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

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Research

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.

Keywords

Feasible Solution Duality Theorem Duality Result Sufficient Optimality Condition Weak Duality 

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 [4, 5, 6, 7, 8, 9, 10]. 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 , Open image in new window
(P)

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 and h ( ) : R n R m Open image in new window are twice differentiable functions. B is an n × n Open image in new window 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 ) Open image in new window-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 ) Open image in new window-type I assumptions. This paper generalizes severalresults that have appeared in the literature [11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22] 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 , Open image in new window
(NP)

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

Let X = { x R n : h ( x ) 0 } Open image in new window denote the set of all feasible solutions of (NP). Anypoint x X Open image in new window is called the feasible point of (NP). For each ( x , y ) X × Y Open image in new window, we define
ψ ( x , y ) = f ( x , y ) + ( x T B x ) 1 / 2 g ( x , y ) ( x T C x ) 1 / 2 Open image in new window
such that for each ( x , y ) X × Y Open image in new window,
f ( x , y ) + ( x T B x ) 1 / 2 0 and g ( x , y ) ( x T C x ) 1 / 2 > 0 . Open image in new window
For each ( x , y ) X × Y Open image in new window, we define
J ( x ) = { j J : h j ( x ) = 0 } , Open image in new window
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 ) } . Open image in new window
Since f and g are continuously differentiable and Y iscompact in R l Open image in new window, it follows that for each x X Open image in new window, Y ( x ) Open image in new window, and for any y ¯ i Y ( x ) Open image in new window, 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 . Open image in new window

Lemma 2.1 (Generalized Schwarz inequality)

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

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

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

    F ( x , x ¯ ; a + b ) F ( x , x ¯ ; a ) + F ( x , x ¯ ; b ) Open image in new window, a , b R n Open image in new window;

     
  2. (ii)

    F ( x , x ¯ ; β a ) = β F ( x , x ¯ ; a ) Open image in new window, β R Open image in new window, β 0 Open image in new window, and a R n Open image in new window.

     

From (ii), it is clear that F ( x , x ¯ ; 0 ) = 0 Open image in new window.

Definition 2.2[15]

For each j J Open image in new window, ( ψ , h j ) Open image in new window is said to be higher-order ( F , α , ρ , d ) Open image in new window-pseudoquasi-type I at x ¯ R n Open image in new window if for all x X Open image in new window, p R n Open image in new window and y ¯ i Y ( x ) Open image in new window,
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 . Open image in new window
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 , Open image in new window

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

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

Theorem 2.1 (Necessary conditions)

If x Open image in new windowis a solution of (NP) satisfying x T B x > 0 Open image in new window, x T C x > 0 Open image in new window, and h j ( x ) Open image in new window, j J ( x ) Open image in new windoware linearly independent, then there exist ( s , t , y ˜ ) S ( x ) Open image in new window, λ 0 R + Open image in new window, w , v R n Open image in new window, and μ R + m Open image in new windowsuch 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 . Open image in new window
In the above theorem, both matrices B and C are positivesemidefinite. If either x T B x Open image in new window or x T C x Open image in new window 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 ) Open image in new window, we define
U y ˜ ( x ) = { u R n : u t h j ( x ) 0 , j J ( x )  satisfying one of the following conditions: Open image in new window
  1. (i)
    x T B x > 0 Open image in new window, x T C x = 0 Open image in new window
    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 , Open image in new window
     
  2. (ii)
    x T B x = 0 Open image in new window, x T C x > 0 Open image in new window
    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 , Open image in new window
     
  3. (iii)
    x T B x = 0 Open image in new window, x T C x = 0 Open image in new window
    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 , Open image in new window
     
  4. (iv)
    x T B x > 0 Open image in new window, x T C x > 0 Open image in new window
    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 } . Open image in new window
     

If in addition, we insert U y ˜ ( x ) = Open image in new window, 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 , Open image in new window
(ND)
where L ( s , t , y ˜ ) Open image in new window denotes the set of all ( z , μ , λ , v , w , p ) R n × R + m × R + × R n × R n × R n Open image in new window 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 , Open image in new window
(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 , Open image in new window
(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 , Open image in new window
(3.3)
w T B w 1 , v T C v 1 , Open image in new window
(3.4)

where F : R n × Y × R n R Open image in new window, G : R n × Y × R n R Open image in new window, J β M = { 1 , 2 , , m } Open image in new window, β = 0 , 1 , 2 , , r Open image in new window with β = 0 r J β = M Open image in new window and J β J α = Open image in new window if β α Open image in new window. If for a triplet ( s , t , y ˜ ) S ( z ) Open image in new window, the set L ( s , t , y ˜ ) = Open image in new window, 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 ) Open image in new windowbe 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 ] Open image in new window
is higher-order ( F , α , ρ , d ) Open image in new window-pseudoquasi-type I at z and
ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) 0 . Open image in new window
Then
sup y Y f ( x , y ) + ( x t B x ) 1 / 2 g ( x , y ) ( x t C x ) 1 / 2 λ . Open image in new window
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 . Open image in new window
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 . Open image in new window
It follows from t i 0 Open image in new window, i = 1 , 2 , , s Open image in new window, 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 , Open image in new window
with at least one strict inequality, since t = ( t 1 , t 2 , , t s ) 0 Open image in new window. Taking summation over i and using i = 1 s t i = 1 Open image in new window, 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 . Open image in new window
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 . Open image in new window
(3.5)
By the feasibility of x for (NP) and μ 0 Open image in new window, we obtain
j J 0 μ j h j ( x ) 0 . Open image in new window
(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 . Open image in new window
(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 ) . Open image in new window
(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 . Open image in new window
(3.9)
The higher second-order ( F , α , ρ , d ) Open image in new window-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 ] Open image in new window
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 . Open image in new window
By using α 1 ( x , z ) > 0 Open image in new window, α 2 ( x , z ) > 0 Open image in new window, and the sublinearity of ℱ 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 ) . Open image in new window
Since ( ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) ) 0 Open image in new window, 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 , Open image in new window

which contradicts (3.1), as F ( x , z ; 0 ) = 0 Open image in new window. □

Theorem 3.2 (Strong duality)

Let x Open image in new windowbe an optimal solution of (NP) and let h j ( x ) Open image in new window, j J ( x ) Open image in new windowbe 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 . Open image in new window

Then there exist ( s , t , y ˜ ) S Open image in new windowand ( x , μ , λ , v , w , p ) L ( s , t , y ˜ ) Open image in new windowsuch that ( x , μ , λ , v , w , s , t , y ˜ , p = 0 ) Open image in new windowis 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 ) Open image in new windowis an optimal solution of (ND).

Proof Since x Open image in new window is an optimal solution of (NP) and h j ( x ) Open image in new window, j J ( x ) Open image in new window are linearly independent, by Theorem 2.1, thereexist ( s , t , y ˜ ) S Open image in new window and ( x , μ , λ , v , w , p ) L ( s , t , y ˜ ) Open image in new window such that ( x , μ , λ , v , w , s , t , y ˜ , p = 0 ) Open image in new window 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 . Open image in new window

 □

Theorem 3.3 (Strict converse duality)

Let x Open image in new windowand ( z , μ , λ , s , t , v , w , y ˜ , p ) Open image in new windowbe 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 ] Open image in new window
is higher-order ( F , α , ρ , d ) Open image in new window-strictly pseudoquasi-type I at z Open image in new windowwith
ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) 0 , Open image in new window

and that h j ( x ) Open image in new window, j J ( x ) Open image in new windoware linearly independent. Then z = x Open image in new window; that is, z Open image in new windowis an optimal solution of (NP).

Proof We assume that z x Open image in new window 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 . Open image in new window
(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 . Open image in new window
(3.11)
The feasibility of x Open image in new window for (NP), μ 0 Open image in new window 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 ) ] , Open image in new window
which along with the second part of higher-order ( F , α , ρ , d ) Open image in new window-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 ] Open image in new window
at z Open image in new window 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 . Open image in new window
As α 2 ( x , z ) > 0 Open image in new window and as ℱ 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 . Open image in new window
(3.12)
From (3.1), (3.12) and the sublinearity of ℱ, 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 ) . Open image in new window
In view of ( ρ 1 1 α 1 ( x , z ) + β = 1 r ρ β 2 α 2 ( x , z ) ) 0 Open image in new window, α 1 ( x , z ) > 0 Open image in new window and the sublinearity of ℱ, 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 ) . Open image in new window
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 ] Open image in new window
at z Open image in new window, 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)) , Open image in new window

which contradicts (3.11). Hence the result. □

4 Special cases

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

5 Conclusion

The notion of higher-order ( F , α , ρ , d ) Open image in new window-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].

Notes

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.

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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.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TabukTabukSaudi Arabia
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of Mathematics and StatisticsKing Fahad University of Petroleum and MineralsDhahranSaudi Arabia
  4. 4.Permanent address: Department of MathematicsAligarh Muslim UniversityAligarhIndia

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