Second-order duality for a nondifferentiable minimax fractional programming under generalized α-univexity

Open AccessResearch

DOI: 10.1186/1029-242X-2012-187

Cite this article as:
Gupta, S., Dangar, D. & Kumar, S. J Inequal Appl (2012) 2012: 187. doi:10.1186/1029-242X-2012-187

Abstract

In this paper, we concentrate our study to derive appropriate duality theorems for two types of second-order dual models of a nondifferentiable minimax fractional programming problem involving second-order α-univex functions. Examples to show the existence of α-univex functions have also been illustrated. Several known results including many recent works are obtained as special cases.

MSC:49J35, 90C32, 49N15.

Keywords

minimax programmingfractional programmingnondifferentiable programmingsecond-order dualityα-univexity

1 Introduction

After Schmitendorf [1], who derived necessary and sufficient optimality conditions for static minimax problems, much attention has been paid to optimality conditions and duality theorems for minimax fractional programming problems [217]. For the theory, algorithms, and applications of some minimax problems, the reader is referred to [18].

In this paper, we consider the following nondifferentiable minimax fractional programming problem:

where Y is a compact subset of R l Open image in new window, f ( , ) : R n × R l R Open image in new window, h ( , ) : R n × R l R Open image in new window are twice continuously differentiable on R n × R l Open image in new window and g ( ) : R n R m Open image in new window is twice continuously differentiable on R n Open image in new window, B, and D are a n × n Open image in new window positive semidefinite matrix, f ( x , y ) + ( x T B x ) 1 / 2 0 Open image in new window, and h ( x , y ) ( x T D x ) 1 / 2 > 0 Open image in new window for each ( x , y ) J × Y Open image in new window, where J = { x R n : g ( x ) 0 } Open image in new window.

Motivated by [7, 14, 15], Yang and Hou [17] formulated a dual model for fractional minimax programming problem and proved duality theorems under generalized convex functions. Ahmad and Husain [5] extended this model to nondifferentiable and obtained duality relations involving ( F , α , ρ , d ) Open image in new window-pseudoconvex functions. Jayswal [11] studied duality theorems for another two duals of (P) under α-univex functions. Recently, Ahmad et al.[4] derived the sufficient optimality condition for (P) and established duality relations for its dual problem under B - ( p , r ) Open image in new window-invexity assumptions. The papers [2, 47, 1115, 17] involved the study of first-order duality for minimax fractional programming problems.

The concept of second-order duality in nonlinear programming problems was first introduced by Mangasarian [19]. One significant practical application of second-order dual over first-order is that it may provide tighter bounds for the value of objective function because there are more parameters involved. Hanson [20] has shown the other advantage of second-order duality by citing an example, that is, if a feasible point of the primal is given and first-order duality conditions do not apply (infeasible), then we may use second-order duality to provide a lower bound for the value of primal problem.

Recently, several researchers [3, 810, 16] considered second-order dual for minimax fractional programming problems. Husain et al.[8] first formulated second-order dual models for a minimax fractional programming problem and established duality relations involving η-bonvex functions. This work was later on generalized in [10] by introducing an additional vector r to the dual models, and in Sharma and Gulati [16] by proving the results under second-order generalized α-type I univex functions. The work cited in [3, 8, 10, 16] involves differentiable minimax fractional programming problems. Recently, Hu et al.[9] proved appropriate duality theorems for a second-order dual model of (P) under η-pseudobonvexity/η-quasibonvexity assumptions. In this paper, we formulate two types of second-order dual models for (P) and then derive weak, strong, and strict converse duality theorems under generalized α-univexity assumptions. Further, examples have been illustrated to show the existence of second-order α-univex functions. Our study extends some of the known results of the literature [5, 6, 11, 12, 14].

2 Notations and preliminaries

For each ( x , y ) R n × R l Open image in new window and M = { 1 , 2 , , m } Open image in new window, we define
Definition 2.1 Let ζ : X R Open image in new window ( X R n Open image in new window) be a twice differentiable function. Then ζ is said to be second-order α-univex at u X Open image in new window, if there exist η : X × X R n Open image in new window, b 0 : X × X R + Open image in new window, ϕ 0 : R R Open image in new window, and α : X × X R + { 0 } Open image in new window such that for all x X Open image in new window and p R n Open image in new window, we have
b 0 ϕ 0 [ ζ ( x ) ζ ( u ) + 1 2 p T 2 ζ ( u ) p ] α ( x , u ) η T ( x , u ) [ ζ ( u ) + 2 ζ ( u ) p ] . Open image in new window
Example 2.1 Let ζ : X R Open image in new window be defined as ζ ( x ) = e x + sin 2 x + x 2 Open image in new window, where X = ( 1 , ) Open image in new window. Also, let ϕ 0 ( t ) = t + 18 Open image in new window, b 0 ( x , u ) = u + 1 Open image in new window, α ( x , u ) = u 2 + 2 x + 1 Open image in new window and η ( x , u ) = x + u Open image in new window. The function ζ is second-order α-univex at u = 1 Open image in new window, since
b 0 ϕ 0 [ ζ ( x ) ζ ( u ) + 1 2 p T 2 ζ ( u ) p ] α ( x , u ) η T ( x , u ) [ ζ ( u ) + 2 ζ ( u ) p ] = 2 ( e x + sin 2 x + x 2 ) + 1.521 + 3.886 ( p 1.5 ) 2 0 for all  x X  and  p R . Open image in new window

But every α-univex function need not be invex. To show this, consider the following example.

Example 2.2 Let Ω : X = ( 0 , ) R Open image in new window be defined as Ω ( x ) = x 2 Open image in new window. Let ϕ 0 ( t ) = t Open image in new window, b 0 ( x , u ) = 1 u Open image in new window, α ( x , u ) = 2 u , Open image in new window and η ( x , u ) = 1 2 u Open image in new window. Then we have
b 0 ϕ 0 [ Ω ( x ) Ω ( u ) + 1 2 p T 2 Ω ( u ) p ] α ( x , u ) η T ( x , u ) [ Ω ( u ) + 2 Ω ( u ) p ] = 1 u [ x 2 + ( p + u ) 2 ] 0 for all  x , u X  and  p R . Open image in new window
Hence, the function Ω is second-order α-univex but not invex, since for x = 3 Open image in new window, u = 2 Open image in new window, and p = 1 Open image in new window, we obtain
Ω ( x ) Ω ( u ) + 1 2 p T 2 Ω ( u ) p η T ( x , u ) [ Ω ( u ) + 2 Ω ( u ) p ] = 4.5 < 0 . Open image in new window

Lemma 2.1 (Generalized Schwartz inequality)

LetBbe a positive semidefinite matrix of ordern. Then, for all x , w R n Open image in new window,
x T B w ( x T B x ) 1 / 2 ( w T B w ) 1 / 2 . Open image in new window

The equality holds if B x = λ B w Open image in new windowfor some λ 0 Open image in new window.

Following Theorem 2.1 ([13], Theorem 3.1) will be required to prove the strong duality theorem.

Theorem 2.1 (Necessary condition)

If x Open image in new windowis an optimal solution of problem (P) satisfying x T B x > 0 Open image in new window, x T D x > 0 Open image in new window, and g j ( x ) Open image in new window, j J ( x ) Open image in new windoware linearly independent, then there exist ( s , t , y ˜ ) K ( x ) Open image in new window, k 0 R + Open image in new window, w , v R n Open image in new windowand μ R + m Open image in new windowsuch that
In the above theorem, both matrices B and D are positive semidefinite at x Open image in new window. If either x T B x Open image in new window or x T D x Open image in new window is zero, then the functions involved in the objective of problem (P) are not differentiable. To derive necessary conditions under this situation, for ( s , t , y ˜ ) K ( x ) Open image in new window, we define
Z y ˜ ( x ) = { z R n : z T g j ( x ) 0 , j J ( x ) , with any one of the next conditions (i)-(iii) holds } . Open image in new window

If in addition, we insert the condition Z y ˜ ( x ) = ϕ Open image in new window, then the result of Theorem 2.1 still holds.

For the sake of convenience, let
ψ 1 ( ) = ξ 1 ( ) + j = 1 m μ j ( g j ( ) g j ( z ) ) Open image in new window
(2.6)
and
ψ 2 ( ) = [ i = 1 s t i ( h ( z , y ˜ i ) z T D v ) ] [ i = 1 s t i ( f ( , y ˜ i ) + ( ) T B w ) + j = 1 m μ j g j ( ) ] [ i = 1 s t i ( f ( z , y ˜ i ) + z T B w ) + j = 1 m μ j g j ( z ) ] [ i = 1 s t i ( h ( , y ˜ i ) ( ) T D v ) ] , Open image in new window
where
ξ 1 ( ) = i = 1 s t i [ ( h ( z , y ˜ i ) z T D v ) ( f ( , y ˜ i ) + ( ) T B w ) ( f ( z , y ˜ i ) + z T B w ) ( h ( , y ˜ i ) ( ) T D v ) ] . Open image in new window

3 Model I

In this section, we consider the following second-order dual problem for (P):
max ( s , t , y ˜ ) K ( z ) sup ( z , μ , w , v , p ) H 1 ( s , t , y ˜ ) F ( z ) , Open image in new window
(DM1)
where F ( z ) = sup y Y f ( z , y ) + ( z T B z ) 1 / 2 h ( z , y ) ( z T D z ) 1 / 2 Open image in new window and H 1 ( s , t , y ˜ ) Open image in new window denotes the set of all ( z , μ , w , v , p ) R n × R + m × R n × R n × R n Open image in new window satisfying

If the set H 1 ( s , t , y ˜ ) = ϕ Open image in new window, we define the supremum of F ( z ) Open image in new window over H 1 ( s , t , y ˜ ) Open image in new window equal to −∞.

Remark 3.1 If p = 0 Open image in new window, then using (3.3), the above dual model reduces to the problems studied in [6, 11, 12]. Further, if B and D are zero matrices of order n, then (DM1) becomes the dual model considered in [14].

Next, we establish duality relations between primal (P) and dual (DM1).

Theorem 3.1 (Weak duality)

Letxand ( z , μ , w , v , s , t , y ˜ , p ) Open image in new windoware feasible solutions of (P) and (DM1), respectively. Assume that
  1. (i)

    ψ 1 ( ) Open image in new windowis second-orderα-univex atz,

     
  2. (ii)

    ϕ 0 ( a ) 0 a 0 Open image in new windowand b 0 ( x , z ) > 0 Open image in new window.

     
Then
sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 F ( z ) . Open image in new window
Proof Assume on contrary to the result that
sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 < F ( z ) . Open image in new window
(3.4)
Since y ˜ i Y ( z ) Open image in new window, i = 1 , 2 , , s Open image in new window, we have
F ( z ) = f ( z , y ˜ i ) + ( z T B z ) 1 / 2 h ( z , y ˜ i ) ( z T D z ) 1 / 2 . Open image in new window
(3.5)
From (3.4) and (3.5), for i = 1 , 2 , , s Open image in new window, we get
f ( x , y ˜ i ) + ( x T B x ) 1 / 2 h ( x , y ˜ i ) ( x T D x ) 1 / 2 sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 < f ( z , y ˜ i ) + ( z T B z ) 1 / 2 h ( z , y ˜ i ) ( z T D z ) 1 / 2 . Open image in new window
This further from t i 0 Open image in new window, i = 1 , 2 , , s Open image in new window, t 0 Open image in new window and y ˜ i Y ( z ) Open image in new window, we obtain
Now,
ξ 1 ( x ) = i = 1 s t i [ ( h ( z , y ˜ i ) z T D v ) ( f ( x , y ˜ i ) + x T B w ) ( f ( z , y ˜ i ) + z T B w ) ( h ( x , y ˜ i ) x T D v ) ] i = 1 s t i [ ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) ( f ( x , y ˜ i ) + ( x T B x ) 1 / 2 ) ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) ( h ( x , y ˜ i ) ( x T D x ) 1 / 2 ) ] ( using Lemma 2.1 and (3.3) ) < 0 ( from (3.6) ) . Open image in new window
Therefore,
ξ 1 ( x ) < 0 = ξ 1 ( z ) . Open image in new window
(3.7)
By hypothesis (i), we have
b 0 ϕ 0 [ ψ 1 ( x ) ψ 1 ( z ) + 1 2 p T 2 ψ 1 ( z ) p ] α ( x , z ) η T ( x , z ) { ψ 1 ( z ) + 2 ψ 1 ( z ) p } . Open image in new window
This follows from (3.1) that
b 0 ϕ 0 [ ψ 1 ( x ) ψ 1 ( z ) + 1 2 p T 2 ψ 1 ( z ) p ] 0 Open image in new window
which using hypothesis (ii) yields
ψ 1 ( x ) ψ 1 ( z ) + 1 2 p T 2 ψ 1 ( z ) p 0 . Open image in new window
This further from (2.6), (3.2), and the feasibility of x implies
ξ 1 ( x ) j = 1 m μ j g j ( x ) 0 = ξ 1 ( z ) . Open image in new window

This contradicts (3.7), hence the result. □

Theorem 3.2 (Strong duality)

Let x Open image in new windowbe an optimal solution for (P) and let g j ( x ) Open image in new window, j J ( x ) Open image in new windowbe linearly independent. Then there exist ( s , t , y ˜ ) K ( x ) Open image in new windowand ( x , μ , w , v , p = 0 ) H 1 ( s , t , y ˜ ) Open image in new window, such that ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new windowis feasible solution of (DM1) and the two objectives have same values. If, in addition, the assumptions of Theorem  3.1 hold for all feasible solutions ( x , μ , w , v , s , t , y ˜ , p ) Open image in new windowof (DM1), then ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new windowis an optimal solution of (DM1).

Proof Since x Open image in new window is an optimal solution of (P) and g j ( x ) Open image in new window, j J ( x ) Open image in new window are linearly independent, then by Theorem 2.1, there exist ( s , t , y ˜ ) K ( x ) Open image in new window and ( x , μ , w , v , p = 0 ) H 1 ( s , t , y ˜ ) Open image in new window such that ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new window is feasible solution of (DM1) and the two objectives have same values. Optimality of ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new window for (DM1), thus follows from Theorem 3.1. □

Theorem 3.3 (Strict converse duality)

Let x Open image in new windowbe an optimal solution to (P) and ( z , μ , w , v , s , t , y ˜ , p ) Open image in new windowbe an optimal solution to (DM1). Assume that
  1. (i)

    ψ 1 ( ) Open image in new windowis strictly second-orderα-univex at z Open image in new window,

     
  2. (ii)

    { g j ( x ) , j J ( x ) } Open image in new window, are linearly independent,

     
  3. (iii)

    ϕ 0 ( a ) > 0 a > 0 Open image in new windowand b 0 ( x , z ) > 0 Open image in new window.

     

Then z = x Open image in new window.

Proof By the strict α-univexity of ψ 1 ( ) Open image in new window at z Open image in new window, we get
which in view of (3.1) and hypothesis (iii) give
ψ 1 ( x ) ψ 1 ( z ) + 1 2 p T 2 ψ 1 ( z ) p > 0 . Open image in new window
Using (2.6), (3.2), and feasibility of x Open image in new window in above, we obtain
ξ 1 ( x ) > 0 = ξ 1 ( z ) . Open image in new window
(3.8)
Now, we shall assume that z x Open image in new window and reach a contradiction. Since x Open image in new window and ( z , μ , w , v , s , t , y ˜ , p ) Open image in new window are optimal solutions to (P) and (DM1), respectively, and { g j ( x ) , j J ( x ) } Open image in new window, are linearly independent, by Theorem 3.2, we get
sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 = F ( z ) . Open image in new window
(3.9)
Since y ˜ i Y ( z ) Open image in new window, i = 1 , 2 , , s Open image in new window, we have
F ( z ) = f ( z , y ˜ i ) + ( z T B z ) 1 / 2 h ( z , y ˜ i ) ( z T D z ) 1 / 2 . Open image in new window
(3.10)
By (3.9) and (3.10), we get
[ ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) ( f ( x , y ˜ i ) + ( x T B x ) 1 / 2 ) ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) ( h ( x , y ˜ i ) ( x T D x ) 1 / 2 ) ] 0 , Open image in new window
for all i = 1 , 2 , , s Open image in new window and y ˜ i Y Open image in new window. From y ˜ i Y ( z ) Y Open image in new window and t R + s Open image in new window, with i = 1 s t i = 1 Open image in new window, we obtain
From Lemma 2.1, (3.3), and (3.11), we have
ξ 1 ( x ) = i = 1 s t i [ ( h ( z , y ˜ i ) z T D v ) ( f ( x , y ˜ i ) + x T B w ) ( f ( z , y ˜ i ) + z T B w ) ( h ( x , y ˜ i ) x T D v ) ] i = 1 s t i [ ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) ( f ( x , y ˜ i ) + ( x T B x ) 1 / 2 ) ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) ( h ( x , y ˜ i ) ( x T D x ) 1 / 2 ) ] 0 = ξ 1 ( z ) , Open image in new window

which contradicts (3.8), hence the result. □

4 Model II

In this section, we consider another dual problem to (P):
max ( s , t , y ˜ ) K ( z ) sup ( z , μ , w , v , p ) H 2 ( s , t , y ˜ ) i = 1 s t i ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) + j = 1 m μ j g j ( z ) i = 1 s t i ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) , Open image in new window
(DM2)
where H 2 ( s , t , y ˜ ) Open image in new window denotes the set of all ( z , μ , w , v , p ) R n × R + m × R n × R n × R n Open image in new window satisfying

If the set H 2 ( s , t , y ˜ ) Open image in new window is empty, we define the supremum in (DM2) over H 2 ( s , t , y ˜ ) Open image in new window equal to −∞.

Remark 4.1 If p = 0 Open image in new window, then using (4.3), the above dual model becomes the dual model considered in [5, 11, 12]. In addition, if B and D are zero matrices of order n, then (DM2) reduces to the problem studied in [14].

Now, we obtain the following appropriate duality theorems between (P) and (DM2).

Theorem 4.1 (Weak duality)

Letxand ( z , μ , w , v , s , t , y ˜ , p ) Open image in new windoware feasible solutions of (P) and (DM2), respectively. Suppose that the following conditions are satisfied:
  1. (i)

    ψ 2 ( ) Open image in new windowis second-orderα-univex atz,

     
  2. (ii)

    ϕ 0 ( a ) 0 a 0 Open image in new windowand b 0 ( x , z ) > 0 Open image in new window.

     
Then
sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 i = 1 s t i ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) + j = 1 m μ j g j ( z ) i = 1 s t i ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) . Open image in new window
Proof Assume on contrary to the result that
sup y ˜ Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 h ( x , y ˜ ) ( x T D x ) 1 / 2 < i = 1 s t i ( f ( z , y ˜ i ) + ( z T B z ) 1 / 2 ) + j = 1 m μ j g j ( z ) i = 1 s t i ( h ( z , y ˜ i ) ( z T D z ) 1 / 2 ) Open image in new window
Using t i 0 Open image in new window, i = 1 , 2 , , s Open image in new window and (4.3) in above, we have
Now,
ψ 2 ( x ) = [ i = 1 s t i ( f ( x , y ˜ i ) + x T B w ) + j = 1 m μ j g j ( x ) ] [ i = 1 s t i ( h ( z , y ˜ i ) z T D v ) ] [ i = 1 s t i ( h ( x , y ˜ i ) x T D v ) ] [ i = 1 s t i ( f ( z , y ˜ i ) + z T B w ) + j = 1 m μ j g j ( z ) ] [ i = 1 s t i ( f ( x , y ˜ i ) + ( x T B x ) 1 / 2 ) + j = 1 m μ j g j ( x ) ] [ i = 1 s t i ( h ( z , y ˜ i ) z T D v ) ] [ i = 1 s t i ( h ( x , y ˜ i ) ( x T D x ) 1 / 2 ) ] [ i = 1 s t i ( f ( z , y ˜ i ) + z T B w ) + j = 1 m μ j g j ( z ) ] ( from Lemma 2.1 and (4.3) ) < i = 1 s t i ( h ( z , y ˜ i ) z T D v ) j = 1 m μ j g j ( x ) ( using (4.4) ) 0 ( since  i = 1 s t i ( h ( z , y ˜ i ) z T D v ) > 0  and  j = 1 m μ j g j ( x ) 0 ) . Open image in new window
Hence,
ψ 2 ( x ) < 0 = ψ 2 ( z ) . Open image in new window
(4.5)
Now, by the second-order α-univexity of ψ 2 ( ) Open image in new window at z, we get
b 0 ϕ 0 [ ψ 2 ( x ) ψ 2 ( z ) + 1 2 p T 2 ψ 2 ( z ) p ] η T ( x , z ) α ( x , z ) { ψ 2 ( z ) + 2 ψ 2 ( z ) p } Open image in new window
which using (4.1) and hypothesis (ii) give
ψ 2 ( x ) ψ 2 ( z ) + 1 2 p T 2 ψ 2 ( z ) p 0 . Open image in new window
This from (4.2) follows that
ψ 2 ( x ) ψ 2 ( z ) Open image in new window

which contradicts (4.5). This proves the theorem. □

By a similar way, we can prove the following theorems between (P) and (DM2).

Theorem 4.2 (Strong duality)

Let x Open image in new windowbe an optimal solution for (P) and let g j ( x ) Open image in new window, j J ( x ) Open image in new windowbe linearly independent. Then there exist ( s , t , y ˜ ) K ( x ) Open image in new windowand ( x , μ , w , v , p = 0 ) H 2 ( s , t , y ˜ ) Open image in new window, such that ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new windowis feasible solution of (DM2) and the two objectives have same values. If, in addition, the assumptions of weak duality hold for all feasible solutions ( x , μ , w , v , s , t , y ˜ , p ) Open image in new windowof (DM2), then ( x , μ , w , v , s , t , y ˜ , p = 0 ) Open image in new windowis an optimal solution of (DM2).

Theorem 4.3 (Strict converse duality)

Let x Open image in new windowand ( z , μ , w , v , s , t , y ˜ , p ) Open image in new windoware optimal solutions of (P) and (DM2), respectively. Assume that
  1. (i)

    ψ 2 ( ) Open image in new windowis strictly second-orderα-univex atz,

     
  2. (ii)

    { g j ( x ) , j J ( x ) } Open image in new windoware linearly independent,

     
  3. (iii)

    ϕ 0 ( a ) > 0 a > 0 Open image in new windowand b 0 ( x , z ) > 0 Open image in new window.

     

Then z = x Open image in new window.

5 Concluding remarks

In the present work, we have formulated two types of second-order dual models for a nondifferentiable minimax fractional programming problems and proved appropriate duality relations involving second-order α-univex functions. Further, examples have been illustrated to show the existence of such type of functions. Now, the question arises whether or not the results can be further extended to a higher-order nondifferentiable minimax fractional programming problem.

Acknowledgements

The authors wish to thank anonymous reviewers for their constructive and valuable suggestions which have considerably improved the presentation of the paper. The second author is also thankful to the Ministry of Human Resource Development, New Delhi (India) for financial support.

Copyright information

© Gupta et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyPatnaIndia
  2. 2.Indian Institute of ManagementUdaipurIndia