On Duality with Support Functions for a Multiobjective Fractional

Programming Problem

Indira P. Debnath and S. K. Gupta

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247 667, India

Keywords:

Nonlinear Multiobjective Programming, Higher Order (K × Q)- F-type I, Support Functions, Efﬁcient

Solutions.

Abstract:

In this article, a different class of function called (K × Q)-F-type I has been introduced. Further, we have

formulated a problem over cones and appropriate duality results have been established taking the concerned

functions to be (K×Q)- F-type I. The results which we have put forward in the paper generalizes some of the

known results appeared in the literature.

1 INTRODUCTION

The mathematical programming problems involving

ratio of two functions in the objective function is

called fractional programming problems. Many types

of optimization problems involves this fascinating

subject. Fractional programming problem emerges in

several types of optimization problems (as for exam-

ple, portfolio selection, production, information the-

ory and numerous decision making problems in man-

agement science). It has also been extensively used in

business and economics situation. Further, these type

of problems are also useful in engineering and eco-

nomics where it is often used to maximize a fractional

function for measuring the efﬁciency or productivity

of a system. (Bector and Chandra, 1987), (Bector

et al., 1993), (Schaible, 1995) etc. have given some

applications of fractional programming problems.

Duality theory happens to be the central concept

in optimization. This theory aids us to experience and

develop new algorithms (numerical algorithms) be-

cause it gives us appropriate stopping rules for a pair

of primal-dual problems. (Dorn, 1960) considered

a convex minimization problem of a differentiable

function subject to some linear constraintsand studied

its duality relations. Over the past few years, many re-

searchers generalized these results to the case of non-

differentiable convex problem (Schechter, 1979) and

differentiable convex problem (Hanson, 1981). As-

suming the functions to be invex, (Hanson, 1981) es-

tablished that KKT conditions are sufﬁcient for op-

timality. Two different types of functions (namely

Type I and Type II) were ﬁrst presented by (Hanson

and Mond, 1987) for a scalar optimization problem.

(Rueda and Hanson, 1988) extended these functions

to pseudo-Type I and quasi-Type I.

The Type I function for a single objective was

extended to a MOPP (multiobjective programming

problem) by (Kaul et al., 1994), where they deﬁned

the Type I, its different generalizations, thus estab-

lished duality relations for the Wolfe and the Mond-

Weir type model. (Kuk and Tanino, 2003) consid-

ered nonsmooth programming problem and derived

the duality results considering the functions to be gen-

eralized Type I. On the other hand, (Suneja et al.,

2008) presented (F, ρ, σ)-type I functions for the case

of higher order. Further, they considered two dual

models (one Mond-Weir and the other Schaible type

both are in higher order case) and obtained their corre-

sponding dual relations (for multiobjective fractional

programs in nondifferentiable case).

Recently, fractional programming duality has be-

come an interesting topic of research. For a con-

vex nondifferentiable fractional problem, (Bector

et al., 1993) established some optimality conditions

(namely Fritz John and KKT necessary and sufﬁ-

cient optimality criteria) and proved some duality re-

sults. Considering a vectorial optimization problem

over cones, (Bhatia, 2012) discussed the sufﬁcient op-

timality conditions and proved some results (duality

theorems) using cone convex and its generalizations

for the case of higher order. (Slimani and Mishra,

2014) introduced a nonlinear multiple objective frac-

tional programming with inequality constraints and

proved duality results for a Mond-Weir type model

using semilocally V-type I-preinvex functions.

Debnath, I. and Gupta, S.

On Duality with Support Functions for a Multiobjective Fractional Programming Problem.

DOI: 10.5220/0005666001150121

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 115-121

ISBN: 978-989-758-171-7

115

(Kim and Lee, 2009) introduced the nondiffer-

entiable multiobjective problem involving cone con-

straints and hence studied their duality relations using

higher order invexity assumptions. On the other hand,

considering the same problem as in (Kim and Lee,

2009), (Ahmad, 2012) formulated a dual program

(uniﬁed and higher order) and discussed some re-

sults in duality (under higher order generalized type-

I functions). In recent times, (Debnath et al., 2015)

constructed a pair of higher order Wolfe type multi-

objective nondifferentiable symmetric dual program

over arbitrary cones and studied duality relations un-

der higher-order K-F convexity assumption.

The paper is arranged as we describe. In section

2, a class of (K × Q)-F-type I function has been put

forward and further some deﬁnitions and terminolo-

gies have been given. In the section 3, we have for-

mulated a dual model (Mond-Weir model) for a frac-

tional problem over arbitrary cones (for nondifferen-

tiable multiobjective case) and proved duality rela-

tions considering the concerned functions as higher

order (K × Q)-F-type I. Section 3 contains the con-

clusion.

2 SOME BASIC DEFINITIONS

Consider the following MOPP (multiobjective pro-

gramming problem):

(VP) K-minimize φ(x)

subject to −ψ(x) ∈ Q,

x ∈ X ⊆ R

where X ⊂ R

be open and φ : X → R

, ψ : X → R

deﬁnes vector valued differentiable functions,

K ⊆ R

and Q ⊆ R

denotes the closed con-

vex pointed cones having non-null interiors. Let

= {x ∈ X : −ψ(x) ∈ Q} denotes the feasible set.

Deﬁnition 2.1 (Agarwal et al., 2010). A point

¯x ∈ X

is a weak efﬁcient solution of (VP) if there

exists no x ∈ X

such that

φ( ¯x) − φ(x) ∈ intK.

Deﬁnition 2.2 (Agarwal et al., 2010). A point ¯x ∈ X

is an efﬁcient solution of (VP) if there exists no x ∈ X

such that

φ( ¯x) − φ(x) ∈ K\{0}.

Deﬁnition 2.3 (Gupta et al., 2012). The positive dual

cone K

∗

of K is deﬁned by

∗

= {y : x

y ≥ 0, for all x ∈ K}.

Deﬁnition 2.4. For all (x, u) ∈ X × X, a functional

H : X ×X ×R

→ R is called sublinear in respect with

the third component, if

(i) H(x, u;b

+ b

) ≤ H(x, u;b

) + H(x, u;b

) for all

, b

∈ R

(ii) H(x, u;βb) = βH(x, u;b), for all β ∈ R

and for

all b ∈ R

Clearly, H(x, u;0) = 0.

Deﬁnition 2.5. Let F : X × X × R

→ R be called a

functional which is sublinear with respect to the third

variable. Also, let H : X × R

→ R

, G : X × R

→ R

be differentiable functions. Then the function (φ, ψ)

will be called higher-order (K × Q)-F type I at u ∈ R

in respect with the functions H and G, if for each

x ∈ X

, p

, q

∈ R

, (i = 1, 2, ..., k, j = 1, 2, ..., m), we

have



(x) − φ

(u) − F(x, u;∇

(u) + ∇

(u, p

)) −

(u, p

) + p

[∇

(u, p

)], ..., φ

(x) − φ

(u) −

F(x, u;∇

(u) + ∇

(u, p

)) − H

(u, p

) +

[∇

(u, p

)]



∈ K.

and



− ψ

(u) − F(x, u;∇

(u) + ∇

(u, q

)) −

(u, q

) + q

∇

(u, q

), ..., −ψ

(u) −

F(x, u;∇

(u) + ∇

(u, q

)) − G

(u, q

) +

∇

(u, q

)



∈ Q.

Deﬁnition 2.6 (Gupta et al., 2012). Let ϕ be a

convex set in R

which is also compact. The support

function of ϕ is given as

τ(x|ϕ) = max{x

y : y ∈ ϕ}.

The subdifferentiable of τ(x|ϕ) is deﬁned by

∂τ(x|ϕ) = {z ∈ ϕ : z

x = τ(x|ϕ)}.

We now present the following problem (KP)

(multiobjective fractional programming problem)

over arbitrary cones containing support functions.

(KP) K-minimize

(x) + τ(x|C

)

(x) − τ(x|D

)

, ...,

(x) + τ(x|C

)

(x) − τ(x|D

)

subject to

−

(x) + τ(x|M

)

∈ Q, j = 1, 2, ..., m.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

116

where

φ : R

→ R

, ψ : R

→ R

and ϕ : R

→ R

are

continuously differentiable functions. Assume that

(.) + τ(.|C

) ≥ 0 and ψ

(.) − τ(.|D

) > 0. C

, D

and

denotes the compact convex sets in R

and their

respective support functions are denoted by τ(x|C

τ(x|D

) and τ(x|M

Throughout the paper, the following notations

have been used:

P(.) + (.)

Q(.) − (.)

(.) + (.)

(.) − (.)

(.) + (.)

(.) − (.)

, ...,

(.) + (.)

(.) − (.)

R(.) + (.)

w =

(.) + (.)

, ϕ

(.) + (.)

, ...,

(.) + (.)

H =



(u, p

), H

(u, p

), ..., H

(u, p

)



G =



(u, q

), G

(u, q

), ..., G

(u, q

)



In the following section, we associate dual model

for the primal problem (KP) and establish duality

relations between them.

3 MOND-WEIR TYPE DUAL

MODEL

Consider the following Mond-Weir type higher order

dual program of the problem (KP):

(MPD) K-maximize

(u) + u

(u) − u

, ...,

(u) + u

(u) − u

subject to

∑

i=1

∇

(u) + u

(u) − u

∑

j=1

∇(ϕ

(u)+u

)

∑

i=1

∇

(u, p

) +

∑

j=1

∇

(u, q

) = 0. (1)

∑

j=1

(ϕ

(u)+u

)+G

(u, q

)−q

∇

(u, q

)

≥ 0.

(2)

∑

i=1

(u, p

) − p

∇

(u, p

)

≥ 0. (3)

∈ C

, v

∈ D

, w

∈ M

, (i = 1, 2, ..., k,

j = 1, 2, ..., m),

λ ∈ intK

∗

, µ ∈ intQ

∗

, (λ, µ) 6= (0, 0).

Remark 3.1. If we consider K = R

and Q = R

the above discussed Mond-Weir model reduces to the

model studied in (Suneja et al., 2008).

Next, we will prove duality results between (KP) and

(MPD).

Theorem 3.1 (Weak Duality Theorem). Con-

sider x be a member of the feasible set for (KP) and

(u, v, w, λ, µ, z, p, q) belongs to the feasible set for

(MPD). Suppose that

(i)

P(.) + (.)

Q(.) − (.)

, R(.) + (.)

is higher order (K ×

Q)-F-type I at u in respect with functions H and

G, where H : X × R

→ R

and G : X × R

→ R

are differentiable functions,

(ii) R

⊆ K and R

⊆ Q.

Then



(u) + u

(u) − u

, ...,

(u) + u

(u) − u



−



(x) + τ(x|C

)

(x) − τ(x|D

)

, ...,

(x) + τ(x|C

)

(x) − τ(x|D

)



6∈ K\{0}.

Proof. We will prove the result by contradic-

tion. Suppose,



(u) + u

(u) − u

, ...,

(u) + u

(u) − u



−



(x) + τ(x|C

)

(x) − τ(x|D

)

, ...,

(x) + τ(x|C

)

(x) − τ(x|D

)



∈ K\{0}.

Since λ ∈ intK

∗

, we have

∑

i=1

h

(u) + u

(v) − u



−



(x) + τ(x|C

)

(x) − τ(x|D

)

i

> 0.

(4)

Now, since x

≤ τ(x|C

), x

≤ τ(x|D

), we obtain

(x) + τ(x|C

)

(x) − τ(x|D

)

−

(x) + x

(x) − x

≥ 0.

Using hypothesis (ii), we have K

∗

⊆ R

⇒ λ ∈

intK

∗

⊆ intR

which yields λ > 0. Therefore, the

above inequality implies,

∑

i=1



(x) + τ(x|C

)

(x) − τ(x|D

)

−

(x) + x

(x) − x



≥ 0. (5)

Further, on adding (4) and (5), we have

∑

i=1

h

(x) + x

(x) − x



−



(u) + u

(v) − u

i

< 0. (6)

On Duality with Support Functions for a Multiobjective Fractional Programming Problem

117

By assumption (i) (since

P(.) + (.)

Q(.) − (.)

, R(.)+(.)

is higher order (K × Q)-F-type I at u in respect with

functions H and G), we thus have



(x) + x

(x) − x

−

(u) + u

(u) − u

− F



x, u;

∇



(u) + u

(u) − u



+ ∇H

(u, p

)



− H

(u, p

) +

∇

(u, p

), ...,

(x) + x

(x) − x

−

(u) + u

(u) − u

−



x, u;∇



(u) + u

(u) − u



+ ∇H

(u, p

)



− H

(u, p

)

∇

(u, p

)



∈ K, (7)

and



− ϕ

(u) − u

− F



x, u;∇(ϕ

(u) + u

)

+ ∇

(u, q

)



− G

(u, q

) + q

∇

(u, q

..., −ϕ

(u) − u

− F



x, u;∇(ϕ

(u) + u

) +

∇

(u, q

)



− G

(u, q

)

∇

(u, q

)



∈ Q. (8)

From (7) and λ ∈ intK

∗

, we obtain

∑

i=1

(x) + x

(x) − x

−

(u) + u

(u) − u

− F



x, u;∇



(u) + u

(u) − u



+ ∇

(u, p

)



− H

(u, p

) + p

∇

(u, p

)

≥ 0.

Using λ > 0 and the fact that F is sublinear, we

get

∑

i=1



(x) + x

(x) − x

−

(u) + u

(u) − u



≥

∑

i=1

(u, p

) − p

∇

(u, p

)



x, u;

∑

i=1



∇



(u) + u

(u) − u



+ ∇

(u, p

)



(9)

Again from (8) and µ ∈ intQ

∗

, we have

∑

j=1



−(ϕ

(u)+u

)−F(x, u;∇(ϕ

(u)+u

∇

(u, q

)) − G

(u, q

) + q

∇

(u, q

)



≥ 0.

Now, by assumption (ii), Q

∗

⊆ R

⇒ µ ∈ intQ

∗

⊆

intR

which yields µ > 0. Therefore, the above

inequality and the sublinearity of F together imply,

∑

j=1

− (ϕ

(u) + u

) − G

(u, q

)

+ q

∇

(u, q

)

≥ F



x, u;

∑

j=1



∇(ϕ

(u)+ u

)+ ∇

(u, q

)



(10)

It follows from the sublinearity of F, (9) and (10) that

∑

i=1

(x) + x

(x) − x

−



(u) + u

(u) − u

i

≥

∑

i=1

(u, p

) − p

∇

(u, p

)

x, u;

∑

i=1



∇



(u) + u

(u) − u



+ ∇

(u, p

)



∑

j=1



∇(ϕ

(u) + u

) + ∇

(u, q

)

i

−

∑

j=1

− (ϕ

(u) + u

) − G

(u, q

) +

∇

(u, q

)

Finally, using dual constraint (1)-(3), we get

∑

i=1

(x) + x

(x) − x

−



(u) + u

(u) − u

i

≥ 0,

which contradicts (6). Hence the result. 

Deﬁnition 3.1 (Clarke, 1983). The function

g : R

→ R will be called a locally Lipschitz at

∈ R

if ∃ k ≥ 0 and a neighbourhood δ(x

) of x

||g(x) − g(y)|| ≤ k||x− y||, ∀ x, y ∈ δ(x

Deﬁnition 3.2 (Husain and Zabeen, 2005). A locally

Lipschitz function g : R

× R

→ R at x

∈ R

in the

direction t ∈ R

is said to have generalized directional

derivative if

′

;t) = lim

y→x

sup

p↓0

g(y+ pt) − g(y)

where y ∈ R

and p > 0.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

118

Deﬁnition 3.3 (Wang, 2005). A function g which is

also locally Lipschitz at x

∈ R

is said to have gener-

alized gradient or the subdifferential if

∂g(x

) = {ζ ∈ R

: g

′

;t) ≥ hζ,ti, ∀ t ∈ R

Remark 3.2.

(i) For a convex function φ, it will be called locally

Lipschitz at x

∈ R

, if

∂φ(x

) = {ζ ∈ R

: φ(x) − φ(x

) ≥ (x− x

)

ζ,

∀ x ∈ R

(ii) If φ at x

is continuously differentiable, then φ

at x

will be locally Lipschitz, and therefore,

∂φ(x

) = {∇φ(x

)}.

Lemma 3.1. Let us deal with the problem:

(P) K − minimize θ(x)

subject to − ω(x) ∈ Q,

where θ : R

→ R

and ω : R

→ R

are locally Lips-

chitz functions. For this problem, suppose ¯x be weak

efﬁcient solution, then ∃ (0, 0) 6= (λ, µ) ∈ intK

∗

intQ

∗

in such a way that

0 ∈ ∂

(λ

θ+ µ

ω)( ¯x), (µ

ω)( ¯x) = 0.

Proof. It follows on the lines of (Craven, 1989) and

(Wang et al., 2008). 

Proposition 3.1 (Husain and Zabeen, 2005).

Let us suppose that φ

: R

→ R and φ

: R

→ R at

∈ R

be locally Lipschitz with φ

(x) 6= 0. Then

will be locally Lipschitz at x

∈ R

With the help of the above lemma and the proposi-

tion, we will now prove the following result, between

(KP) and (MPD).

Theorem 3.2 (Strong Duality). If ¯x is a weak efﬁ-

cient of (KP), then ∃ (0, 0) 6= (

λ, ¯µ) ∈ intK

∗

× intQ

∗

¯z

∈ C

, v

∈ D

, w

∈ M

; i belongs to the index set

{1, 2, ..., k}, and j belongs to {1, 2, ..., m} in such

a way that, ( ¯x, ¯v, ¯w,

λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) belongs

to the feasible set for (MPD) and their respective

objective functions possesses same values pro-

vided H( ¯x, 0) = 0, G( ¯x, 0) = 0, ∇

( ¯x, 0) = 0,

∇

( ¯x, 0) = 0, i = 1, 2, ..., k, j = 1, 2, ..., m.

Moreover, if the supposition of Theorem

3.1 are fulﬁlled for every feasible solution x

of (KP) and (u, v, w, λ, µ, z, p, q) of (MPD), then

( ¯x, ¯v, ¯w,

λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) is an efﬁcient for (MPD).

Proof. Let ¯x ∈ X

be a weak efﬁcient of (KP)

and suppose that θ : R

→ R

, ϕ : R

→ R

be taken

θ(x) =



(x) + τ(x|C

)

(x) − τ(x|D

)

, ...,

(x) + τ(x|C

)

(x) − τ(x|D

)



and ω(x) =



(x)+ τ(x|M

), ..., ϕ

(x)+ τ(x|M

)



The functions τ(x|C

), τ(x|D

) and τ(x|M

(i = 1, 2, ..., k, j = 1, 2, ..., m) are locally Lips-

chitz, since each of them are convex. Also, φ , ψ and

ϕ are functions which are continuously differentiable,

hence the above functions are also locally Lipschitz

and as a result, φ

(x) + τ(x|C

), ψ

(x) − τ(x|D

), i

takes values from 1, 2, ..., k and ϕ

(x) + τ(x|M

), j

takes values from 1, 2, ..., m are locally Lipschitz.

Using the Proposition 3.1, we thus conclude

that θ(x) and ω(x) are also locally Lipschitz.

Following the Lemma 3.1, ∃

λ ∈ intK

∗

and ¯µ ∈ intQ

∗

(

λ, ¯µ) not equal to (0, 0) in such a way that

0 ∈ ∂

∑

i=1



( ¯x) + τ( ¯x|C

)

( ¯x) − τ( ¯x|D

)



∑

j=1

¯µ



( ¯x)+

τ(¯x|M

)

i

and

∑

j=1

¯µ



( ¯x) + τ( ¯x|M

)



= 0,

which implies

0 ∈

∑

i=1



∂



( ¯x) + τ( ¯x|C

)

( ¯x) − τ( ¯x|D

)



∑

j=1

¯µ

(∇ϕ

( ¯x))

∑

j=1

¯µ

∂

S( ¯x|M

As we know that the support functions are convex, we

have

∂

τ(¯x|C

) = ∂τ( ¯x|C

), ∂

τ(¯x|D

) = ∂τ( ¯x|D

) and

∂

τ(¯x|M

) = ∂τ( ¯x|M

)

Therefore, there exist ¯z

∈ ∂τ( ¯x|C

), ¯v

∈ ∂τ( ¯x|D

) and

¯w

∈ ∂τ( ¯x|M

) just as if

¯x

¯z

= τ( ¯x|C

), ¯x

¯v

= τ( ¯x|D

) and ¯x

¯w

= τ( ¯x|M

(11)

Hence,

∑

i=1



∇



( ¯x) + ¯x

¯z

( ¯x) − ¯x

¯v



∑

j=1

¯µ

(∇ϕ

( ¯x)+ ¯x

¯w

) = 0,

and

∑

j=1

¯µ

(∇ϕ

( ¯x) + ¯x

¯w

) = 0.

On Duality with Support Functions for a Multiobjective Fractional Programming Problem

119

Using H(¯x, 0) = G( ¯x, 0) = 0, ∇

( ¯x, 0) = 0 and

∇

( ¯x, 0) = 0, (i = 1, 2, ..., k, j = 1, 2, ..., m), we

obtain (¯x, ¯v, ¯w,

λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) belongs to the

domain feasible for (MPD) and also the respective

values of the objectives are equivalent.

We now claim that for (MPD) ( ¯x, ¯v, ¯w,

λ, ¯µ, ¯z, ¯p =

0, ¯q = 0) is efﬁcient.

On the contrary, let us assume that

( ¯x, ¯v, ¯w,

λ, ¯µ, ¯z, ¯p = 0, ¯q = 0) be efﬁcient for (MPD),

therefore ∃ (u, v, w, λ, µ, z, p, q), which is in the feasible

domain for (MPD) such that

(u)+ u

(u)− u

, ...,

(u)+ u

(u)− u

−

( ¯x) + ¯x

¯z

( ¯x) − ¯x

¯v

, ...,

( ¯x) + ¯x

¯z

( ¯x) − ¯x

¯v

∈ K\{0}

which using (11) imply

(u)+ u

(u)− u

, ...,

(u)+ u

(u)− u

−

( ¯x) + τ( ¯x|C

)

( ¯x) − τ( ¯x|D

)

, ...,

( ¯x) + τ( ¯x|C

)

( ¯x) − τ( ¯x|D

)

∈ K\{0},

a contradiction to the Theorem 3.1. Therefore,

the required result. 

4 CONCLUSIONS

In this paper, we have presented a current class of

higher order (K × Q)- F-type I function. A Mond-

Weir type higher order multiobjective fractional prob-

lem (which is also nondiifferentiable) over cone has

been constructed. Considering this dual program, we

have established the corresponding duality relation un-

der higher order (K × Q)- F-type I function. The re-

sults which we have put forward in this paper are ex-

tension of some previously studied results appearing in

the literature. It is to be noted that, researchers can

further extend our work for different types of duality

problems for fractional problems, such as, mixed type

duality etc.

ACKNOWLEDGEMENTS

The ﬁrst author is grateful to the Ministry of Human

Resource and Development, India for ﬁnancial support,

to carry out this work.

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