Optimality and Duality for Minimax Fractional Semi-Infinite Programming
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
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Optimality and Duality for Minimax Fractional Semi-Infinite Programming Xiaoyan Gao School of Science, Xi’an University of Science and Technology, Xi’an, China Corresponding author, Email: [email protected]
Abstract—The purpose of this paper is to consider a class of nonsmooth minimax fractional semi-infinite programming problem. Based on the concept of H − tangent derivative, a new generalization of convexity, namely generalized uniform ( BH , ρ ) − invexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Weak, strong and reverse duality theorems are also derived for dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable minimax fractional programming problems turn out to be special cases for the results described in the paper Index Terms— H − tangent derivative, generalized convexity, minimax fractional semi-infinite programming, optimality conditions, duality
I. INTRODUCTION In recent years, the concept of convexity and generalized convexity is well known in optimization theory and plays a central role in mathematical economics, management science, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematics programming. To relax convexity assumptions imposed on theorems on optimality conditions for generalized mathematical programming problems, various generalized convexity notations have been introduced. In particular, the concept of generalized ( F , ρ ) − convexity, introduced by Preda [1] is in turn an extension of the convexity and was used by several authors to obtain relevant results. In [2, 3], the concept of V − ρ − invexity and ( F , α , ρ , d ) − convexity were introduced, respectively. Other classes of generalized type I functions have been discussed in [4, 5]. On the other hand, a large literature was developed around generalized convexity and its applications in mathematical programming. Many authors investigated the optimality conditions and duality results for min-max programming problems under the conditions of generalized convexity. In particular, Aparna Mehra [6] employed various optimality conditions and duality © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.3.739-746
results under arcwise connectedness and generalized arcwise connectedness assumptions for a static minimax programming problem. Lin [7] and Wu [8] derived the sufficient optimality conditions for the generalized minmax fractional programming in the framework of ( F , ρ ) − convex functions and invex functions. In [9], the Karush-Kuhn-Tucker-type sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem under the assumptions of alpha-univex and related functions were derived. Hang-Chin Lai [10] established the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities. Lai and Liu [11] employed the elementary method and technique to prove the necessary and sufficient optimality conditions for nondifferentiable minimax fractional programming problem involving convexity. In [12], a unified higherorder dual for a nondifferentiable minimax programming problem was formulated involving generalized higherorder (F, α, ρ, d)-Type I functions. Semi-infinite programming have been a subject of wide interest since they play a key role in a particular physical or social science situation, i.e., control of robots, mechanical stress of materials, and air pollution abatement etc. Recently, Qingxiang zhang [13] obtained the necessary and sufficient optimality conditions for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected functions. In [14, 15, 16, 17], the optimality conditions under various constraints qualification for semi-infinite programming problems were established. In this paper, motivated by the above work, we first define a kind of generalize convexity about the H-tangent derivative. Then, the sufficient optimality conditions are obtained for a class of min-max fractional semi-infinite programming problem involving the new generalized convexity. Further, we develop duality theory. Several duality results are established for the optimization problem. II. DEFINITIONS AND PRELIMINARIES Let X ⊂ R n be a nonempty set, x0 ∈ X , d ∈ R n and H ( x0 , f ( x0 )) be f : X → R {+∞} be a function, Tepif +
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H − tangent cone of epif + with respect to ( x0 , f ( x0 )) . We say that f ( x ; d ) is H − tangent derivative of f at H
0
In this section, we consider the following minimax fractional semi-infinite programming problem: ( SIFP ) minimizeF ( x ) = sup
x 0 along the direction d , where
y∈Y
H = f H ( x 0 ; d ) inf{η ( d ,η ) ∈ Tepif ( x0 , f ( x0 ))} +
subject to g ( x, u ) ≤ 0 , u ∈ U , x ∈ X , where X is a
To define a class of new functions, we suppose that X is nonempty open subset of R n , real valued function f : X → R is H − tangent derivable at x 0 ∈ X , ρ ∈ R ,
b : X × X × [0,1] → R+ , lim+ b= ( x, x 0 ; λ ) b( x, x 0 ), ϕ : R → R λ →0
, η : X × X → R n , θ : X × X → R n , where η and θ are vectori- al application. Definition2.1. f is said to be generalized uniform
( BH , ρ ) − invex function at x ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that 0
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≥ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 )
2
nonempty open subset of R n , Y is compact subset of R m ; f (⋅ , ⋅) : X × Y → R , h(⋅ , ⋅) : X × Y → R , f ( x, ⋅), h( x, ⋅) are continuous on Y for every x ∈ X ; g : X × U → R r and U ⊂ R r is an infinite index set; f ( x, y ) ≥ 0 and h( x, y ) > 0 for each ( x, y ) ∈ X × Y . We assume that f ( x, ⋅), h( x, ⋅) and g (⋅, u ) are
H − tangent derivable at x ∈ X . We put X 0 = {x g ( x, u ) ≤ 0, u ∈ U } for the feasible set of problem (SIFP). For each x ∈ X 0 , we define
= ∆ { j g ( x, u j ) ≤ 0, x ∈ X , u j ∈ U }, J ( x 0 ) ={ j g ( x 0 , u j ) =0, x 0 ∈ X , u j ∈ U },
Definition2.2. f is said to be strictly generalized uniform ( BH , ρ ) − invex function at x 0 ∈ X , if for any
x ∈ X and x ≠ x , there exists b, ϕ ,η , θ and ρ , such that 0
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] > f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 )
2
Definition2.3. f is said to be generalized uniform
( BH , ρ ) − pseudoinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that b( x, x )ϕ [ f ( x ) − f ( x )] < 0 0
* U= {u j ∈ U g ( x, u j ) ≤ 0, x ∈ X , j ∈ ∆},
= Λ {µ j µ j ≥ 0, j ∈ ∆} Where U * is a countable subset of U , in the set Λ , every µ j ≥ 0 , for all j ∈ ∆ , and only finitely many are strictly positive.
Y ( x) = {y ∈Y
0
f ( x, y ) f ( x, z ) < sup h( x, y ) h( x, z ) z∈Y
}
= + 1, λ (λ1 , λ2 ,, λs ) ∈ R+s Q {( s, λ , y ) ∈ N × R+s × R ms 1 ≤ s ≤ n =
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) < 0
s
Definition2.4. f is said to be strictly generalized uniform ( BH , ρ ) − pseudoinvex function at x ∈ X , if 0
for any x ∈ X and x ≠ x 0 , there exists b, ϕ ,η , θ and ρ , such that
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≤ 0 2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) < 0 Definition2.5. f is said to be generalized uniform
( BH , ρ ) − quasiinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that
with ∑ λi = 1,and y = ( y1 , y 2 ,, y s ) with y i ∈ Y ( x), i = 1,, s} i =1
In view of the continuity of f ( x, ⋅) and h( x, ⋅) on Y and compactness of Y , it is clear that Y ( x ) is nonempty compact subset of Y for each x ∈ X , and for any f ( x0 , yi ) , which is always a y i ∈ Y ( x 0 ) , we let q* = h( x 0 , y i ) constant. Definition3.1. For the problem (SIFP), a point x 0 ∈ X 0 is said to be an optimal solution, if for any x ∈ X 0 such that
f ( x0 , y)
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≤ 0
f ( x, y )
sup h( x , y ) ≤ sup h( x, y ) 0
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) ≤ 0 Definition2.6. f is said to be weakly generalized uniform ( BH , ρ ) − quasiinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that
y∈Y
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) ≤ 0 III. SUFFICIENT OPTIMALITY CONDITIONS
y∈Y
Definition3.2. It is said that x * satisfies the Kuhn-Tuker constraint qualification for (SIFP), if there exists s* > 0, λi* ≥ 0,1 ≤ i ≤ s* , µ *j ∈ Λ, j ∈ ∆, y i ∈ Y ( x * ),1 ≤ i ≤ s* and q* ∈ R , such that
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] < 0
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f ( x, y ) , h( x, y )
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
s*
∑ λ ( f − q h) i =1
* i
*
∑µ g * j
j∈∆
H x
f ( x, y i ) − q*h( x, y= f ( x* , y i ) − q*h( x* , y i ), i) < 0
( x * , y i ;η ( x, x * )) +
H x
( x , u ;η ( x, x )) ≥ 0, ∀u ∈ U j
*
741
j
*
f ( x * , y i ) − q* h ( x * , y i ) = 0, i = 1,, s*
i = 1,, s* From (vi), we get
*
b0 ( x, x * )φ0 [( f ( x, y i ) − q*h( x, y i ))
(1)
−( f ( x * , y i ) − q*h( x * , y i ))] < 0
j µ *j g ( x* , u= ) 0, j ∈ ∆ s*
∑λ + ∑ µ
=i 1
* i
j∈∆
s* * j =i 1
Then from (i), we have
≠ 0, ∑ λ = 1 * i
2
( f − q*h ) Hx ( x * , y i ;η ( x, x * )) + ρ i* θ ( x, x * ) < 0
Theorem3.1. Let x * ∈ X 0 and for any x ∈ X 0 , we assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ , j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R ,τ ∈ R s*
*
*
(∆)
s*
Since λi* ≥ 0 and ∑ λi* = 1 , we have i =1
s*
, such that
∑ λ ( f − q h)
y i ∈ Y ( x= ), i 1,, s ,( f − q h )(⋅, y i ) is *
(i)For any
*
*
i =1
(ii) For any u ∈ U , j ∈ J ( x ), g (⋅, u ) is generalized j
*
j
*
uniform ( BH ,τ *j ) − invex at x * with respect to b1 and φ1 ; s*
(iii) ∑ λ ( f − q h ) ( x , y i ;η ( x, x )) i =1
* i
*
H x
*
*
H x
( x * , y i ;η ( x, x * ))
s*
genralized uniform ( BH , ρ ) − invex at x with respect to b0 and φ0 ;
+ ∑ λi* ρ i* θ ( x, x * ) < 0
*
* i
* i
2
i =1
Now from (iii) and (vii), we get
∑µ g j∈∆
* j
H x
( x * , u j ;η ( x, x * )) + ∑ µ *jτ *j θ ( x, x * ) > 0 2
j∈∆
By (iv), we know that as j ∈ ∆ \ J ( x * ), µ *j = 0, always
*
+ ∑ µ *j g xH ( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆; j∈∆
holds for any u j ∈ U * . Hence, as j ∈ J ( x * ), we also have
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
∑
j∈∆
µ *j g xH ( x* , u j ;η ( x, x* ))
j∈J ( x ) *
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0)= 0, a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, x * ) > 0, b1 ( x, x * ) ≥ 0 ; s*
(vii)
∑λ ρ + ∑ µ τ * i
=i 1
* i
j∈∆
* * j j
+
∑
(2)
2
µ *jτ *j θ ( x, x* ) > 0
j∈J ( x* )
But as j ∈ J ( x * ), we know
≥0.
g ( x= , u j ) ≤ 0 g ( x* , u j ), u j ∈ U *
Then x * is an optimal solution of (SITP). Proof: Suppose that x * is not an optimal solution of (SITP). Then there exists x ∈ X , such that 0
From (vi), we get
b1 ( x, x * )φ1[( g ( x, u j ) − g ( x* , u j )] ≤ 0, ∀u j ∈ U * By (ii), we have
f ( x, y ) f ( x* , y ) sup h( x, y ) < sup h( x* , y ) y∈Y y∈Y
2
g xH ( x * , u j ;η ( x, x* )) + τ *j θ ( x, x* ) ≤ 0, ∀u j ∈ U * , j ∈ J ( x * )
Also
f ( x* , y i ) = q* , ∀ y i ∈ Y ( x * ), i = 1,, s* h( x* , y i )
f ( x* , y )
sup h( x , y ) = *
y∈Y
Since µ *j ∈ Λ, j ∈ J ( x * ) , we get
∑
µ *j g xH ( x* , u j ;η ( x, x* ))
j∈J ( x ) *
Further
+
f ( x, y i ) f ( x, y ) ≤ sup h( x, y i ) h ( x, y ) y∈Y
∑
2
µ *jτ *j θ ( x, x* ) ≤ 0, ∀u j ∈ U *
j∈J ( x* )
Finally, we have a contradiction. Thus the theorem is proved and x * is an optimal solution of (SITP). Theorem3.2. Let x * ∈ X 0 and for any x ∈ X 0 , we
Thus, we have
f ( x, y i ) ≤ q* , i = 1,, s*. h( x, y i )
assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ,
That is
j ∈ ∆ , and b0 , φ0 , b1 , φ1 ,η , θ , ρ * ∈ R s ,τ * ∈ R ( ∆ ) , such that *
f ( x, y i ) − q h( x, y i ) < 0, i = 1,, s *
By (v), we obtain
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* (i) For any y i ∈ Y ( x= ), i 1,, s* ,( f − q*h )(⋅, y i ) is
s*
∑ λ ( f − q h)
generalized uniform ( BH , ρ ) − pseudoinvex at x with *
* i
i =1
respect to b0 and φ0 ; uniform ( BH ,τ ) − quasiinvex at x with respect to b1 and * j
*
i =1
(3) * i
* i
2
*
j g ( x, u= ) ≤ 0 g ( x * , u j ), ∀u j ∈ U *
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * )) +
Then using (vi), we obtain b1 ( x, x * )φ1[( g ( x, u j ) − g ( x* , u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( x * ) Now by (ii), we have
i =1
H x
( x * , y i ;η ( x, x * ))
Also, as j ∈ J ( x * ), we have
s*
* j
H x
+ ∑ λ ρ θ ( x, x ) < 0
φ1 ;
j∈∆
*
s*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is generalized
∑µ g
* i
( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆;
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
2
g xH ( x * , u j ;η ( x, x* )) + τ *j θ ( x, x* ) ≤ 0
j∈∆
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
Since µ *j ∈ Λ, j ∈ J ( x * ) , it follows that
∑
b0 ( x, x ) > 0 , b1 ( x, x ) ≥ 0; *
*
j∈J ( x* )
2
µ *jτ *j θ ( x, x* ) ≤ 0
j∈J ( x* )
Also by (iv), as j ∈ ∆ \ J ( x ), we have µ *j = 0 . So
s*
*
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
=i 1
∑
µ *j g xH ( x* , u j ;η ( x, x* )) +
j∈∆
*
Then x is an optimal solution of (SIFP). Proof: Suppose that x * is not an optimal solution of (SIFP). Then there exists x ∈ X 0 , such that
∑µ g j∈∆
* j
H x
( x * , u j ;η ( x, x * )) + ∑ µ *jτ *j θ ( x, x * ) ≤ 0 2
j∈∆
(4)
Now, adding (3) and (4), then from (vii), we have s*
∑ λ ( f − q h)
f ( x, y ) f ( x* , y ) sup h( x, y ) < sup h( x* , y ) y∈Y y∈Y
* i
i =1
Also f ( x* , y ) f ( x* , y ) sup h( x* , y ) = h( x* , y i ) = q* , ∀ y i ∈ Y ( x* ), i = 1,, s*. y∈Y i Further
f ( x, y i ) f ( x, y ) ≤ sup h( x, y i ) h ( x, y ) y∈Y Thus, we have
*
H x
( x * , y i ;η ( x, x * ))
+ ∑ µ *j g xH ( x * , u j ;η ( x, x* )) j∈∆
s*
< −( ∑ λi* ρ i* + ∑ µ *jτ *j ) θ ( x, x* ) ≤ 0, ∀u j ∈ U *
=i 1
2
j∈∆
Finally, we have a contradiction. Hence x * is an optimal solution of (SIFP). Theorem3.3. Let x * ∈ X 0 and for any x ∈ X 0 , we assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ , j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ * ∈ R s ,τ * ∈ R ( ∆ ) , such that *
f ( x, y i ) ≤ q* , i = 1,, s* . h( x, y i )
* (i) For any y i ∈ Y ( x= ), i 1,, s* , ( f − q*h )(⋅ , y i ) is
strictly generalized uniform ( BH , ρ i* ) − invex at x * with respect to b0 and φ0 ;
Which is equivalent to
f ( x, y i ) − q*h( x, y i ) < 0, i = 1,, s*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is strictly
By (v), we get
f ( x, y i ) − q*h( x, y= f ( x* , y i ) − q*h( x* , y i ), i) < 0 i = 1,, s*
b1 and φ1 ; s*
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * ))
From (vi), we get
i =1
b0 ( x, x * )φ0 [( f ( x, y i ) − q*h( x, y i ))
+ ∑ µ *j g xH ( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆ ;
−( f ( x * , y i ) − q*h( x * , y i ))] < 0
j∈∆
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
Then by (i), we have
j∈∆
2
( f − q h ) ( x , y i ;η ( x, x )) + ρ θ ( x, x ) < 0 *
generalized uniform ( BH ,τ *j ) − invex at x * with respect to
H x
*
*
s*
* i
Since λi* ≥ 0 and ∑ λi* = 1 , we get i =1
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*
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, x * ) > 0, b1 ( x, x * ) ≥ 0 ;
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
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s*
s
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
λ ∑=
Then x * is an optimal solution of (SIFP). Theorem3.4. Let x * ∈ X 0 and for any x ∈ X 0 , we
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that
=i 1
assume that there exists ( s , λ , y ) ∈ Q , q ∈ R+ , µ ∈ Λ , *
*
*
j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R ,τ ∈ R *
s*
*
(∆)
* j
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
j∈∆
(i)For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − invex with respect to b0 and φ0 at z ; (ii) For all u j ∈ U * , j ∈ J ( z ), g (⋅, u j ) is generalized
, such that
* (i) For any y i ∈ Y ( x= ), i 1,, s* , ( f − q*h )(⋅ , y i ) is
strictly generalized uniform ( BH , ρ ) − pseudoinvex at * i
uniform ( BH ,τ j ) − invex with respect to b1 and φ1 at z ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
x with respect to b0 and φ0 ;
b0 ( x, z ) > 0, b1 ( x, z ) ≥ 0
*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is generalized uniform ( BH ,τ *j ) − quasiinvex at x * with respect to b1 and φ1 ;
s
∑λ ρ + ∑ µ τ
(iv)
i
=i 1
i
j∈∆
j
j
≥0.
Then s
*
f ( x, y )
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * ))
sup h( x, y ) ≥ q y∈Y
i =1
+ ∑ µ g ( x , u ;η ( x, x )) ≥ 0, ∀u ∈ U , j ∈ ∆ ; H x
* j
j∈∆
j
*
*
j
*
Proof: Suppose contrary that
f ( x, y )
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
sup h( x, y ) < q y∈Y
j∈∆
(v) f ( x , y i ) − q h( x , y i ) = 0, i = 1,, s ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0, *
*
*
*
Then, we have f ( x, y ) − q hx ( , y ) < 0, ∀y ∈ Y Using the constraint condition (5), it follows that for all y i , i = 1,, s , we get
b0 ( x, x * ) ≥ 0, b1 ( x, x * ) ≥ 0 ; s*
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
=i 1
f ( x, y i ) − qh( x, y i ) < 0 ≤ f ( z, y i ) − qh( z, y i )
j∈∆
By (iii), we get
Then x * is an optimal solution of (SIFP).
( , y i )) − b0 ( x, z )φ0 [( f ( x, y i ) − q hx
IV. DUALITY THEOREMS In this section, we formulate a dual problem to the minmax problem (SIVP). (SIFD)
max
sup
( s ,λ , y )∈Q ( s ,λ , q )∈D ( s ,λ , y ) s*
∑ λ ( f − qh) i =1
i
H x
q
j∈∆
2
( f − q h) Hx ( z, y i ;η ( x, z )) + ρ i θ ( x, z ) < 0 s
i =1
( z, y i ;η ( x, z )) j
Then (i) yields
Since λi ≥ 0 and ∑ λi = 1 , it follows that s
+ ∑ µ j g ( z, u ;η ( x, z )) ≥ 0, ∀u ∈ U , j ∈ ∆; H x
( f ( z, y i ) − q hz ( , y i ))] < 0
s
∑ λi ( f − q h) Hx ( z, y i ;η ( x, z )) + ∑ λi ρi θ ( x, z ) < 0
(5)
f ( z, y i ) − q hz ( , y i ) ≥ 0, i = 1, 2,, s;
µ j g ( z, u j ) ≥ 0, ∀u j ∈U * , j ∈ ∆; ( s, λ , y ) ∈ Q . Where D ( s, λ , y ) denotes the set of all ( z, µ , q) ∈
R n × Λ × R+ to satisfy relations (5). If for a triplet ( s, λ , y ) ∈ Q , the set D ( s, λ , y ) = Φ , then we define the supremum over D ( s, λ , y ) to be −∞ . Theorem4.1. (Weak duality) Let x be a feasible solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution of (SIFD). For λi ≥ 0, i = 1, 2,, s with
Hence by the constraint condition (5) and (iv), we get
∑µ g j∈∆
j
H x
( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) > 0 2
j∈∆
For j ∈ J ( z ) , using the constraint condition (5), then we get
g ( x, u j ) ≤ 0 ≤ g ( z, u j ), ∀u j ∈ U * From (iii), we get b1 ( x, z )φ1 [( g ( x, u j ) − g ( z, u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( z ) By (ii), we have 2
g xH ( z, u j ;η ( x, z )) + τ j θ ( x, z ) ≤ 0 Since µ j ∈ Λ, j ∈ J ( z ) , it follows that
∑
j∈J ( z )
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2
j * =i 1 =i 1
µ j g xH ( z, u j ;η ( x, z )) +
∑
j∈J ( z )
2
µ jτ j θ ( x , z ) ≤ 0
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JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
s s According to the constraint condition (5), for all 2 H λ ( f − q h ) ( z , y ; η ( x , z )) + λi ρ i θ ( x, z ) < 0 * ∑ ∑ i i x u ∈ U , we let µ j = 0 , as j ∈ ∆ \ J (= z) . i 1 =i 1 Thus, we have (6) Hence by the constraint condition (5) and (iv), we get 2 µ j g xH ( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) ≤ 0 ∑ 2 H j j∈∆ j∈∆ ∑ µ j g x ( z, u ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) > 0 j
We have a contradiction. Therefore, we conclude f ( x, y ) that sup ≥ q . Hence, the proof of the theorem is y∈Y h ( x , y )
j∈∆
g ( x, u j ) ≤ 0 ≤ g ( z, u j ), ∀u j ∈ U *
complete. Theorem4.2. (Weak duality) Let x be a feasible solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution s
λ ∑=
(SIFD).
For
From (iii), we get
b1 ( x, z )φ1 [( g ( x, u j ) − g ( z, u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( z )
λi ≥ 0, i = 1, 2,, s with
By (ii), we have
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
of
2
g xH ( z, u j ;η ( x, z )) + τ j θ ( x, z ) ≤ 0 Since µ j ∈ Λ, j ∈ J ( z ) , it follows that
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that (i)For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − pseudoinvex with respect to b0 and φ0 at z ; (ii) For all u ∈ U , j ∈ J ( z ), g (⋅, u ) is generalized j
*
j
∑
j∈J ( z )
∑λ ρ + ∑ µ τ
=i 1
i
i
j∈∆
j
∑µ g j
j∈∆
j∈J ( z )
2
µ jτ j θ ( x , z ) ≤ 0
H x
( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) ≤ 0 2
j∈∆
(7)
Adding (6) and (7), then from (iv), we obtain s
≥0.
j
∑
Thus, we have
b0 ( x, z ) > 0, b1 ( x, z ) ≥ 0 s
µ j g xH ( z, u j ;η ( x, z )) +
According to the constraint condition (5), for all u j ∈ U * , we let µ j = 0 , as j ∈ ∆ \ J ( z ) .
uniform ( BH ,τ j ) − quasiinvex with respect to b1 and φ1 at z; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
(iv)
j∈∆
For j ∈ J ( z ) , using the constraint condition (5), then we get
∑λ
=i 1
Then
( f − q h) Hx ( z, y i ;η ( x, z )) + ∑ µ j g xH ( z, u j ;η ( x, z ))
i
j∈∆
s
< −( ∑ λi ρ i + ∑ µ jτ j ) θ ( x, z ) ≤ 0
f ( x, y )
sup h( x, y ) ≥ q
=i 1
y∈Y
f ( x, y )
sup h( x, y ) < q y∈Y
solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution of (SIFD). For λi ≥ 0, i = 1, 2,, s with
Then, we have f ( x, y ) − q hx ( , y ) < 0, ∀y ∈ Y
s
Using the constraint condition (5), it follows that for all y i , i = 1,, s , we get
f ( x, y i ) − qh( x, y i ) < 0 ≤ f ( z, y i ) − qh( z, y i ) By (iii), we get
b0 ( x, z )φ0 [( f ( x, y i ) − q hx ( , y i )) − ( f ( z, y i ) − q hz ( , y i ))] < 0 2
( f − q h) Hx ( z, y i ;η ( x, z )) + ρ i θ ( x, z ) < 0
i =1
λ ∑=
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that (i) For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − quasiinvex with respect to b0 and φ0 at z ; (ii) For all u j ∈ U * , j ∈ J ( z ), g (⋅, u j ) is strictly generalized uniform ( BH ,τ j ) − pseudoinvex with respect
Then (i) yields
s
to b1 and φ1 at z ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, z ) ≥ 0, b1 ( x, z ) ≥ 0 s
(iv)
∑λ ρ + ∑ µ τ
=i 1
Then
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j∈∆
We have a contradiction. Hence, the proof of the theorem is complete. Similarly, we can derive the following theorems. Theorem4.3. (Weak duality) Let x be a feasible
Proof: Suppose contrary that
Since λi ≥ 0 and ∑ λi = 1 , it follows that
2
i
i
j∈∆
j
j
≥0.
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745
Proof: Suppose on the contrary that x * ≠ z . From
f ( x, y )
sup h( x, y) ≥ q
*
Theorem 4.4, we know that there exists ( s* , λ * , y ) ∈ Q
y∈Y
Theorem4.4. (Strong duality) Let x * be an optimal solution of problem (SIFP). Assume that x * satisfies K − T constraint qualification for (SIFP). Then there *
*
exists ( s* , λ * , y ) ∈ Q and ( x * , µ * , q* ) ∈ D ( s* , λ * , y ) , *
such that ( x * , µ * , q* , s* , λ * , y ) is a feasible solution of (SIFD). If the hypothesis of theorem 4.1 is also satisfied, *
then ( x * , µ * , q* , s* , λ * , y ) is an optimal of (SIFD), furthermore, the two problems (SIFP) and (SIFD) have the same optimal value. Proof: Since x * is an optimal solution of problem (SIFP), and x * satisfies K − T constraint qualification
*
and ( x * , µ * , q* ) ∈ D ( s* , λ * , y ) , such that ( x * , µ * , q* , s* , *
λ * , y ) is an optimal solution of (SIFD) with the optimal value.
q* = sup y∈Y
Now using the conditions (i)- (iv) and like the proof of theorem 4.1 (by replacing x by x * and ( z, µ , q, s, λ , y ) by
( z, µ , q, s, λ , y ) ), we arrive at the strict inequality. sup y∈Y
*
for (SIFP), then there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ and
µ *j ∈ Λ , j ∈ ∆ , such that the relations (3.1)- (3.4) hold. *
Therefore, ( x * , µ * , q* , s* , λ * , y ) is a feasible solution of (SIFD), and we have *
q* =
f ( x* , y )
f ( x* , y ) h( x* , y )
f ( x* , y ) >q h( x* , y )
This contradicts the fact
sup y∈Y
f ( x* , y ) * q = q= h( x* , y )
Therefore, we conclude that x * = z . Hence, the proof of the theorem is complete.
*
h( x* , y )
V. CONCLUSION
The optimality of this feasible solution for (SIFD) follows from theorem 4.1. It is clear that the two problems have the same optimal values. Remark 4.1. the result of strong duality under the hypothesis of theorem 4.2 (or 4.3) follows with the same lines as the argument given in theorem 4.1. Theorem 4.5. (Strict reverse duality) Let x * and
( z, µ , q, s, λ , y ) be an optimal solution of problem (SIFP) and (SIFD), respectively. Assume that x * satisfies K − T s
constraint qualification for (SIFP). And for
∑λ
i
=1 ,
Throughout this paper, we have defined a new generalized convex function, extending many wellknown classes of generalized convex functions. Furthermore, we have achieved some sufficient optimality conditions for a class of multiobjective semiinfinite programming problem. Finally, we have formulated the multiobjective dual problem and proved the results concerning weak and strong duality between the primal (SIVP) and the dual (SIVD), there should be further opportunities for exploiting this structure of the semi-infinite programming problem.
i =1
µ i ∈ Λ , j ∈ ∆ and q ∈ R+ , there exists b0 , φ0 , b1 , φ1 ,η , θ ,
ρ ∈ R s ,τ ∈ R ( ∆ ) , the following conditions are fulfilled: (i)For all y , f (⋅, y ) − qh )(⋅, y )e is strictly generalized uniform ( BH , ρ i ) − invex with respect to b0 and φ0 at z ; (ii)
For
all
u j ∈ U * , j ∈ J ( z ), g (⋅, u j )
is strictly
generalized uniform ( BH ,τ j ) − invex with respect to b1 and φ1 at z ; (iii) a ≤ 0 ⇒ φ0 ( a ) ≤ 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x * , z ) > 0, b1 ( x * , z ) ≥ 0 s
(iv)
∑λ ρ + ∑ µ τ
=i 1
i
i
j∈∆
j
j
≥0.
Then, x * = z ; that is, z is also an optimal solution of f ( z, y ) (SIFP) and q = sup . y∈Y h ( z , y )
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ACKNOWLEDGMENT This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 08JK237). REFERENCES [1] V. Preda, “On efficiency and duality for multiobjective programs”, J. Math. Anal. Appl., vol. 42, no. 3, pp. 234240, 1992. [2] Kuk, H., Lee, G.M., Kim, D.S., Nonsmooth multiobjective programs with-invexity, Ind. J. Pure. Appl. Math., vol. 29, no. 2, pp. 405-412, 1998. [3] Liang Z.A., Huang H.X., Pardalos P.M., “Optimality conditions and duality for a class of nonlinear fractional programming problems”, J. Optim. Theory Appl., vol. 110, no. 1, pp. 611-619, 2001. [4] Hanson M.A., Pini R., Singh C., “Multiobjective programming under generalized type I invexity”, J. Math. Anal. Appl., vol. 261, no. 2, pp. 562-577, 2002. [5] Suneja S.K., Srivastava M.K., “Optimality and duality in nondifferentiable multiobjective optimization involving dtype I and related functions”, J. Math. Anal. Appl., vol. 206, no. 2, pp. 465-479, 1997.
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[6] Aparna Mehra, Davinder Bhatia, “Optimality and duality for minimax problems involving arcwise connected and generalized arcwise connected functions”, J. Math. Anal. Appl., vol. 231, no. 2, pp. 425-445, 1999. [7] J.C. Liu, C.S. Wu, “On minimax fractional optimality conditions withconvex”, J. Math. Anal. Appl., vol. 219, no. 3, pp. 36-51, 1998. [8] J.C. Liu, C.S. Wu, “On minimax fractional optimality conditions with invexity”, J. Math. Anal. Appl., vol. 219, no. 3, pp. 21-35, 1998. [9] Mishra S.K., Pant R.P., Rautela J.S., “Generalized alphaunivexity and duality for nondifferentiable minimax fractional programming”, Nonlinear analysis theory method and applications, vol. 70, no. 1, pp. 144-158, 2009. [10] Lai Hangchen, Huang Toneyau, “Optimality conditions for nondifferentiable minimax fractional programming with complex variables”, J. Math. Anal. Appl., vol. 359, no. 1, pp. 229-239, 2009. [11] Lai Hangchin, Liu Jenchwan, “A new characterization on optimality and duality for nondifferentiable minimax fractional programming problems”, J. Math. Anal. Appl., vol. 12, no. 3, pp. 69-80, 2011. [12] Ahmal L., Husain Z., Sharma S., “Higher-order duality in nondifferentiable minimax programming with generalized type I functios”, J. Optim. Theory Appl., vol. 141, no. 1, pp. 1-12, 2009. [13] Qingxiang Zhang, “Optimality conditions and duality for semi-infinite programming involving B-arcwise connected
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[14]
[15]
[16]
[17]
functions”, J. Glob. Optim., vol. 45, no. 4, pp. 615-629, 2009. N. Kanzi, S. Nobakhtian, “Nonsmooth semi-infinite programming problems with mixed constraints”, J. Math. Anal. Appl., vol. 351, no. 1, pp. 170-181, 2009. N. Kanzi, S. Nobakhtian, “Optimality conditions for nonsmooth semi-infinite programming”, Optimization, vol. 59, no. 5, pp. 717-727, 2010. Shapiro A., “Semi-infinite programming duality discretization and optimality condition”, Optimization, vol. 58, no. 2, pp. 133-161, 2009. N. Kanzi, “Necessary optimality conditions for nonsmooth semi-infinite programming problems”, J. Glob. Optim., vol. 49, no. 1, pp. 713-725, 2011.
Xiaoyan Gao, female, was born in shanxi Province, China on August 28, 1979. She received her master’s degree in optimization theory and applications from Yan’an University, China in 2005. Now she is a lecturer of School of Science, Xi’an University of Science and Technology, Xi’an, China. Her main research fields include the generalized convexity, the optimization theory and applications for semi-infinite and multiobjective programming, etc.
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Optimality and Duality for Minimax Fractional Semi-Infinite Programming Xiaoyan Gao School of Science, Xi’an University of Science and Technology, Xi’an, China Corresponding author, Email: [email protected]
Abstract—The purpose of this paper is to consider a class of nonsmooth minimax fractional semi-infinite programming problem. Based on the concept of H − tangent derivative, a new generalization of convexity, namely generalized uniform ( BH , ρ ) − invexity, is defined for this problem. For such semi-infinite programming problem, several sufficient optimality conditions are established and proved by utilizing the above defined new classes of functions. The results extend and improve the corresponding results in the literature. Subsequently, these optimality conditions are utilized as a basis for formulating dual problems. Weak, strong and reverse duality theorems are also derived for dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable minimax fractional programming problems turn out to be special cases for the results described in the paper Index Terms— H − tangent derivative, generalized convexity, minimax fractional semi-infinite programming, optimality conditions, duality
I. INTRODUCTION In recent years, the concept of convexity and generalized convexity is well known in optimization theory and plays a central role in mathematical economics, management science, and optimization theory. Therefore, the research on convexity and generalized convexity is one of the most important aspects in mathematics programming. To relax convexity assumptions imposed on theorems on optimality conditions for generalized mathematical programming problems, various generalized convexity notations have been introduced. In particular, the concept of generalized ( F , ρ ) − convexity, introduced by Preda [1] is in turn an extension of the convexity and was used by several authors to obtain relevant results. In [2, 3], the concept of V − ρ − invexity and ( F , α , ρ , d ) − convexity were introduced, respectively. Other classes of generalized type I functions have been discussed in [4, 5]. On the other hand, a large literature was developed around generalized convexity and its applications in mathematical programming. Many authors investigated the optimality conditions and duality results for min-max programming problems under the conditions of generalized convexity. In particular, Aparna Mehra [6] employed various optimality conditions and duality © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.3.739-746
results under arcwise connectedness and generalized arcwise connectedness assumptions for a static minimax programming problem. Lin [7] and Wu [8] derived the sufficient optimality conditions for the generalized minmax fractional programming in the framework of ( F , ρ ) − convex functions and invex functions. In [9], the Karush-Kuhn-Tucker-type sufficient optimality conditions and duality theorems for a nondifferentiable minimax fractional programming problem under the assumptions of alpha-univex and related functions were derived. Hang-Chin Lai [10] established the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities. Lai and Liu [11] employed the elementary method and technique to prove the necessary and sufficient optimality conditions for nondifferentiable minimax fractional programming problem involving convexity. In [12], a unified higherorder dual for a nondifferentiable minimax programming problem was formulated involving generalized higherorder (F, α, ρ, d)-Type I functions. Semi-infinite programming have been a subject of wide interest since they play a key role in a particular physical or social science situation, i.e., control of robots, mechanical stress of materials, and air pollution abatement etc. Recently, Qingxiang zhang [13] obtained the necessary and sufficient optimality conditions for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected functions. In [14, 15, 16, 17], the optimality conditions under various constraints qualification for semi-infinite programming problems were established. In this paper, motivated by the above work, we first define a kind of generalize convexity about the H-tangent derivative. Then, the sufficient optimality conditions are obtained for a class of min-max fractional semi-infinite programming problem involving the new generalized convexity. Further, we develop duality theory. Several duality results are established for the optimization problem. II. DEFINITIONS AND PRELIMINARIES Let X ⊂ R n be a nonempty set, x0 ∈ X , d ∈ R n and H ( x0 , f ( x0 )) be f : X → R {+∞} be a function, Tepif +
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H − tangent cone of epif + with respect to ( x0 , f ( x0 )) . We say that f ( x ; d ) is H − tangent derivative of f at H
0
In this section, we consider the following minimax fractional semi-infinite programming problem: ( SIFP ) minimizeF ( x ) = sup
x 0 along the direction d , where
y∈Y
H = f H ( x 0 ; d ) inf{η ( d ,η ) ∈ Tepif ( x0 , f ( x0 ))} +
subject to g ( x, u ) ≤ 0 , u ∈ U , x ∈ X , where X is a
To define a class of new functions, we suppose that X is nonempty open subset of R n , real valued function f : X → R is H − tangent derivable at x 0 ∈ X , ρ ∈ R ,
b : X × X × [0,1] → R+ , lim+ b= ( x, x 0 ; λ ) b( x, x 0 ), ϕ : R → R λ →0
, η : X × X → R n , θ : X × X → R n , where η and θ are vectori- al application. Definition2.1. f is said to be generalized uniform
( BH , ρ ) − invex function at x ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that 0
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≥ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 )
2
nonempty open subset of R n , Y is compact subset of R m ; f (⋅ , ⋅) : X × Y → R , h(⋅ , ⋅) : X × Y → R , f ( x, ⋅), h( x, ⋅) are continuous on Y for every x ∈ X ; g : X × U → R r and U ⊂ R r is an infinite index set; f ( x, y ) ≥ 0 and h( x, y ) > 0 for each ( x, y ) ∈ X × Y . We assume that f ( x, ⋅), h( x, ⋅) and g (⋅, u ) are
H − tangent derivable at x ∈ X . We put X 0 = {x g ( x, u ) ≤ 0, u ∈ U } for the feasible set of problem (SIFP). For each x ∈ X 0 , we define
= ∆ { j g ( x, u j ) ≤ 0, x ∈ X , u j ∈ U }, J ( x 0 ) ={ j g ( x 0 , u j ) =0, x 0 ∈ X , u j ∈ U },
Definition2.2. f is said to be strictly generalized uniform ( BH , ρ ) − invex function at x 0 ∈ X , if for any
x ∈ X and x ≠ x , there exists b, ϕ ,η , θ and ρ , such that 0
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] > f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 )
2
Definition2.3. f is said to be generalized uniform
( BH , ρ ) − pseudoinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that b( x, x )ϕ [ f ( x ) − f ( x )] < 0 0
* U= {u j ∈ U g ( x, u j ) ≤ 0, x ∈ X , j ∈ ∆},
= Λ {µ j µ j ≥ 0, j ∈ ∆} Where U * is a countable subset of U , in the set Λ , every µ j ≥ 0 , for all j ∈ ∆ , and only finitely many are strictly positive.
Y ( x) = {y ∈Y
0
f ( x, y ) f ( x, z ) < sup h( x, y ) h( x, z ) z∈Y
}
= + 1, λ (λ1 , λ2 ,, λs ) ∈ R+s Q {( s, λ , y ) ∈ N × R+s × R ms 1 ≤ s ≤ n =
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) < 0
s
Definition2.4. f is said to be strictly generalized uniform ( BH , ρ ) − pseudoinvex function at x ∈ X , if 0
for any x ∈ X and x ≠ x 0 , there exists b, ϕ ,η , θ and ρ , such that
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≤ 0 2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) < 0 Definition2.5. f is said to be generalized uniform
( BH , ρ ) − quasiinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that
with ∑ λi = 1,and y = ( y1 , y 2 ,, y s ) with y i ∈ Y ( x), i = 1,, s} i =1
In view of the continuity of f ( x, ⋅) and h( x, ⋅) on Y and compactness of Y , it is clear that Y ( x ) is nonempty compact subset of Y for each x ∈ X , and for any f ( x0 , yi ) , which is always a y i ∈ Y ( x 0 ) , we let q* = h( x 0 , y i ) constant. Definition3.1. For the problem (SIFP), a point x 0 ∈ X 0 is said to be an optimal solution, if for any x ∈ X 0 such that
f ( x0 , y)
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] ≤ 0
f ( x, y )
sup h( x , y ) ≤ sup h( x, y ) 0
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) ≤ 0 Definition2.6. f is said to be weakly generalized uniform ( BH , ρ ) − quasiinvex function at x 0 ∈ X , if for any x ∈ X , there exists b, ϕ ,η , θ and ρ , such that
y∈Y
2
⇒ f H ( x 0 ,η ( x, x 0 )) + ρ θ ( x, x 0 ) ≤ 0 III. SUFFICIENT OPTIMALITY CONDITIONS
y∈Y
Definition3.2. It is said that x * satisfies the Kuhn-Tuker constraint qualification for (SIFP), if there exists s* > 0, λi* ≥ 0,1 ≤ i ≤ s* , µ *j ∈ Λ, j ∈ ∆, y i ∈ Y ( x * ),1 ≤ i ≤ s* and q* ∈ R , such that
b( x, x 0 )ϕ [ f ( x ) − f ( x 0 )] < 0
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f ( x, y ) , h( x, y )
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
s*
∑ λ ( f − q h) i =1
* i
*
∑µ g * j
j∈∆
H x
f ( x, y i ) − q*h( x, y= f ( x* , y i ) − q*h( x* , y i ), i) < 0
( x * , y i ;η ( x, x * )) +
H x
( x , u ;η ( x, x )) ≥ 0, ∀u ∈ U j
*
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j
*
f ( x * , y i ) − q* h ( x * , y i ) = 0, i = 1,, s*
i = 1,, s* From (vi), we get
*
b0 ( x, x * )φ0 [( f ( x, y i ) − q*h( x, y i ))
(1)
−( f ( x * , y i ) − q*h( x * , y i ))] < 0
j µ *j g ( x* , u= ) 0, j ∈ ∆ s*
∑λ + ∑ µ
=i 1
* i
j∈∆
s* * j =i 1
Then from (i), we have
≠ 0, ∑ λ = 1 * i
2
( f − q*h ) Hx ( x * , y i ;η ( x, x * )) + ρ i* θ ( x, x * ) < 0
Theorem3.1. Let x * ∈ X 0 and for any x ∈ X 0 , we assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ , j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R ,τ ∈ R s*
*
*
(∆)
s*
Since λi* ≥ 0 and ∑ λi* = 1 , we have i =1
s*
, such that
∑ λ ( f − q h)
y i ∈ Y ( x= ), i 1,, s ,( f − q h )(⋅, y i ) is *
(i)For any
*
*
i =1
(ii) For any u ∈ U , j ∈ J ( x ), g (⋅, u ) is generalized j
*
j
*
uniform ( BH ,τ *j ) − invex at x * with respect to b1 and φ1 ; s*
(iii) ∑ λ ( f − q h ) ( x , y i ;η ( x, x )) i =1
* i
*
H x
*
*
H x
( x * , y i ;η ( x, x * ))
s*
genralized uniform ( BH , ρ ) − invex at x with respect to b0 and φ0 ;
+ ∑ λi* ρ i* θ ( x, x * ) < 0
*
* i
* i
2
i =1
Now from (iii) and (vii), we get
∑µ g j∈∆
* j
H x
( x * , u j ;η ( x, x * )) + ∑ µ *jτ *j θ ( x, x * ) > 0 2
j∈∆
By (iv), we know that as j ∈ ∆ \ J ( x * ), µ *j = 0, always
*
+ ∑ µ *j g xH ( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆; j∈∆
holds for any u j ∈ U * . Hence, as j ∈ J ( x * ), we also have
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
∑
j∈∆
µ *j g xH ( x* , u j ;η ( x, x* ))
j∈J ( x ) *
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0)= 0, a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, x * ) > 0, b1 ( x, x * ) ≥ 0 ; s*
(vii)
∑λ ρ + ∑ µ τ * i
=i 1
* i
j∈∆
* * j j
+
∑
(2)
2
µ *jτ *j θ ( x, x* ) > 0
j∈J ( x* )
But as j ∈ J ( x * ), we know
≥0.
g ( x= , u j ) ≤ 0 g ( x* , u j ), u j ∈ U *
Then x * is an optimal solution of (SITP). Proof: Suppose that x * is not an optimal solution of (SITP). Then there exists x ∈ X , such that 0
From (vi), we get
b1 ( x, x * )φ1[( g ( x, u j ) − g ( x* , u j )] ≤ 0, ∀u j ∈ U * By (ii), we have
f ( x, y ) f ( x* , y ) sup h( x, y ) < sup h( x* , y ) y∈Y y∈Y
2
g xH ( x * , u j ;η ( x, x* )) + τ *j θ ( x, x* ) ≤ 0, ∀u j ∈ U * , j ∈ J ( x * )
Also
f ( x* , y i ) = q* , ∀ y i ∈ Y ( x * ), i = 1,, s* h( x* , y i )
f ( x* , y )
sup h( x , y ) = *
y∈Y
Since µ *j ∈ Λ, j ∈ J ( x * ) , we get
∑
µ *j g xH ( x* , u j ;η ( x, x* ))
j∈J ( x ) *
Further
+
f ( x, y i ) f ( x, y ) ≤ sup h( x, y i ) h ( x, y ) y∈Y
∑
2
µ *jτ *j θ ( x, x* ) ≤ 0, ∀u j ∈ U *
j∈J ( x* )
Finally, we have a contradiction. Thus the theorem is proved and x * is an optimal solution of (SITP). Theorem3.2. Let x * ∈ X 0 and for any x ∈ X 0 , we
Thus, we have
f ( x, y i ) ≤ q* , i = 1,, s*. h( x, y i )
assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ,
That is
j ∈ ∆ , and b0 , φ0 , b1 , φ1 ,η , θ , ρ * ∈ R s ,τ * ∈ R ( ∆ ) , such that *
f ( x, y i ) − q h( x, y i ) < 0, i = 1,, s *
By (v), we obtain
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*
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* (i) For any y i ∈ Y ( x= ), i 1,, s* ,( f − q*h )(⋅, y i ) is
s*
∑ λ ( f − q h)
generalized uniform ( BH , ρ ) − pseudoinvex at x with *
* i
i =1
respect to b0 and φ0 ; uniform ( BH ,τ ) − quasiinvex at x with respect to b1 and * j
*
i =1
(3) * i
* i
2
*
j g ( x, u= ) ≤ 0 g ( x * , u j ), ∀u j ∈ U *
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * )) +
Then using (vi), we obtain b1 ( x, x * )φ1[( g ( x, u j ) − g ( x* , u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( x * ) Now by (ii), we have
i =1
H x
( x * , y i ;η ( x, x * ))
Also, as j ∈ J ( x * ), we have
s*
* j
H x
+ ∑ λ ρ θ ( x, x ) < 0
φ1 ;
j∈∆
*
s*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is generalized
∑µ g
* i
( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆;
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
2
g xH ( x * , u j ;η ( x, x* )) + τ *j θ ( x, x* ) ≤ 0
j∈∆
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
Since µ *j ∈ Λ, j ∈ J ( x * ) , it follows that
∑
b0 ( x, x ) > 0 , b1 ( x, x ) ≥ 0; *
*
j∈J ( x* )
2
µ *jτ *j θ ( x, x* ) ≤ 0
j∈J ( x* )
Also by (iv), as j ∈ ∆ \ J ( x ), we have µ *j = 0 . So
s*
*
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
=i 1
∑
µ *j g xH ( x* , u j ;η ( x, x* )) +
j∈∆
*
Then x is an optimal solution of (SIFP). Proof: Suppose that x * is not an optimal solution of (SIFP). Then there exists x ∈ X 0 , such that
∑µ g j∈∆
* j
H x
( x * , u j ;η ( x, x * )) + ∑ µ *jτ *j θ ( x, x * ) ≤ 0 2
j∈∆
(4)
Now, adding (3) and (4), then from (vii), we have s*
∑ λ ( f − q h)
f ( x, y ) f ( x* , y ) sup h( x, y ) < sup h( x* , y ) y∈Y y∈Y
* i
i =1
Also f ( x* , y ) f ( x* , y ) sup h( x* , y ) = h( x* , y i ) = q* , ∀ y i ∈ Y ( x* ), i = 1,, s*. y∈Y i Further
f ( x, y i ) f ( x, y ) ≤ sup h( x, y i ) h ( x, y ) y∈Y Thus, we have
*
H x
( x * , y i ;η ( x, x * ))
+ ∑ µ *j g xH ( x * , u j ;η ( x, x* )) j∈∆
s*
< −( ∑ λi* ρ i* + ∑ µ *jτ *j ) θ ( x, x* ) ≤ 0, ∀u j ∈ U *
=i 1
2
j∈∆
Finally, we have a contradiction. Hence x * is an optimal solution of (SIFP). Theorem3.3. Let x * ∈ X 0 and for any x ∈ X 0 , we assume that there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ , µ *j ∈ Λ , j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ * ∈ R s ,τ * ∈ R ( ∆ ) , such that *
f ( x, y i ) ≤ q* , i = 1,, s* . h( x, y i )
* (i) For any y i ∈ Y ( x= ), i 1,, s* , ( f − q*h )(⋅ , y i ) is
strictly generalized uniform ( BH , ρ i* ) − invex at x * with respect to b0 and φ0 ;
Which is equivalent to
f ( x, y i ) − q*h( x, y i ) < 0, i = 1,, s*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is strictly
By (v), we get
f ( x, y i ) − q*h( x, y= f ( x* , y i ) − q*h( x* , y i ), i) < 0 i = 1,, s*
b1 and φ1 ; s*
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * ))
From (vi), we get
i =1
b0 ( x, x * )φ0 [( f ( x, y i ) − q*h( x, y i ))
+ ∑ µ *j g xH ( x * , u j ;η ( x, x * )) ≥ 0, ∀u j ∈ U * , j ∈ ∆ ;
−( f ( x * , y i ) − q*h( x * , y i ))] < 0
j∈∆
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
Then by (i), we have
j∈∆
2
( f − q h ) ( x , y i ;η ( x, x )) + ρ θ ( x, x ) < 0 *
generalized uniform ( BH ,τ *j ) − invex at x * with respect to
H x
*
*
s*
* i
Since λi* ≥ 0 and ∑ λi* = 1 , we get i =1
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*
(v) f ( x * , y i ) − q*h( x * , y i ) = 0, i = 1,, s* ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, x * ) > 0, b1 ( x, x * ) ≥ 0 ;
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
743
s*
s
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
λ ∑=
Then x * is an optimal solution of (SIFP). Theorem3.4. Let x * ∈ X 0 and for any x ∈ X 0 , we
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that
=i 1
assume that there exists ( s , λ , y ) ∈ Q , q ∈ R+ , µ ∈ Λ , *
*
*
j ∈ ∆ and b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R ,τ ∈ R *
s*
*
(∆)
* j
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
j∈∆
(i)For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − invex with respect to b0 and φ0 at z ; (ii) For all u j ∈ U * , j ∈ J ( z ), g (⋅, u j ) is generalized
, such that
* (i) For any y i ∈ Y ( x= ), i 1,, s* , ( f − q*h )(⋅ , y i ) is
strictly generalized uniform ( BH , ρ ) − pseudoinvex at * i
uniform ( BH ,τ j ) − invex with respect to b1 and φ1 at z ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
x with respect to b0 and φ0 ;
b0 ( x, z ) > 0, b1 ( x, z ) ≥ 0
*
(ii) For any u j ∈ U * , j ∈ J ( x * ), g (⋅, u j ) is generalized uniform ( BH ,τ *j ) − quasiinvex at x * with respect to b1 and φ1 ;
s
∑λ ρ + ∑ µ τ
(iv)
i
=i 1
i
j∈∆
j
j
≥0.
Then s
*
f ( x, y )
(iii) ∑ λi* ( f − q*h ) Hx ( x * , y i ;η ( x, x * ))
sup h( x, y ) ≥ q y∈Y
i =1
+ ∑ µ g ( x , u ;η ( x, x )) ≥ 0, ∀u ∈ U , j ∈ ∆ ; H x
* j
j∈∆
j
*
*
j
*
Proof: Suppose contrary that
f ( x, y )
(iv) ∑ µ *j g ( x * , u j )= 0, ∀u j ∈ U * , j ∈ ∆ ;
sup h( x, y ) < q y∈Y
j∈∆
(v) f ( x , y i ) − q h( x , y i ) = 0, i = 1,, s ; (vi) a < 0 ⇒ φ0 ( a ) < 0 , a ≤ 0 ⇒ φ1 ( a ) ≤ 0, *
*
*
*
Then, we have f ( x, y ) − q hx ( , y ) < 0, ∀y ∈ Y Using the constraint condition (5), it follows that for all y i , i = 1,, s , we get
b0 ( x, x * ) ≥ 0, b1 ( x, x * ) ≥ 0 ; s*
(vii) ∑ λi* ρ i* + ∑ µ *jτ *j ≥ 0 .
=i 1
f ( x, y i ) − qh( x, y i ) < 0 ≤ f ( z, y i ) − qh( z, y i )
j∈∆
By (iii), we get
Then x * is an optimal solution of (SIFP).
( , y i )) − b0 ( x, z )φ0 [( f ( x, y i ) − q hx
IV. DUALITY THEOREMS In this section, we formulate a dual problem to the minmax problem (SIVP). (SIFD)
max
sup
( s ,λ , y )∈Q ( s ,λ , q )∈D ( s ,λ , y ) s*
∑ λ ( f − qh) i =1
i
H x
q
j∈∆
2
( f − q h) Hx ( z, y i ;η ( x, z )) + ρ i θ ( x, z ) < 0 s
i =1
( z, y i ;η ( x, z )) j
Then (i) yields
Since λi ≥ 0 and ∑ λi = 1 , it follows that s
+ ∑ µ j g ( z, u ;η ( x, z )) ≥ 0, ∀u ∈ U , j ∈ ∆; H x
( f ( z, y i ) − q hz ( , y i ))] < 0
s
∑ λi ( f − q h) Hx ( z, y i ;η ( x, z )) + ∑ λi ρi θ ( x, z ) < 0
(5)
f ( z, y i ) − q hz ( , y i ) ≥ 0, i = 1, 2,, s;
µ j g ( z, u j ) ≥ 0, ∀u j ∈U * , j ∈ ∆; ( s, λ , y ) ∈ Q . Where D ( s, λ , y ) denotes the set of all ( z, µ , q) ∈
R n × Λ × R+ to satisfy relations (5). If for a triplet ( s, λ , y ) ∈ Q , the set D ( s, λ , y ) = Φ , then we define the supremum over D ( s, λ , y ) to be −∞ . Theorem4.1. (Weak duality) Let x be a feasible solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution of (SIFD). For λi ≥ 0, i = 1, 2,, s with
Hence by the constraint condition (5) and (iv), we get
∑µ g j∈∆
j
H x
( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) > 0 2
j∈∆
For j ∈ J ( z ) , using the constraint condition (5), then we get
g ( x, u j ) ≤ 0 ≤ g ( z, u j ), ∀u j ∈ U * From (iii), we get b1 ( x, z )φ1 [( g ( x, u j ) − g ( z, u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( z ) By (ii), we have 2
g xH ( z, u j ;η ( x, z )) + τ j θ ( x, z ) ≤ 0 Since µ j ∈ Λ, j ∈ J ( z ) , it follows that
∑
j∈J ( z )
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2
j * =i 1 =i 1
µ j g xH ( z, u j ;η ( x, z )) +
∑
j∈J ( z )
2
µ jτ j θ ( x , z ) ≤ 0
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JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
s s According to the constraint condition (5), for all 2 H λ ( f − q h ) ( z , y ; η ( x , z )) + λi ρ i θ ( x, z ) < 0 * ∑ ∑ i i x u ∈ U , we let µ j = 0 , as j ∈ ∆ \ J (= z) . i 1 =i 1 Thus, we have (6) Hence by the constraint condition (5) and (iv), we get 2 µ j g xH ( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) ≤ 0 ∑ 2 H j j∈∆ j∈∆ ∑ µ j g x ( z, u ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) > 0 j
We have a contradiction. Therefore, we conclude f ( x, y ) that sup ≥ q . Hence, the proof of the theorem is y∈Y h ( x , y )
j∈∆
g ( x, u j ) ≤ 0 ≤ g ( z, u j ), ∀u j ∈ U *
complete. Theorem4.2. (Weak duality) Let x be a feasible solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution s
λ ∑=
(SIFD).
For
From (iii), we get
b1 ( x, z )φ1 [( g ( x, u j ) − g ( z, u j )] ≤ 0, ∀u j ∈ U * , j ∈ J ( z )
λi ≥ 0, i = 1, 2,, s with
By (ii), we have
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
of
2
g xH ( z, u j ;η ( x, z )) + τ j θ ( x, z ) ≤ 0 Since µ j ∈ Λ, j ∈ J ( z ) , it follows that
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that (i)For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − pseudoinvex with respect to b0 and φ0 at z ; (ii) For all u ∈ U , j ∈ J ( z ), g (⋅, u ) is generalized j
*
j
∑
j∈J ( z )
∑λ ρ + ∑ µ τ
=i 1
i
i
j∈∆
j
∑µ g j
j∈∆
j∈J ( z )
2
µ jτ j θ ( x , z ) ≤ 0
H x
( z, u j ;η ( x, z )) + ∑ µ jτ j θ ( x, z ) ≤ 0 2
j∈∆
(7)
Adding (6) and (7), then from (iv), we obtain s
≥0.
j
∑
Thus, we have
b0 ( x, z ) > 0, b1 ( x, z ) ≥ 0 s
µ j g xH ( z, u j ;η ( x, z )) +
According to the constraint condition (5), for all u j ∈ U * , we let µ j = 0 , as j ∈ ∆ \ J ( z ) .
uniform ( BH ,τ j ) − quasiinvex with respect to b1 and φ1 at z; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
(iv)
j∈∆
For j ∈ J ( z ) , using the constraint condition (5), then we get
∑λ
=i 1
Then
( f − q h) Hx ( z, y i ;η ( x, z )) + ∑ µ j g xH ( z, u j ;η ( x, z ))
i
j∈∆
s
< −( ∑ λi ρ i + ∑ µ jτ j ) θ ( x, z ) ≤ 0
f ( x, y )
sup h( x, y ) ≥ q
=i 1
y∈Y
f ( x, y )
sup h( x, y ) < q y∈Y
solution of (SIFP) and ( z, µ , q, s, λ , y ) be a feasible solution of (SIFD). For λi ≥ 0, i = 1, 2,, s with
Then, we have f ( x, y ) − q hx ( , y ) < 0, ∀y ∈ Y
s
Using the constraint condition (5), it follows that for all y i , i = 1,, s , we get
f ( x, y i ) − qh( x, y i ) < 0 ≤ f ( z, y i ) − qh( z, y i ) By (iii), we get
b0 ( x, z )φ0 [( f ( x, y i ) − q hx ( , y i )) − ( f ( z, y i ) − q hz ( , y i ))] < 0 2
( f − q h) Hx ( z, y i ;η ( x, z )) + ρ i θ ( x, z ) < 0
i =1
λ ∑=
1, µ j ∈ Λ , j ∈ ∆ and q ∈ R+ , assume there exists
i
i =1
b0 , φ0 , b1 , φ1 ,η , θ , ρ ∈ R s and τ ∈ R ( ∆ ) , such that (i) For all = y i , i 1,, s,( f − qh )(⋅, y i ) is generalized uniform ( BH , ρ i ) − quasiinvex with respect to b0 and φ0 at z ; (ii) For all u j ∈ U * , j ∈ J ( z ), g (⋅, u j ) is strictly generalized uniform ( BH ,τ j ) − pseudoinvex with respect
Then (i) yields
s
to b1 and φ1 at z ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x, z ) ≥ 0, b1 ( x, z ) ≥ 0 s
(iv)
∑λ ρ + ∑ µ τ
=i 1
Then
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j∈∆
We have a contradiction. Hence, the proof of the theorem is complete. Similarly, we can derive the following theorems. Theorem4.3. (Weak duality) Let x be a feasible
Proof: Suppose contrary that
Since λi ≥ 0 and ∑ λi = 1 , it follows that
2
i
i
j∈∆
j
j
≥0.
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745
Proof: Suppose on the contrary that x * ≠ z . From
f ( x, y )
sup h( x, y) ≥ q
*
Theorem 4.4, we know that there exists ( s* , λ * , y ) ∈ Q
y∈Y
Theorem4.4. (Strong duality) Let x * be an optimal solution of problem (SIFP). Assume that x * satisfies K − T constraint qualification for (SIFP). Then there *
*
exists ( s* , λ * , y ) ∈ Q and ( x * , µ * , q* ) ∈ D ( s* , λ * , y ) , *
such that ( x * , µ * , q* , s* , λ * , y ) is a feasible solution of (SIFD). If the hypothesis of theorem 4.1 is also satisfied, *
then ( x * , µ * , q* , s* , λ * , y ) is an optimal of (SIFD), furthermore, the two problems (SIFP) and (SIFD) have the same optimal value. Proof: Since x * is an optimal solution of problem (SIFP), and x * satisfies K − T constraint qualification
*
and ( x * , µ * , q* ) ∈ D ( s* , λ * , y ) , such that ( x * , µ * , q* , s* , *
λ * , y ) is an optimal solution of (SIFD) with the optimal value.
q* = sup y∈Y
Now using the conditions (i)- (iv) and like the proof of theorem 4.1 (by replacing x by x * and ( z, µ , q, s, λ , y ) by
( z, µ , q, s, λ , y ) ), we arrive at the strict inequality. sup y∈Y
*
for (SIFP), then there exists ( s* , λ * , y ) ∈ Q , q* ∈ R+ and
µ *j ∈ Λ , j ∈ ∆ , such that the relations (3.1)- (3.4) hold. *
Therefore, ( x * , µ * , q* , s* , λ * , y ) is a feasible solution of (SIFD), and we have *
q* =
f ( x* , y )
f ( x* , y ) h( x* , y )
f ( x* , y ) >q h( x* , y )
This contradicts the fact
sup y∈Y
f ( x* , y ) * q = q= h( x* , y )
Therefore, we conclude that x * = z . Hence, the proof of the theorem is complete.
*
h( x* , y )
V. CONCLUSION
The optimality of this feasible solution for (SIFD) follows from theorem 4.1. It is clear that the two problems have the same optimal values. Remark 4.1. the result of strong duality under the hypothesis of theorem 4.2 (or 4.3) follows with the same lines as the argument given in theorem 4.1. Theorem 4.5. (Strict reverse duality) Let x * and
( z, µ , q, s, λ , y ) be an optimal solution of problem (SIFP) and (SIFD), respectively. Assume that x * satisfies K − T s
constraint qualification for (SIFP). And for
∑λ
i
=1 ,
Throughout this paper, we have defined a new generalized convex function, extending many wellknown classes of generalized convex functions. Furthermore, we have achieved some sufficient optimality conditions for a class of multiobjective semiinfinite programming problem. Finally, we have formulated the multiobjective dual problem and proved the results concerning weak and strong duality between the primal (SIVP) and the dual (SIVD), there should be further opportunities for exploiting this structure of the semi-infinite programming problem.
i =1
µ i ∈ Λ , j ∈ ∆ and q ∈ R+ , there exists b0 , φ0 , b1 , φ1 ,η , θ ,
ρ ∈ R s ,τ ∈ R ( ∆ ) , the following conditions are fulfilled: (i)For all y , f (⋅, y ) − qh )(⋅, y )e is strictly generalized uniform ( BH , ρ i ) − invex with respect to b0 and φ0 at z ; (ii)
For
all
u j ∈ U * , j ∈ J ( z ), g (⋅, u j )
is strictly
generalized uniform ( BH ,τ j ) − invex with respect to b1 and φ1 at z ; (iii) a ≤ 0 ⇒ φ0 ( a ) ≤ 0 ; a ≤ 0 ⇒ φ1 ( a ) ≤ 0,
b0 ( x * , z ) > 0, b1 ( x * , z ) ≥ 0 s
(iv)
∑λ ρ + ∑ µ τ
=i 1
i
i
j∈∆
j
j
≥0.
Then, x * = z ; that is, z is also an optimal solution of f ( z, y ) (SIFP) and q = sup . y∈Y h ( z , y )
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ACKNOWLEDGMENT This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 08JK237). REFERENCES [1] V. Preda, “On efficiency and duality for multiobjective programs”, J. Math. Anal. Appl., vol. 42, no. 3, pp. 234240, 1992. [2] Kuk, H., Lee, G.M., Kim, D.S., Nonsmooth multiobjective programs with-invexity, Ind. J. Pure. Appl. Math., vol. 29, no. 2, pp. 405-412, 1998. [3] Liang Z.A., Huang H.X., Pardalos P.M., “Optimality conditions and duality for a class of nonlinear fractional programming problems”, J. Optim. Theory Appl., vol. 110, no. 1, pp. 611-619, 2001. [4] Hanson M.A., Pini R., Singh C., “Multiobjective programming under generalized type I invexity”, J. Math. Anal. Appl., vol. 261, no. 2, pp. 562-577, 2002. [5] Suneja S.K., Srivastava M.K., “Optimality and duality in nondifferentiable multiobjective optimization involving dtype I and related functions”, J. Math. Anal. Appl., vol. 206, no. 2, pp. 465-479, 1997.
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Xiaoyan Gao, female, was born in shanxi Province, China on August 28, 1979. She received her master’s degree in optimization theory and applications from Yan’an University, China in 2005. Now she is a lecturer of School of Science, Xi’an University of Science and Technology, Xi’an, China. Her main research fields include the generalized convexity, the optimization theory and applications for semi-infinite and multiobjective programming, etc.
