Advanced
OPTIMALITY AND DUALITY IN NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMMING USING α-UNIVEXITY
OPTIMALITY AND DUALITY IN NONDIFFERENTIABLE MULTIOBJECTIVE FRACTIONAL PROGRAMMING USING α-UNIVEXITY
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 359-375
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : September 10, 2013
  • Accepted : December 24, 2014
  • Published : September 28, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
REKHA GUPTA
MANJARI SRIVASTAVA

Abstract
In this paper, a multiobjective nondifferentiable fractional programming problem (MFP) is considered where the objective function contains a term involving the support function of a compact convex set. A vector valued (generalized) α -univex function is defined to extend the concept of a real valued (generalized) α -univex function. Using these functions, sufficient optimality criteria are obtained for a feasible solution of (MFP) to be an efficient or weakly efficient solution of (MFP). Duality results are obtained for a Mond-Weir type dual under (generalized) α -univexity assumptions. AMS Mathematics Subject Classification : 90C30, 90C32, 90C46.
Keywords
1. Introduction
Most of the real world problems which arise in the areas of portfolio selection, stock cutting, game theory and many decision making problems in management science etc. are (multiobjective) fractional programming problems. Extensive researches have been reported in the literature for the multiobjective nonlinear (nondifferentiable) fractional programming problems involving generalized convex functions by various authors, for details see ( [1 , 4 , 6 - 13 , 15 , 18 - 21] ) and references therein. The areas which have been explored are mainly to weaken the convexity and to relax the differentiability assumption of the functions used in developing optimality and duality of the above programming problems. Bector et al. [3] introduced univex functions by relaxing the definition of an invex function and obtained optimality and duality results for a nonlinear programming problem. Jayswal [8] defined α -univexity and its generalizations for a real valued function and proved duality theorems for a nondifferentiable generalized fractional programming problem.
Different authors have used different forms of nondifferentiability to obtain optimality conditions and duality theory for fractional programming problem under generalized convexity assumptions. Authors like Mond [15] , Singh [19] , Zhang and Mond [21] and in the references cited therein considered a class of nondifferentiable fractional programming problems containing square root terms in the objective function and derived optimality criteria and discussed duality theory. Non smooth optimization involves functions for which subderivatives exist [5] . Square root of a positive semidefinite quadratic form is one of the few types of a nondifferentiable function whose subdifferential can be written explicitly. Square root of a quadratic form can be replaced by a more general function, namely, the support function of a compact convex set, whose subdifferential can be simply expressed. For these considerations Mond and Schechter [16] considered programs which contain support function in objective function and studied symmetric duality. Kim et al. [9] established necessary and sufficient optimality conditions and proved duality results for weakly efficient solutions of multiobjective fractional programming problem containing support functions under the assumption of ( V, ρ ) invex functions. Later in [10] , Kim et al. established duality results using ( V, ρ ) invexity for the same problem with cone constraints.
Motivated by the above researches, in this paper, we consider a nondifferentiable multiobjective fractional programming problem (MFP) over cones with objective function containing support function of a compact convex set. We introduce the concept of α -univexity and its various generalizations for a vector valued function. This generalizes the concept of α -univexity for a scalar valued function [8] . We also give the examples to show the existence of above defined classes of functions. Sufficient optimality conditions for a (weakly) efficient solution of (MFP) are derived using these newly defined classes of (generalized) α -univex functions. A Mond-Weir type dual is proposed for (MFP) and standard duality theorems are proved assuming the functions to be (generalized) α -univex.
2. Preliminaries
Let Rn be the n -dimensional Euclidean space and let
PPT Slide
Lager Image
be its non negative orthant. The following convention for inequalities will be used in this paper. If x, u Rn , then
PPT Slide
Lager Image
Note : For x, u R , we use x u to denote x is less than or equal to u .
Definition 2.1 ( [17] ). A non empty set C in Rn is said to be a cone if for each x C and λ ≥ 0, λx C . If in addition C is convex then C is called a convex cone.
Definition 2.2 ( [17] ). Let C Rn be a cone. The set
PPT Slide
Lager Image
is called the polar cone of C .
We now consider the following nondifferentiable multiobjective fractional programming problem:
PPT Slide
Lager Image
subject to
PPT Slide
Lager Image
where ƒ : X Rk, g : X Rk and h : X Rm are continuously differentiable functions over an open subset X of Rn . C 1 X and C 2 are closed convex cones with non empty interiors in Rn and Rm respectively, Di ( i = 1, 2, . . . , k ) are compact convex sets in Rn and s ( x | Di ) = max {< x, y > | y Di } denotes the support function of Di . Let
PPT Slide
Lager Image
be the set of all feasible solutions of (MFP) and
PPT Slide
Lager Image
For any w = ( w 1 , w 2 , . . . , wk ) ∈ Rn × Rn × . . . × Rn and x Rn , xTw = ( xTw 1 , xTw 2 , . . . , xTwk ).
We now review some known facts about support functions. The support function s ( x | C ) of compact convex set C Rn , being convex and everywhere finite, has a subgradient at every x in the sense of Rockafellar [17] , that is, there exists z C such that
PPT Slide
Lager Image
Equivalently,
PPT Slide
Lager Image
The subdifferential of s ( x | C ) is given by
PPT Slide
Lager Image
For any set D Rn , the normal cone to D at any point x D is defined by
PPT Slide
Lager Image
If C is a compact convex set then y NC ( x ) iff
PPT Slide
Lager Image
or equivalently x ∂s ( y | C ).
Definition 2.3. A feasible solution
PPT Slide
Lager Image
is said to be a weakly efficient solution of (MFP) if there does not exist any x X 0 such that
PPT Slide
Lager Image
Definition 2.4. A feasible solution
PPT Slide
Lager Image
is said to be an efficient solution of (MFP) if there does not exist any x X 0 such that
PPT Slide
Lager Image
We recall the definition of α-univexity for a differentiable real valued function ƒ : X R .
Definition 2.5 ( [8] ). The function f is said to be α- univex
PPT Slide
Lager Image
with respect to α : X × X → R+ \ {0}, b : X × X R + , ϕ : R R and η : X × X Rn if for every x X , we have
PPT Slide
Lager Image
Now to extend the above concept of α -univexity to multiobjective programming we give the following definitions for a vector valued differentiable function ƒ : X Rk . Assume that
PPT Slide
Lager Image
and η : X × X Rn .
Definition 2.6. The function ƒ : X Rk is said to be α - univex at
PPT Slide
Lager Image
with respect to α, b, ϕ , and η if for every x X and for each i = 1, 2, . . . , k , we have
PPT Slide
Lager Image
If (2.1) is a strict inequality for all
PPT Slide
Lager Image
then ƒ is said to be strict α -univex function.
Definition 2.7. The function ƒ : X Rk is said to be pseudo α - univex at
PPT Slide
Lager Image
with respect to α, b, ϕ , and η if for every x X and for each i = 1, 2, . . . , k , we have
PPT Slide
Lager Image
Definition 2.8. The function ƒ : X Rk is said to be strict pseudo α - univex at
PPT Slide
Lager Image
with respect to α, b, ϕ , and η if for every
PPT Slide
Lager Image
and for each i = 1, 2, . . . , k , we have
PPT Slide
Lager Image
Definition 2.9. The function ƒ : X Rk is said to be quasi α - univex at
PPT Slide
Lager Image
with respect to α, b, ϕ , and η if for every x X and for each i = 1, 2, . . . , k , we have
PPT Slide
Lager Image
ƒ is said to be (strict) α -univex, (strict) pseudo α -univex and quasi α -univex on X if it is (strict) α -univex, (strict) pseudo α -univex and quasi α -univex respectively at every x X .
We now give the following example to show the existence of vector valued α -univex function.
Example 2.10. Let X = ]0, 1[ and ƒ : X R 2 is given by
PPT Slide
Lager Image
Also let,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Then ƒ is α -univex on X with respect to α, b, ϕ and η .
We note that every α-univex function is pseudo α -univex as well as quasi α -univex but the converse is not true. To illustrate this fact, we give the following examples of pseudo α -univex and quasi α -univex functions which are not α -univex.
Example 2.11. Let
PPT Slide
Lager Image
and ƒ : X R 2 is given by
PPT Slide
Lager Image
Also let, ϕ ( x ) = 2 x , η ( x, u ) = u x, α ( x, u ) = x 2 + u
PPT Slide
Lager Image
Then ƒ is pseudo α -univex on X with respect to α, b, ϕ and η but it is not α -univex on X because for
PPT Slide
Lager Image
PPT Slide
Lager Image
Example 2.12. Let
PPT Slide
Lager Image
and ƒ : X R 2 is given by
PPT Slide
Lager Image
Also let, ϕ ( x ) = 2 x , α ( x, u ) = x + u ,
PPT Slide
Lager Image
Then ƒ is quasi α -univex on X with respect to α, b, ϕ and η but it is not α -univex on X because for
PPT Slide
Lager Image
PPT Slide
Lager Image
Now, we give the following lemma.
Lemma 2.13. Assume that ƒ and g are differentiable functions defined from X to Rk, where X is an open subset of Rn and g ( x ) > 0 for all x X. If for w = ( w 1 , w 2 , . . . , wk ) ∈ Rn × Rn ×. . .× Rn , ƒ (·)+(·) Tw and g (·) are α-univex at
PPT Slide
Lager Image
with respect to α, b, ϕ and η and ϕ is linear, then
PPT Slide
Lager Image
is α-univex at
PPT Slide
Lager Image
and η,where
PPT Slide
Lager Image
Proof . Consider for each i = 1, 2, . . . , k and x X ,
PPT Slide
Lager Image
As ϕ is linear, we have that
PPT Slide
Lager Image
Since ƒ (·)+(·) Tw and − g (·) are α -univex at
PPT Slide
Lager Image
with respect to α, b, ϕ and η , we have for each i = 1, 2, . . . , k ,
PPT Slide
Lager Image
Therefore,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Therefore,
PPT Slide
Lager Image
is α -univex at
PPT Slide
Lager Image
Remark 2.1. The following are satisfied:
  • (1) If in Lemma 2.13 the functionsƒ(·)+(·)Twand −g(·) are assumed to be strictα-univex andα-univex respectively atwith respect toα, b, ϕandη, then moving on the similar lines as in Lemma 2.13, it can be shown thatis strictα-univex atwith respect toandη.
  • (2) Since everyα-univex function is pseudoα-univex, therefore assuming all the conditions of Lemma 2.13,is also pseudoα-univex at.
  • (3) Again by using the fact that every strictα-univex function is strict pseudoα-univex, the functionis strict pseudoα-univex at x if the functionsƒ(·) + (·)Twand −gg(·) are assumed to be strictα-univex andα-univex respectively at x in Lemma 2.13.
We now give an example which illustrates the above Lemma 2.13 and Remark 2.14(2).
Example 2.14. Let X =] − 1, 1[ and ƒ : X R 2 , g : X R 2 are defined by
PPT Slide
Lager Image
Also let,
PPT Slide
Lager Image
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
is α -univex and hence pseudo α -univex at
PPT Slide
Lager Image
with respect to
PPT Slide
Lager Image
as ƒ (·) + (·) Tw and − g (·) are α -univex at
PPT Slide
Lager Image
with respect to α, b, ϕ and η , where
PPT Slide
Lager Image
3. Optimality Conditions
The following lemma giving necessary optimality conditions will be used in the sequel. The lemma is cited in [10] and can be obtained from [2] and [9] .
Lemma 3.1. ( Necessary Optimality Conditions ) Let
PPT Slide
Lager Image
be a weakly efficient solution of ( MFP ) at which a suitable constraint qualification [14] be satisfied, then there exist
PPT Slide
Lager Image
and
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
We now establish some sufficient optimality conditions for
PPT Slide
Lager Image
to be a (weakly) efficient solution of (MFP) under (generalized) α -univexity defined in the previous section.
Theorem 3.2. Let
PPT Slide
Lager Image
be a feasible solution of ( MFP ) and that there exist
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Further assume that all the conditions of Lemma 2.13 are satisfied at
PPT Slide
Lager Image
and a < 0 ⇒ ϕ ( a ) < 0. Also if any one of the following conditions hold:
  • (a)is α-univex atwith respect to α,b0,ϕ0andηwithand ϕ0(a) > 0 ⇒a> 0,
  • (b)is quasi α-univex atwith respect to α,b0,ϕ0and η with a≤ 0 ⇒ϕ0(a) ≤ 0,
then
PPT Slide
Lager Image
is a weakly efficient solution of ( MFP ).
Proof . Suppose that
PPT Slide
Lager Image
is not a weakly efficient solution of (MFP). Then there exists some x X 0 such that
PPT Slide
Lager Image
Using the fact that
PPT Slide
Lager Image
we have for each i = 1, 2, . . . , k that
PPT Slide
Lager Image
Now assume that ( a ) holds. By Lemma 2.13,
PPT Slide
Lager Image
is α -univex at
PPT Slide
Lager Image
with respect to
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Since a < 0 ⇒ ϕ ( a ) < 0 and
PPT Slide
Lager Image
(3.7) gives
PPT Slide
Lager Image
Using the definition of α -univexity of
PPT Slide
Lager Image
, (3.8) implies
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
therefore multiplying each of the above inequalities by
PPT Slide
Lager Image
and summing over i = 1, 2, . . . , k , we get that
PPT Slide
Lager Image
As (3.4) holds for all x Rn , we have
PPT Slide
Lager Image
Now using (3.10) in (3.9) we get that
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
is α -univex at
PPT Slide
Lager Image
, the above inequality implies
PPT Slide
Lager Image
Using (3.5), we get
PPT Slide
Lager Image
As
PPT Slide
Lager Image
and ϕ 0 ( a ) > 0 ⇒ a > 0, we have
PPT Slide
Lager Image
But as x is feasible for (MFP) and
PPT Slide
Lager Image
we get that
PPT Slide
Lager Image
which contradicts (3.11). Hence
PPT Slide
Lager Image
is a weakly efficient solution of (MFP).
Assume that ( b ) holds. Using Remark 2.14(2), we have that
PPT Slide
Lager Image
is pseudo α -univex at
PPT Slide
Lager Image
with respect to
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Also as a < 0 ⇒ ϕ ( a ) < 0, (3.7) gives
PPT Slide
Lager Image
Now
PPT Slide
Lager Image
being pseudo α -univex at
PPT Slide
Lager Image
, (3.12) implies
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
therefore multiplying each of the above inequalities by
PPT Slide
Lager Image
and summing over i = 1, 2, . . . , k , we get
PPT Slide
Lager Image
As (3.4) holds for all x Rn , we have
PPT Slide
Lager Image
Because x is feasible for (MFP) and
PPT Slide
Lager Image
we get that
PPT Slide
Lager Image
Using ( b ), (3.5) and above inequality, we obtain
PPT Slide
Lager Image
As
PPT Slide
Lager Image
is quasi α -univex at
PPT Slide
Lager Image
, we obtain from above inequality that
PPT Slide
Lager Image
Adding (3.13) and (3.15), we get that
PPT Slide
Lager Image
which contradicts (3.14). Hence
PPT Slide
Lager Image
is a weakly efficient solution of (MFP).
Theorem 3.3. Let
PPT Slide
Lager Image
be a feasible solution of ( MFP ) and that there exist
PPT Slide
Lager Image
such that the conditions (3.4) - (3.6) hold. Assume that all the conditions of Lemma 2.13 are satisfied at
PPT Slide
Lager Image
for all i = 1, 2, . . . , k and a < 0 ⇒ ϕ ( a ) < 0. Also assume that
PPT Slide
Lager Image
is α-univex at
PPT Slide
Lager Image
with respect to α , b 0 , ϕ 0 and η with
PPT Slide
Lager Image
and ϕ 0 ( a ) > 0 ⇒ a > 0. Then
PPT Slide
Lager Image
is an efficient solution of ( MFP ).
Proof . Suppose that
PPT Slide
Lager Image
is not an efficient solution of (MFP). Then there exists some x X 0 such that
PPT Slide
Lager Image
Using (3.6) and the fact that
PPT Slide
Lager Image
we have for all i = 1, 2, . . . , k , i j that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
That is,
PPT Slide
Lager Image
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
and − g (·) are α -univex at
PPT Slide
Lager Image
with respect to α, b, ϕ and η , therefore by Lemma 2.13,
PPT Slide
Lager Image
is α -univex at
PPT Slide
Lager Image
with respect to
PPT Slide
Lager Image
where
PPT Slide
Lager Image
for all i = 1, 2, . . . , k . Using the assumption that a < 0 ⇒ ϕ ( a ) < 0 where ϕ is linear, (3.16) and (3.17) give
PPT Slide
Lager Image
Using α -univexity of
PPT Slide
Lager Image
in last two inequalities, we get
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
therefore multiplying each of the above inequalities by
PPT Slide
Lager Image
and summing over i = 1, 2, . . . , k , we get
PPT Slide
Lager Image
Rest of the proof follows on the lines of proof of part (a) of Theorem 3.2.
Theorem 3.4. Let
PPT Slide
Lager Image
be a feasible solution of ( MFP ) and that there exist Let
PPT Slide
Lager Image
such that the conditions (3.4) - (3.6) hold. Assume that all the conditions of Lemma 2.13 are satisfied at
PPT Slide
Lager Image
being strict α-univex at
PPT Slide
Lager Image
. Further assume that
PPT Slide
Lager Image
is quasi α-univex at
PPT Slide
Lager Image
with respect to α , b 0 , ϕ 0 and η . Also let a ≤ 0 ⇒ ϕ ( a ) ≤ 0 and a ≤ 0 ⇒ ϕ 0 (a) ≤ 0. Then x is an efficient solution of ( MFP ).
Proof . Suppose that
PPT Slide
Lager Image
is not an efficient solution of (MFP). Then there exists some x X 0 such that
PPT Slide
Lager Image
Using (3.6) and the fact that
PPT Slide
Lager Image
we have for all i = 1, 2, . . . , k, i j that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
That is,
PPT Slide
Lager Image
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
is strict α -univex and − g (·) is α -univex at
PPT Slide
Lager Image
with respect to α, b, ϕ and η , therefore by Remark 2.14(3),
PPT Slide
Lager Image
is strict pseudo α -univex at
PPT Slide
Lager Image
with respect to
PPT Slide
Lager Image
where
PPT Slide
Lager Image
for all i = 1, 2, . . . , k . Thus by using the assumption that a ≤ 0 ⇒ ϕ ( a ) ≤ 0, (3.18) and (3.19) give
PPT Slide
Lager Image
Using the definition of strict pseudo α -univexity of
PPT Slide
Lager Image
, (3.20) implies
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
therefore multiplying each of the above inequalities by
PPT Slide
Lager Image
and summing over i = 1, 2, . . . , k , we get
PPT Slide
Lager Image
Rest of the proof follows on the lines of proof of part (b) of Theorem 3.2.
4. Duality
Now we consider the following Mond-Weir type dual of (MFP).
(MFD) Maximize
PPT Slide
Lager Image
subject to
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
where wi ( i = 1, 2, . . . , k ) is a vector in Rn and uTw = ( uTw 1 , . . . , uTwk ).
Theorem 4.1. ( Weak Duality ) Let x be feasible for ( MFP ) and ( u, y, λ,w ) be feasible for ( MFD ). Assume that
  • (a)ƒ(·) + (·)Twand−g(·)are α-univex at u with respect to α, b, ϕ, η with ϕ linear andyTh(·)+vT(·)is α-univex at u with respect to α,b0,ϕ0and η for all
  • (b)a≤ 0 ⇒ϕ0(a) ≤ 0,a< 0 ⇒ϕ(a) < 0andbi(x, u) > 0for all i= 1, 2, . . . ,k,
then
PPT Slide
Lager Image
Proof . Let us suppose on the contrary that
PPT Slide
Lager Image
that is,
PPT Slide
Lager Image
Using the fact that s ( x | Di ) ≥ xTwi for all i = 1, 2, . . . , k , we get that
PPT Slide
Lager Image
By ( a ) as
PPT Slide
Lager Image
and − g (·) are α -univex at u with respect to α, b, ϕ and η , therefore by Lemma 2.13,
PPT Slide
Lager Image
is α -univex at u with respect to
PPT Slide
Lager Image
where
PPT Slide
Lager Image
for all i = 1, 2, . . . , k . Thus by using the assumption that a < 0 ⇒ ϕ ( a ) < 0, (4.4) gives
PPT Slide
Lager Image
Now as
PPT Slide
Lager Image
is α - univex at u , we get
PPT Slide
Lager Image
Since λ ≥ 0 by (4.3), therefore multiplying above inequality for each i = 1, 2, . . . , k by λi and summing over i = 1, 2, . . . , k , we get that
PPT Slide
Lager Image
From the dual constraint (4.1), we have
PPT Slide
Lager Image
therefore there exist
PPT Slide
Lager Image
such that
PPT Slide
Lager Image
On using (4.6) in (4.2), we obtain
PPT Slide
Lager Image
Since x is feasible for (MFP), y C 2 and
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
therefore (4.7) and (4.8) together give