −

i=1

λi[∇xfi(u,v) + ∇rigi(u,v,ri)].

It follows from (8) that

k −

i=1

λi(fi(x,v) − fi(u,v) − gi(u,v,ri) + rT

i∇rigi(u,v,ri)) ? 0.

(9)

Page 4

3190

S.K. Gupta, A. Jayswal / Computers and Mathematics with Applications 60 (2010) 3187–3192

Similarly, using higher-order K-preinvexity of−f(x,.) w.r.t.−h(x,y,p) andη2, primal constraints (1), (2) andη2(v,y)+y ∈

C2, we obtain

k −

i=1

λi(fi(x,y) − fi(x,v) + hi(x,y,pi) − pT

i∇pihi(x,y,pi)) ? 0.

(10)

Now, adding (9) and (10), we get

k −

i=1

λi(fi(x,y) + hi(x,y,pi) − pT

i∇pihi(x,y,pi)) ?

k −

i=1

λi(fi(u,v) + gi(u,v,ri) − rT

i∇rigi(u,v,ri)).

This contradicts (7). Hence the result.

?

Since every invex functions are pseudoinvex, therefore the following weak duality theorem for the dual pair (MP) and

(MD) can also be obtained under higher-order K-pseudoinvexity assumptions.

Theorem 3.2 (Weak Duality). Let (x,y,λ,p) and (u,v,λ,r) be feasible solutions to (MP) and (MD), respectively. Suppose that

f(.,v) is higher-order K-pseudoinvex with respect to g(u,v,r) and η1for fixed v and −f(x,.) is higher-order K-pseudoinvex

with respect to −h(x,y,p) and η2for fixed x. Moreover, if η1(x,u) + u ∈ C1and η2(v,y) + y ∈ C2, then

(f1(x,y) + h1(x,y,p1) − pT

−(f1(u,v) + g1(u,v,r1) − rT

1[∇p1h1(x,y,p1)],...,fk(x,y) + hk(x,y,pk) − pT

1[∇r1g1(u,v,r1)],...,fk(u,v) + gk(u,v,rk) − rT

k[∇pkhk(x,y,pk)])

k[∇rkgk(u,v,rk)]) ̸∈ −K \ {0}.

Theorem 3.3 (Strong Duality). Let (¯ x, ¯ y,¯λ, ¯ p) be an efficient solution for (MP), fi: S1× S2→ R be twice differentiable at (¯ x, ¯ y),

hi: S1× S2× Rm→ R be differentiable at (¯ x, ¯ y, ¯ pi) and gi: S1× S2× Rn→ R be differentiable at (¯ x, ¯ y, ¯ ri) for i = 1,2,...,k.

If the following conditions hold:

(I) hi(¯ x, ¯ y,0) = 0,gi(¯ x, ¯ y,0) = 0, ∇xhi(¯ x, ¯ y,0) = ∇rigi(¯ x, ¯ y,0), ∇pihi(¯ x, ¯ y,0) = 0,∇yhi(¯ x, ¯ y,0) = 0,i = 1,2,...,k,

(II) for all i = 1,2,...,k, the Hessian matrix ∇pipihi(¯ x, ¯ y, ¯ pi) is positive or negative definite,

(III) the set of vectors {∇yfi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)}k

(IV) the set of vectors {∇yfi(¯ x, ¯ y) + ∇yhi(¯ x, ¯ y, ¯ pi),∇yfi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)}k

(V) for some ξ ∈ int K∗and ¯ pi∈ Rm, ¯ pi̸= 0 (i = 1,2,...,k) implies that

k −

(VI) Rk

i=1is linearly independent,

i=1is linearly independent,

i=1

ξi¯ pi

T[∇yfi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)] ̸= 0,

+⊆ K.

Then ¯ p = 0, (¯ x, ¯ y,¯λ, ¯ r = 0) is a feasible solution for (MD) and the functional values of two objectives are equal. Furthermore, if

the hypotheses in Theorem 3.1 or 3.2 are satisfied, then (¯ x, ¯ y,¯λ, ¯ r = 0) is an efficient solution of (MD).

Proof. It follows on the lines of Theorem 3.2 in [15], taking ¯ zi= 0 and ξi= 0 for i = 1,2,...,k.

We now state a converse duality theorem whose proof follows on the lines of Theorem 3.3.

?

Theorem 3.4 (Converse Duality). Let (¯ u, ¯ v,¯λ, ¯ r) be an efficient solution for (MD), fi : S1× S2 → R be twice differentiable

at (¯ u, ¯ v),hi : S1× S2× Rm→ R be differentiable at (¯ u, ¯ v, ¯ pi) and gi : S1× S2× Rn→ R be differentiable at (¯ u, ¯ v, ¯ ri) for

i = 1,2,...,k. If

(I) hi(¯ u, ¯ v,0) = 0,gi(¯ u, ¯ v,0) = 0, ∇xgi(¯ u, ¯ v,0) = 0,∇rigi(¯ u, ¯ v,0) = 0, ∇pihi(¯ u, ¯ v,0) = ∇ygi(¯ u, ¯ v,0),i = 1,2,...,k,

(II) for all i = 1,2,...,k, the Hessian matrix ∇ririgi(¯ u, ¯ v, ¯ ri) is positive or negative definite,

(III) the set of vectors {∇xfi(¯ u, ¯ v) + ∇rigi(¯ u, ¯ v, ¯ ri)}k

(IV) the set of vectors {∇xfi(¯ u, ¯ v) + ∇xgi(¯ u, ¯ v, ¯ ri),∇xfi(¯ u, ¯ v) + ∇rigi(¯ u, ¯ v, ¯ ri)}k

(V) for some ξ ∈ int K∗and ¯ ri∈ Rn, ¯ ri̸= 0 (i = 1,2,...,k) implies that

k −

(VI) Rk

i=1is linearly independent,

i=1is linearly independent,

i=1

ξi¯ ri

T[∇xfi(¯ u, ¯ v) + ∇rigi(¯ u, ¯ v, ¯ ri)] ̸= 0,

+⊆ K.

Then ¯ r = 0, (¯ u, ¯ v,¯λ, ¯ p = 0) is a feasible solution for (MP) and the functional values of two objectives are equal. Furthermore, if

the hypotheses in Theorem 3.1 or 3.2 are satisfied, then (¯ u, ¯ v,¯λ, ¯ r = 0) is an efficient solution of (MP).

Page 5

S.K. Gupta, A. Jayswal / Computers and Mathematics with Applications 60 (2010) 3187–3192

3191

4. Minimax mixed integer programming

In this section, we constrain some of the components of x and y belonging to arbitrary sets of integers. Suppose that the

first n1(0 ≤ n1 ≤ n) components of x belong to U and the first m1(0 ≤ m1 ≤ m) components of y belongs to V, then

we write (x,y) = (x1,x2,y1,y2) where x1= (x1,x2,...,xn1) and y1= (y1,y2,...,ym1), x2∈ Rn−n1and y2∈ Rm−m1,

respectively.

Now, we consider the following minimax mixed integer higher-order symmetric dual programs:

Primal Problem (MMP).

Maxx1Minx2,y(f1(x,y) + h1(x,y,p1) − pT

subject to

k −

yT

2

i=1

λ ∈ int K∗,

x1∈ U,

Dual Problem (MMD).

Minv1Maxu,v2(f1(u,v) + g1(u,v,r1) − rT

subject to

k −

uT

2

i=1

λ ∈ int K∗,

u1∈ U,

1[∇p1h1(x,y,p1)],...,fk(x,y) + hk(x,y,pk) − pT

k[∇pkhk(x,y,pk)])

−

i=1

k −

λi[∇y2fi(x,y) + ∇pihi(x,y,pi)] ∈ C∗

2,

λi[∇y2fi(x,y) + ∇pihi(x,y,pi)] ? 0,

x2∈ C1,

y1∈ V,

p ∈ Rm−m1.

1[∇r1g1(u,v,r1)],...,fk(u,v) + gk(u,v,rk) − rT

k[∇rkgk(u,v,rk)])

i=1

λi[∇x2fi(u,v) + ∇rigi(u,v,ri)] ∈ C∗

1,

k −

λi[∇x2fi(u,v) + ∇rigi(u,v,ri)] ? 0,

v2∈ C2,

v1∈ V,

r ∈ Rn−n1.

Theorem 4.1 (Symmetric Duality). Let (¯ x, ¯ y,¯λ, ¯ p) be an efficient solution for (MMP). Let for i = 1,2,...,k, the following

conditions hold:

(I) fi(x,y) be additively separable with respect to x1or y1,

(II) f(u,v) be higher-order K-preinvex with respect to g(u,v,r) and η1in u2for each (u1,v) and −f(x,y) be higher-order

K-preinvex with respect to −h(x,y,p) and η2in y2for each (x,y1),

(III) ∇pipihi(¯ x, ¯ y, ¯ pi) be positive or negative definite,

(IV) the set of vectors {∇y2fi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)}k

(V) the set of vectors {∇y2fi(¯ x, ¯ y) + ∇y2hi(¯ x, ¯ y, ¯ pi),∇y2fi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)}k

(VI) hi(¯ x, ¯ y,0) = 0,gi(¯ x, ¯ y,0) = 0, ∇x2hi(¯ x, ¯ y,0) = ∇rigi(¯ x, ¯ y,0), ∇pihi(¯ x, ¯ y,0) = 0,∇y2hi(¯ x, ¯ y,0) = 0,i = 1,2,...,k,

(VII) for some ξ ∈ int K∗and ¯ pi∈ Rm−m1, ¯ pi̸= 0,(i = 1,2,...,k) implies that

k −

(VIII) Rk

Then ¯ p = 0, (¯ x, ¯ y,¯λ, ¯ r = 0) is a feasible solution for (MMD) and the functional values of two objectives are equal. Furthermore,

if the hypotheses in Theorem 3.1 or 3.2 are satisfied, then (¯ x, ¯ y,¯λ, ¯ r = 0) is an efficient solution of (MMD).

Proof. The proof follows along the lines of Theorem 1 in [16] by using Theorems 3.1 and 3.3.

i=1is linearly independent,

i=1is linearly independent,

i=1

ξi¯ pT

i[∇y2fi(¯ x, ¯ y) + ∇pihi(¯ x, ¯ y, ¯ pi)] ̸= 0,

+⊆ K.

?

5. Special cases

In this section, we consider some of the special cases of the programs studied in Section 3.

(i) If K = R1

Gupta [6].

+, then our higher-order dual programs reduce to the programs (MHP) and (MHD) studied in Gulati and