Open Access

Non-differentiable minimax fractional programming with higher-order type Ifunctions

Journal of Inequalities and Applications20132013:484

https://doi.org/10.1186/1029-242X-2013-484

Received: 25 May 2013

Accepted: 9 September 2013

Published: 7 November 2013

Abstract

In this article, we are concerned with a class of non-differentiable minimaxfractional programming problems and their higher-order dual model. Weak, strongand converse duality theorems are discussed involving generalized higher-ordertype I functions. The presented results extend some previously known results onnon-differentiable minimax fractional programming.

1 Introduction

In nonlinear optimization, problems, where minimization and maximization process areperformed together, are called minimax (minmax) problems. Frequently, problems ofthis type arise in many areas like game theory, Chebychev approximation, economics,financial planning and facility location [1].

The optimization problems in which the objective function is a ratio of two functionsare commonly known as fractional programming problems. In the past few years, manyauthors have shown interest in the field of minimax fractional programming problems.Schmittendorf [2] first developed necessary and sufficient optimality conditions for aminimax programming problem. Tanimoto [3] applied the necessary conditions in [2] to formulate a dual problem and discussed the duality results, which wereextended to a fractional analogue of the problem considered in [2, 3] by several authors [410]. Liu [11] proposed the second-order duality theorems for a minimax programmingproblem under generalized second-order B-invex functions. Husain etal.[12] formulated two types of second-order dual models for minimax fractionalprogramming and derived weak, strong and converse duality theorems underη-convexity assumptions.

Ahmad et al.[13] and Husain et al.[14] discussed the second-order duality results for the followingnon-differentiable minimax programming problem:
Minimize  sup y Y f ( x , y ) + ( x T B x ) 1 / 2 , subject to  h ( x ) 0 , x R n ,
(P)

where Y is a compact subset of R l , f ( , ) : R n × R l R and h ( ) : R n R m are twice differentiable functions. B is an n × n positive semidefinite symmetric matrices. Ahmadet al.[15] formulated a unified higher-order dual of (P) and established appropriateduality theorems under higher-order ( F , α , ρ , d ) -type I assumptions. Recently, Jayswal andStancu-Minasian [16] obtained higher-order duality results for (P).

In this paper, we formulate a higher-order dual for a non-differentiable minimaxfractional programming problem and establish weak, strong and strict converseduality theorems under generalized higher-order ( F , α , ρ , d ) -type I assumptions. This paper generalizes severalresults that have appeared in the literature [11, 12, 1422] and references therein.

2 Preliminaries

The problem to be considered in the present analysis is the followingnon-differentiable minimax fractional problem:
Minimize  sup y Y f ( x , y ) + ( x T B x ) 1 / 2 g ( x , y ) ( x T C x ) 1 / 2 , subject to  h ( x ) 0 , x R n ,
(NP)

where Y is a compact subset of R l , f ( , ) , g ( , ) : R n × R l R and h ( ) : R n R m are differentiable functions. B andC are n × n positive semidefinite symmetric matrices. It isassumed that for each ( x , y ) in R n × R l , f ( x , y ) + ( x T B x ) 1 2 0 and g ( x , y ) ( x T C x ) 1 2 > 0 .

Let X = { x R n : h ( x ) 0 } denote the set of all feasible solutions of (NP). Anypoint x X is called the feasible point of (NP). For each ( x , y ) X × Y , we define
ψ ( x , y ) = f ( x , y ) + ( x T B x ) 1 / 2 g ( x , y ) ( x T C x ) 1 / 2
such that for each ( x , y ) X × Y ,
f ( x , y ) + ( x T B x ) 1 / 2 0 and g ( x , y ) ( x T C x ) 1 / 2 > 0 .
For each ( x , y ) X × Y , we define
J ( x ) = { j J : h j ( x ) = 0 } ,
where
J = { 1 , 2 , , m } , Y ( x ) = { y Y : f ( x , y ) + ( x T B x ) 1 / 2 g ( x , y ) ( x T C x ) 1 / 2 = sup z Y f ( x , z ) + ( x T B x ) 1 / 2 g ( x , z ) ( x T C x ) 1 / 2 } , S ( x ) = { ( s , t , y ˜ ) N × R + s × R l s : 1 s n + 1 , t = ( t 1 , t 2 , , t s ) R + s S ( x ) = with  i = 1 s t i = 1 , y ˜ = ( y ¯ 1 , y ¯ 2 , , y ¯ s )  with  y ¯ i Y ( x ) ( i = 1 , 2 , , s ) } .
Since f and g are continuously differentiable and Y iscompact in R l , it follows that for each x X , Y ( x ) , and for any y ¯ i Y ( x ) , we have
λ = ϕ ( x , y ¯ i ) = f ( x , y ¯ i ) + ( x T B x ) 1 / 2 g ( x , y ¯ i ) ( x T C x ) 1 / 2 .

Lemma 2.1 (Generalized Schwarz inequality)

Let A be a positive-semidefinite matrix of order n. Then, for all x , w R n ,
x T A w ( x T A x ) 1 2 ( w T A w ) 1 2 .
(2.1)
The equality A x = ξ A w holds for some ξ 0 . Clearly, if ( w T A w ) 1 2 1 , we have
x T A w ( x T A x ) 1 2 .

Let be a sublinear functional, and let d ( , ) : R n × R n R . Let ρ = ( ρ 1 , ρ 2 ) , where ρ 1 = ( ρ 1 1 , ρ 2 1 , , ρ s 1 ) R s and ρ 2 = ( ρ 1 2 , ρ 2 2 , , ρ m 2 ) R m , and let α = ( α 1 , α 2 ) : R n × R n R + { 0 } . Let ψ ( , ) : R n × Y R , h ( ) : R n R m , K : R n × Y × R n R and H j : R n × Y × R n R , j = 1 , 2 , , m , be differentiable functions at x ¯ R n .

Definition 2.1 A functional F : R n × R n × R n R is said to be sublinear in its third argument if forall x , x ¯ R n ,
  1. (i)

    F ( x , x ¯ ; a + b ) F ( x , x ¯ ; a ) + F ( x , x ¯ ; b ) , a , b R n ;

     
  2. (ii)

    F ( x , x ¯ ; β a ) = β F ( x , x ¯ ; a ) , β R , β 0 , and a R n .

     

From (ii), it is clear that F ( x , x ¯ ; 0 ) = 0 .

Definition 2.2[15]

For each j J , ( ψ , h j ) is said to be higher-order ( F , α , ρ , d ) -pseudoquasi-type I at x ¯ R n if for all x X , p R n and y ¯ i Y ( x ) ,
ψ ( x , y ¯ i ) < ψ ( x ¯ , y ¯ i ) + K ( x ¯ , y ¯ i , p ) p T p K ( x ¯ , y ¯ i , p ) F ( x , x ¯ ; α 1 ( x , x ¯ ) ( p K ( x ¯ , y ¯ i , p ) ) ) < ρ i 1 d 2 ( x , x ¯ ) , i = 1 , 2 , , s , [ h j ( x ¯ ) + H j ( x ¯ , p ) p T p H j ( x ¯ , p ) ] 0 F ( x , x ¯ ; α 2 ( x , x ¯ ) ( p H j ( x ¯ , p ) ) ) ρ j 2 d 2 ( x , x ¯ ) , j = 1 , 2 , , m .
In the above definition, if
F ( x , x ¯ ; α 1 ( x , x ¯ ) ( p K ( x ¯ , y ¯ i , p ) ) ) ρ i 1 d 2 ( x , x ¯ ) ψ ( x , y ¯ i ) > ψ ( x ¯ , y ¯ i ) + K ( x ¯ , y ¯ i , p ) p T p K ( x ¯ , y ¯ i , p ) , i = 1 , 2 , , s ,

then we say that ( ψ , g j ) is higher-order ( F , α , ρ , d ) -strictly pseudoquasi-type I at x ¯ .

If the functions f, g and h in problem (NP) arecontinuously differentiable with respect to x R n , then Liu [11] derived the following necessary conditions for optimalityof (NP).

Theorem 2.1 (Necessary conditions)

If x is a solution of (NP) satisfying x T B x > 0 , x T C x > 0 , and h j ( x ) , j J ( x ) are linearly independent, then there exist ( s , t , y ˜ ) S ( x ) , λ 0 R + , w , v R n , and μ R + m such that
i = 1 s t i { f ( x , y ¯ i ) + B w λ 0 ( g ( x , y ¯ i ) C v ) } + j = 1 m μ j h j ( x ) = 0 , f ( x , y ¯ i ) + ( x T B x ) 1 2 λ 0 ( g ( x , y ¯ i ) ( x T C x ) 1 2 ) = 0 , i = 1 , 2 , , s , j = 1 m μ j h j ( x ) = 0 , t i 0 ( i = 1 , 2 , , s ) , i = 1 s t i = 1 , w T B w 1 , v T C v 1 , ( x T B x ) 1 / 2 = x T B w , ( x T C x ) 1 / 2 = x T C v .
In the above theorem, both matrices B and C are positivesemidefinite. If either x T B x or x T C x is zero, then the functions involved in the objectivefunction of problem (NP) are not differentiable. To derive these necessaryconditions under this situation, for ( s , t , y ˜ ) S ( x ) , we define
U y ˜ ( x ) = { u R n : u t h j ( x ) 0 , j J ( x )  satisfying one of the following conditions:
  1. (i)
    x T B x > 0 , x T C x = 0
    u T ( i = 1 s t i { f ( x , y ¯ i ) + B x ( x t B x ) 1 2 λ g ( x , y ¯ i ) } ) + ( u T ( λ 2 C ) u ) 1 2 < 0 ,
     
  2. (ii)
    x T B x = 0 , x T C x > 0
    u T ( i = 1 s t i { f ( x , y ¯ i ) λ ( g ( x , y ¯ i ) C x ( x T C x ) 1 2 ) } ) + ( u T B u ) 1 2 < 0 ,
     
  3. (iii)
    x T B x = 0 , x T C x = 0
    u T ( i = 1 s t i { f ( x , y ¯ i ) λ g ( x , y ¯ i ) } ) + ( u T ( λ 2 C ) u ) 1 2 + ( u T B u ) 1 2 < 0 ,
     
  4. (iv)
    x T B x > 0 , x T C x > 0
    u T ( i = 1 s t i { f ( x , y ¯ i ) λ g ( x , y ¯ i ) } ) + ( u T ( λ 2 C ) u ) 1 2 + ( u T B u ) 1 2 < 0 } .
     

If in addition, we insert U y ˜ ( x ) = , then the results of Theorem 2.1 still hold.

3 Higher-order non-differentiable fractional duality

In this section, we consider the following dual problem to (NP):
max ( s , t , y ˜ ) S ( z ) sup ( z , μ , λ , v , w , p ) L ( s , t , y ˜ ) λ ,
(ND)
where L ( s , t , y ˜ ) denotes the set of all ( z , μ , λ , v , w , p ) R n × R + m × R + × R n × R n × R n satisfying
i = 1 s t i [ p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) ] + B w + λ C v + j = 1 m μ j p H j ( z , p ) = 0 ,
(3.1)
i = 1 s t i [ f ( z , y ¯ i ) + z T B w λ ( g ( z , y ¯ i ) z T C v ) + F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) + j J 0 μ j h j ( z ) p T p { F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) } ] + j J 0 μ j H j ( z , p ) p T j J 0 μ j p H j ( z , p ) 0 ,
(3.2)
j J β μ j [ h j ( z ) + H j ( z , p ) p T p H j ( z , p ) ] 0 , β = 1 , 2 , , r ,
(3.3)
w T B w 1 , v T C v 1 ,
(3.4)

where F : R n × Y × R n R , G : R n × Y × R n R , J β M = { 1 , 2 , , m } , β = 0 , 1 , 2 , , r with β = 0 r J β = M and J β J α = if β α . If for a triplet ( s , t , y ˜ ) S ( z ) , the set L ( s , t , y ˜ ) = , then we define the supremum over it to be∞.

Theorem 3.1 (Weak duality)

Let x and ( z , μ , λ , s , t , v , w , y ˜ , p ) be feasible solutions of (NP) and (ND), respectively.Suppose that
[ i = 1 s t i { f ( , y ¯ i ) + ( ) T B w λ ( g ( , y ¯ i ) z T C v ) } + j J 0 μ j h j ( ) , j j β μ j h j ( ) , β = 1 , 2 , , r ]
is higher-order ( F , α , ρ , d ) -pseudoquasi-type I at z and
ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) 0 .
Then
sup y Y f ( x , y ) + ( x t B x ) 1 / 2 g ( x , y ) ( x t C x ) 1 / 2 λ .
Proof Suppose to the contrary that
sup y Y f ( x , y ) + ( x T B x ) 1 / 2 g ( x , y ) ( x T C x ) 1 / 2 < λ .
Then we have
f ( x , y ¯ i ) + ( x T B x ) 1 / 2 λ ( g ( x , y ¯ i ) ( x T C x ) 1 / 2 ) < 0 , for all  y ¯ i Y , i = 1 , 2 , , s .
It follows from t i 0 , i = 1 , 2 , , s , that
t i [ f ( x , y ¯ i ) + ( x T B x ) 1 / 2 λ ( g ( x , y ¯ i ) ( x T C x ) 1 / 2 ) ] 0 , i = 1 , 2 , , s ,
with at least one strict inequality, since t = ( t 1 , t 2 , , t s ) 0 . Taking summation over i and using i = 1 s t i = 1 , we have
i = 1 s t i [ f ( x , y ¯ i ) + ( x T B x ) 1 / 2 λ ( g ( x , y ¯ i ) ( x T C x ) 1 / 2 ) ] < 0 .
It follows from the generalized Schwarz inequality and (3.4) that
i = 1 s t i [ f ( x , y ¯ i ) + x T B w λ ( g ( x , y ¯ i ) x T C v ) ] < 0 .
(3.5)
By the feasibility of x for (NP) and μ 0 , we obtain
j J 0 μ j h j ( x ) 0 .
(3.6)
The above inequality with (3.5) gives
i = 1 s t i [ f ( x , y ¯ i ) + x T B w λ ( g ( x , y ¯ i ) x T C v ) ] + j J 0 μ j h j ( x ) < 0 .
(3.7)
From (3.2) and (3.7), we have
i = 1 s t i [ f ( x , y ¯ i ) + x T B w λ ( g ( x , y ¯ i ) x T C v ) ] + j J 0 μ j h j ( x ) < i = 1 s t i [ f ( z , y ¯ i ) + z T B w λ ( g ( z , y ¯ i ) z T C v ) + F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) + j J 0 μ j h j ( z ) p T p { F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) } ] + j J 0 μ j H j ( z , p ) p T j J 0 μ j p H j ( z , p ) .
(3.8)
Also, from (3.3), we have
j j β μ j [ h j ( z ) + H j ( z , p ) p T p H j ( z , p ) ] 0 , β = 1 , 2 , , r .
(3.9)
The higher second-order ( F , α , ρ , d ) -pseudoquasi-type I assumption on
[ i = 1 s t i { f ( , y ¯ i ) + ( ) T B w λ ( g ( , y ¯ i ) ( ) T C v ) } + j J 0 μ j h j ( ) , j j β μ j h j ( ) , β = 1 , 2 , , r ]
at z, with (3.8) and (3.9), implies
F ( x , z ; α 1 ( x , z ) i = 1 s t i { p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) } + B w + λ C v ) < ρ 1 1 d 2 ( x , z ) , F ( x , z ; α 2 ( x , z ) j j β μ j p H j ( z , p ) ) ρ β 2 d 2 ( x , z ) , β = 1 , 2 , , r .
By using α 1 ( x , z ) > 0 , α 2 ( x , z ) > 0 , and the sublinearity of in the aboveinequalities, we summarize to get
F ( x , z ; i = 1 s t i { p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) } + B w + λ C v + β = 1 r j j β μ j p H j ( z , p ) ) < ( ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) ) d 2 ( x , z ) .
Since ( ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) ) 0 , therefore
F ( x , z ; i = 1 s t i { p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) } + B w + λ C v + j = 1 m μ j p H j ( z , p ) ) < 0 ,

which contradicts (3.1), as F ( x , z ; 0 ) = 0 . □

Theorem 3.2 (Strong duality)

Let x be an optimal solution of (NP) and let h j ( x ) , j J ( x ) be linearly independent. Assume that
F ( x , y ¯ i , 0 ) = 0 ; p F ( x , y ¯ i , 0 ) = f ( x , y ¯ i ) , i = 1 , 2 , , s , G ( x , y ¯ i , 0 ) = 0 ; p G ( x , y ¯ i , 0 ) = g ( x , y ¯ i ) , i = 1 , 2 , , s , H j ( x , 0 ) = 0 ; p H j ( x , 0 ) = h j ( x ) , j J .

Then there exist ( s , t , y ˜ ) S and ( x , μ , λ , v , w , p ) L ( s , t , y ˜ ) such that ( x , μ , λ , v , w , s , t , y ˜ , p = 0 ) is a feasible solution of (ND) and the two objectives have the samevalues. Furthermore, if the assumptions of weak duality(Theorem  3.1) hold for all feasible solutions of (NP)and (ND), then ( x , μ , λ , v , w , s , t , y ˜ , p = 0 ) is an optimal solution of (ND).

Proof Since x is an optimal solution of (NP) and h j ( x ) , j J ( x ) are linearly independent, by Theorem 2.1, thereexist ( s , t , y ˜ ) S and ( x , μ , λ , v , w , p ) L ( s , t , y ˜ ) such that ( x , μ , λ , v , w , s , t , y ˜ , p = 0 ) is a feasible solution of (ND) and problems (NP) and(ND) have the same objectives values and
λ = f ( x , y ¯ i ) + ( x T B x ) 1 / 2 g ( x , y ¯ i ) ( x T C x ) 1 / 2 .

 □

Theorem 3.3 (Strict converse duality)

Let x and ( z , μ , λ , s , t , v , w , y ˜ , p ) be the optimal solutions of (NP) and (ND), respectively.Suppose that
[ i = 1 s t i { f ( , y ¯ i ) + ( ) T B w λ ( g ( , y ¯ i ) ( ) T C v ) } + j J 0 μ j h j ( ) , j j β μ j h j ( ) , β = 1 , 2 , , r ]
is higher-order ( F , α , ρ , d ) -strictly pseudoquasi-type I at z with
ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) 0 ,

and that h j ( x ) , j J ( x ) are linearly independent. Then z = x ; that is, z is an optimal solution of (NP).

Proof We assume that z x and reach a contradiction. From the strong dualitytheorem (Theorem 3.2), it follows that
sup y Y f ( x , y ˜ ) + ( x T B x ) 1 / 2 g ( x , y ˜ ) ( x T C x ) 1 / 2 = λ .
(3.10)
Now, proceeding as in Theorem 3.1, we get
i = 1 s t i [ f ( x , y ¯ i ) + x T B w λ ( g ( x , y ¯ i ) x T C v ) ] + j J 0 μ j h j ( x ) < 0 .
(3.11)
The feasibility of x for (NP), μ 0 and (3.3) imply
j J β μ j h j ( x ) 0 j J β μ j [ h j ( z ) + H j ( z , p ) p T p H j ( z , p ) ] ,
which along with the second part of higher-order ( F , α , ρ , d ) -strictly pseudoquasi-type I assumption on
[ i = 1 s t i { f ( , y ¯ i ) + ( ) T B w λ ( g ( , y ¯ i ) ( ) T C v ) } + j J 0 μ j h j ( ) , j j β μ j h j ( ) , β = 1 , 2 , , r ]
at z gives
F ( x , z ; α 2 ( x , z ) j j β μ j p H j ( z , p ) ) < ρ β 2 d 2 ( x , z ) , β = 1 , 2 , , r .
As α 2 ( x , z ) > 0 and as is sublinear, it follows that
F ( x , z ; j j β μ j p H j ( z , p ) ) < ρ β 2 α 2 ( x , z ) d 2 ( x , z ) , β = 1 , 2 , , r .
(3.12)
From (3.1), (3.12) and the sublinearity of , we have
F ( x , z ; i = 1 s t i [ p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) ] + B w + λ C v + j j 0 μ j p H j ( z , p ) ) β = 1 r ρ β 2 α 2 ( x , z ) d 2 ( x , z ) .
In view of ( ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) ) 0 , α 1 ( x , z ) > 0 and the sublinearity of , the above inequalitybecomes
F ( x , z ; α 1 ( x , z ) i = 1 s t i [ p ( F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) ) ] + B w + λ C v + j j 0 μ j p H j ( z , p ) ) ρ 1 1 d 2 ( x , z ) .
By using the first part of the said assumption imposed on
[ i = 1 s t i { f ( , y ¯ i ) + ( ) T B w λ ( g ( , y ¯ i ) ( ) T C v ) } + j J 0 μ j h j ( ) , j j β μ j h j ( ) , β = 1 , 2 , , r ]
at z , it follows that
i = 1 s t i [ f ( x , y ¯ i ) + x T B w λ ( g ( x , y ¯ i ) x T C v ) ] + j J 0 μ j h j ( x ) > i = 1 s t i [ f ( z , y ¯ i ) + z T B w λ ( g ( z , y ¯ i ) z T C v ) + F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) + j J 0 μ j h j ( z ) p T p { F ( z , y ¯ i , p ) λ G ( z , y ¯ i , p ) } ] + j J 0 μ j H j ( z , p ) p T j J 0 μ j p H j ( z , p ) 0 (by (3.2)) ,

which contradicts (3.11). Hence the result. □

4 Special cases

Let J 0 = , F ( z , y ¯ i , p ) = p T f ( z , y ¯ i ) + 1 2 p T 2 f ( z , y ¯ i ) , G ( z , y ¯ i , p ) = p T g ( z , y ¯ i ) + 1 2 p T 2 g ( z , y ¯ i ) , i = 1 , 2 , , s and H j ( z , p ) = p T h j ( z ) + 1 2 p T 2 h j ( z ) p , j = 1 , 2 , , m . Then (ND) becomes the second-order dual studied in [17, 22]. If, in addition, p = 0 , then we obtain the dual formulated by Ahmad etal.[20].

5 Conclusion

The notion of higher-order ( F , α , ρ , d ) -pseudoquasi-type I is adopted, which includes manyother generalized convexity concepts in mathematical programming as special cases.This concept is appropriate to discuss the weak, strong and strict converse dualitytheorems for a higher-order dual (ND) of a non-differentiable minimax fractionalprogramming problem (NP). The results of this paper can be discussed by formulatinga unified higher-order dual involving support functions on the lines of Ahmad [23].

Declarations

Acknowledgements

This work was partially supported by the Deanship of Scientific Research Unit,University of Tabuk, Tabuk, Kingdom of Saudi Arabia. The authors are grateful tothe anonymous referee for a careful checking of the details and for helpfulcomments that improved this paper.

Authors’ Affiliations

(1)
Department of Mathematics, University of Tabuk
(2)
Department of Mathematics, King Abdulaziz University
(3)
Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals
(4)
Permanent address: Department of Mathematics, Aligarh Muslim University

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© Muhiuddin et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly cited.