Optimality and Duality for Non-Smooth Multiple Objective Semi-Infinite ...
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
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Optimality and Duality for Non-Smooth Multiple Objective 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 non-smooth multiobjective semi-infinite programming problem. Based on the concepts of local cone approximation, K − directional derivative and K − subdifferential, a new generalization of convexity, namely generalized uniform K − ( F , α , ρ , d ) − convexity, 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 Mond-Weir type multiobjective dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases for the results described in the paper. Index Terms—local cone approximation, K − directional derivative, K − subdifferential, semi-infinite programming, sufficient optimality condition, duality
I. INTRODUCTION In recent years, there has been considerable interest in so-called semi-infinite programming problem--the optimization of an objective function in finitely many variables over a feasible region defined by an infinite number of constraints, since this model arises in a large number of applications in different fields of mathematics, economics and engineering. We can see in [1, 2]. To date, many authors investigated the optimality conditions and duality results for semi-infinite programming problems. In particular, Kanzi and Nobakhtian [3] established some alternative theorems and several necessary optimality conditions of Fritz-John and Karush-Kuhn-Tucher type for nonsmooth semi-infinite programming problem. In [4], they also established necessary and sufficient optimality conditions under various constraints qualifications for nonsmooth semi-infinite programming problem using Clarke subdifferential. We also refer [5, 6] to understand different aspects of semi-infinite programming. On the other hand, the concept of convexity and generalized convexity plays a central role in mathematical economics, management science, and optimization theory. Therefore, the research on convexity and generalized © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.2.413-420
convexity is one of the most important aspects in mathematical programming. Recently, various generalized convexity notations have been introduced. In particular, the concept of generalized ( F , ρ ) − convexity, introduced by Preda [7] was in turn an extension of the convexity and was used by several authors to obtain relevant results. In [8, 9], 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 [10, 11]. In [12, 13], the sufficient optimality conditions and duality results were obtained under the generalized convex functions. For details, the readers are advised to consult [14, 15, 16]. In this paper, motivated by the above work, we first define a kind of generalize convex functions about the local cone approximation, K − directional derivative and K − subdifferential. Then, the sufficient optimality conditions are obtained for a class of multiobjective semiinfinite 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 be a nonempty set of R n . The epigragh of a real-valued function f : X → R is the following subset of X ×R: epi= f {( x, r ) ∈ X × R f ( x ) ≤ r} Definition2.1. Let K (⋅, ⋅) be a local cone approximation. Then, f k ( x; ⋅) : X × X → R {+∞} is said to be K-direc tional deriveative at x , where f K ( x; y ) = inf{ξ ∈ R ( y , ξ ) ∈ K ( epi f , ( x, f ( x )))} Definition2.2. [17] f : X → R is said to be K-subdi fferentiable, if there exists convex compact set ∂ K f ( x ) , such that = f K ( x; y ) max ξ , y , ∀y ∈ R n K ξ ∈∂ f ( x )
Where ∂ f ( x )= {x ∈ X * y , x * ≤ f K ( x; y ), ∀y ∈ R n } is K
*
K -subdifferential of f at x .
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Definition2.3.[17] A functional F : X × X × R n → R ( X ⊂ R n ) is said to be sublinear about the third variable, if for ∀( x1 , x2 ) ∈ X × X , it satisfies (i) F ( x1 , x2 ; α1 + α 2 ) ≤ F ( x1 , x2 ; α1 ) + F ( x1 , x2 ; α 2 ),
∀α1 , α 2 ∈ R n . (ii) F= ( x1 , x2 ; rα ) rF ( x1 , x2 ; α1 ), ∀r ∈ R+ , α ∈ R n . We suppose that X ⊂ R n is nonempty; f : X → R is local Lipschitz function; F : X × X × R n → R is sublinear; φ : R → R; b : X × X × [0,1] → R+ , lim+ b( x, x 0 ; λ ) = b( x, x 0 ); λ →0
α : X × X → R+ \ {0}; d : X × X → R is a pseudometric n
Assumption A0 Let the local cone approximation K be one among the tangent cone, arrival directional cone, Clarke tangent cone, and feasible directional cone. Lemma2.1.[18] (i)The f K ( x, ⋅) is positively homogeneous and subadditive function. (ii) f K ( x, ⋅) is convex function. Lemma2.2. [18] 0 ∈ ∂ K f ( x ) ⇔ f K ( x; y ) ≥ 0, ∀y ∈ R n . Theorem2.1. If x is a local minimum of f ( x ) on X , and satisfies the assumption A0 , then there exists 0 ∈
∂ K f ( x) . The result can be obtained easily.
on R n , ρ ∈ R . Definition2.4. f is said to be generalized uniform
III. SUFFICIENT OPTIMALITY CONDITIONS
K − ( F , α , ρ , d ) − convex at x 0 ∈ X , if for all x ∈ X , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≥ F ( x, x 0 , α ( x, x 0 )ξ )
In this paper, we consider the following multiobjective semi-infinite programming problem: ( SIVP ) min imize f ( x ) = ( f1 ( x ), f 2 ( x ),, f p ( x )) ,
+ ρ d 2 ( x, x 0 ), ∀ξ ∈ ∂ K f ( x0 ) Definition2.5. f is said to be strictly generalized uniform K − ( F , α , ρ , d ) − convex at x 0 ∈ X , if for all x ∈ X ,
x ≠ x0 , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )]
> F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ), ∀ξ ∈ ∂ K f ( x0 )
subject to g t ( x ) ≦ 0 , t ∈ T , x ∈ X . Where X ⊂ R n is a nonempty open subset, f : x →
R n , g t : x → R, t ∈ T and T is an infinite compact index set. We put X 0 = {x ∈ X g t ( x ) ≦ 0, t ∈ T } for the feasible set of problem (SIVP). = f i : x → R(i 1, 2,, p )and g t : x → R(t ∈ T ) are local Lipschitz and K-subdifferentiable at x ∈ X . Then we define p
Definition2.6. f is said to be generalized uniform K − ( F , α , ρ , d ) − pseudoconvex at x 0 ∈ X , if for all
Λ + ={ λ =(λ1 , λ2 ,, λ p )T λi ≧ 0, i = 1, 2, p, ∑ λi = 1},
x ∈ X , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] < 0
Λ ++ ={ λ =(λ1 , λ2 ,, λ p )T λi > 0, i = 1, 2, p, ∑ λi = 1},
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) < 0, ∀ξ ∈ ∂ K f ( x0 ) Definition2.7. f is said to be strictly generalized uniform K − ( F , α , ρ , d ) − pseudoconvex at x 0 ∈ X , if for all x ∈ X , x ≠ x0 , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≤ 0 ⇒
F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) < 0, ∀ξ ∈ ∂ K f ( x0 ) Definition2.8. f is said to be generalized uniform K − ( F , α , ρ , d ) − quasiconvex at x 0 ∈ X , if for all x ∈ X , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≤ 0
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) ≤ 0, ∀ξ ∈ ∂ K f ( x0 )
i =1 p
i =1
T ( x ) =∈ {t T g t ( x ) = 0}, R
(T ) +
= {µ : T → R+ t ∈ T }.
Where T ( x ) is active constraint set; the set R+(T ) denotes that for all t ∈ T , µt ≧ 0 , and only finitely many are strictly positive. The following notation conventions are used in this paper: For x, y ∈ R n , x = ( x1 ,, xn )T , y = ( y1 ,, yn )T , where the superscript T denotes the transpose of a vector, (1) x < y ⇔ xi < yi , i = 1,, n ; (2) x ≤ y ⇔ xi ≦ yi , i = 1,, n , and at least one
xi0 < yi0 holds for some i0 ; (3) x ≦≦ y ⇔ xi yi , i = 1,, n ; (4) x ≧≧ y ⇔ xi yi , i = 1,, n . Throughout this paper, we denote
Definition2.9. f is said to be generalized uniform K − ( F , α , ρ , d ) − weakly quasiconvex at x 0 ∈ X , if for
λ T f ( x ) = ∑ λi f i ( x )
all x ∈ X , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] < 0
Definition3.1. A point x * ∈ X 0 is said to be an efficient solution of (SIVP), if there exists no x ∈ X 0 such that f ( x ) ≤ f ( x* )
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) ≤ 0, ∀ξ ∈ ∂ K f ( x0 )
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Definition3.2. A point x * ∈ X 0 is said to be a weakly µt* F ( x, x* ; α ( x, x* )ζ t ) + ∑ µt* ρ t*d 2 ( x, x* ) ∑ 0 t T t∈T ∈ efficient solution of (SIVP), if there exists no x ∈ X (2) * * * = ∑ F ( x, x ; α ( x, x ) µt ζ t ) + ∑ µt* ρ t*d 2 ( x, x * ) ≦ 0 such that t∈T t∈T f ( x ) < f ( x* ) Adding (1) and (2), then by the sublinearity of F and Definition3.3. It is said that x * satisfies the (vi), we have Kuhn-Tuker constraint qualification for (SIVP), if there exists λ * ∈ Λ + ( Λ ++ ) and ( µt* )t∈T ∈ R+(T ) , such that
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t ))
0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) t∈T * * g ( x ) 0, t T µ = ∈ t t * * (λ , µ ) ≥ 0 Theorem3.1. Let x * ∈ X . Assume that for all x ∈ X , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
convex at x with respect to b0 and φ0 ; *
(ii) For any t ∈ T ( x ), g t is generalized uniform K t −
( F , α , ρ , d ) − convex at x with respect to b1 and φ1 ; *
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
a ≦≦≧ 0 ⇒ φ1 ( a )
0;
(vi) ρ 0 + ∑ µ ρ ≧ 0 . t∈T
* t
t∈T
Then x * is a weakly efficient solution of (SIVP). Proof: Suppose that x * is not weakly efficient solution of (SIVP). Then there exists x ∈ X 0 , such that
0;
Then x * is a weakly efficient solution of (SIVP). Proof: Suppose that x * is not weakly efficient solution
Since λ * ∈ Λ + , it follows that
λ f ( x) < λ f ( x )
λ *T f ( x ) < λ *T f ( x* )
*
Condition (v) implies b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] < 0 Then from (i), we get
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, ∀ξ ∈ ∂ K0 (λ *T f )( x * )
By (v), we obtain b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] < 0 Then (i) implies (1)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) By (v), we obtain b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0, ∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * ) From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
* t
f ( x ) < f ( x* )
Since λ * ∈ Λ + , it follows that *T
* t
of (SIVP). Then there exists x ∈ X 0 , such that
f ( x ) < f ( x* )
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0; b0 ( x, x * ) > 0, b1 ( x, x * )
(vi) ρ 0 + ∑ µ ρ ≧ 0 .
* t
*T
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; t∈T
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
0; b0 ( x, x * ) > 0, b1 ( x, x * )
pseudoconvex at x * with respect to b0 and φ0 ;
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
t∈T
a ≦≦≧ 0 ⇒ φ1 ( a )
t∈T
Finally, we have a contradiction. Hence x * is a weakly efficient solution of (SIVP). Theorem3.2. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − (ii) For any t ∈ T ( x * ), g t is generalized uniform K t −
*
* t
t∈T
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, ∀ξ ∈ ∂ K0 (λ *T f )( x * )
(3)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) From (v), we get b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0,
∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * ) From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
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∑ µ F ( x, x ;α ( x, x )ζ ) + ∑ µ ρ d t∈T
* t
*
*
t
t∈T
* t
* t
2
∑ µ F ( x, x ;α ( x, x )ζ ) + ∑ µ ρ d
( x, x * ) (4)
t∈T
* t
*
*
t
* t
t∈T
* t
2
( x, x * )
( x, x ) ≦ 0 ∑ F ( x, x ; α ( x, x ) µ ζ ) + ∑ µ ρ d ∑ F ( x, x ;α ( x, x )µ ζ ) + ∑ µ ρ d= *
*
* t
t∈T
t
* t
t∈T
* t
2
*
*
* t
t∈T
Adding (3) and (4), then by the sublinearity of F and (vi), we get
t
t∈T
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t ))
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
t∈T
(ii) For any t ∈ T ( x * ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; (iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
b0 ( x, x ) ≧≧ 0, b1 ( x, x )
0 ⇒ φ1 ( a )
0;
0;
(ii) For any t ∈ T ( x * ), g t is strictly generalized uniform K t − ( F , α , ρ t* , d ) − pseudoconvex at x * with respect to b1 and φ1 ;
Then x is an efficient solution of (SIVP). Proof: Suppose that x * is not an efficient solution of (SIVP). Then there exists x ∈ X , such that 0
f ( x ) ≤ f ( x* ) Since λ * ∈ Λ + , it follows that
λ *T f ( x ) ≦ λ *T f ( x* ) By (v), we obtain b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] ≦ 0 Then (i) implies
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, *
(5)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) Using (v), we get b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0, ∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * )
(vi) ρ 0 + ∑ µt* ρ t* ≧ 0 . t∈T
*
Then x is a weakly efficient solution of (SIVP). Remark3.1. If the assumption λ * ∈ Λ + in the above theorem 3.1 and theorem 3.2 is replaced by the assumption λ * ∈ Λ ++ , we get the stronger conclusion that x * is an efficient solution of (SIVP). The proof follows on the same line. Theorem3.5. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , the following conditions. (i) f i ( x )(i = 1, 2,, p )
is
*
))
, satisfying
generalized
uniform
α , ρ i , d ) − convex at x with respect to b0
Ki − ( F ,
*
and φ0 ; (ii) For any t ∈ T ( x * ), g t is generalized uniform K t* −
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; p
(iii) 0 ∈ ∑ λi*∂ Ki f i ( x * ) + i =1
∑
µt*∂ K g t ( x* ) ; * t
t∈T ( x* )
(iv) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
a ≦≦≧ 0 ⇒ φ1 ( a )
From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
*
there exists λ * ∈ Λ ++ and ( µt* )t∈T ( x* ) ∈ R+(T ( x
f )( x )
p
(v)
∑λ ρ i =1
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0;
b0 ( x, x ) > 0, b1 ( x, x ) ≧ 0 ; *
t∈T
∀ξ ∈ ∂ (λ
weakly quasiconvex at x * with respect to b0 and φ0 ;
t∈T
*
*T
(6)
Finally, we have a contradiction. Hence x * is an efficient solution of (SIVP). Similarly, we can derive the following theorem easily. Theorem3.4. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0; a ≦≦ 0 ⇒ φ1 ( a )
(vi) ρ 0 + ∑ µt* ρ t* ≧ 0 .
K0
( x, x * ) ≦ 0
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
t∈T
*
*
2
t∈T
, d ) − pseudoconvex at x * with respect to b0 and φ0 ;
*
* t
t∈T
Finally, we have a contradiction. Hence x * is a weakly efficient solution of (SIVP). Theorem3.3. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the follow- ing conditions. (i) λ *T f is strictly generalized uniform K 0 − ( F , α , ρ 0
(iv) µ g t ( x = ) 0, t ∈ T ; (v) a ≦≦≦≦ 0 ⇒ φ0 ( a ) 0; a
* t
Adding (5) and (6), then by the sublinearity of F and (vi), we obtain
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t )) t∈T
* t
*
* i
i
+
∑
t∈T ( x* )
0; b0 ( x, x * ) > 0, b1 ( x, x * )
µt* ρ t* ≧ 0 .
0;
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F ( x 0 , u; α ( x 0 , u ) µt0 ζ t0 ) + µt0 ρ t*0 d 2 ( x 0 , u ) < 0,
Then x * is an efficient solution of (SIVP).
∀ζ t0 ∈ ∂ t0 g t0 (u ) K
IV. DUALITY THEOREMS In this section, we consider the following Mond-Weir type dual model for (SIVP): (SIVD) max f (u )
s.t. 0 ∈ ∂ K0 (λ T f )(u ) + ∑ µt ∂ Kt g t (u );
(7)
t∈T
µt gt (u ) ≧ 0, t ∈ T ;
(8)
By µt ≧ 0, t ∈ T (u ), t ≠ t0 and (4.2), we have
g t ( x 0 ) ≦≦ 0 g t (u ) Then from (iv) and (ii), for any ζ t ∈ ∂ Kt g t (u ) , we have
F ( x 0 , u; α ( x 0 , u ) µt ζ t ) + µt ρ t*d 2 ( x 0 , u ) ≦ 0, t ∈ T (u ), t ≠ t0 Now, together with t ∈ T (u ), t ≠ t0 implies
F ( x 0 , u; α ( x 0 , u )
λ ∈ Λ ( Λ ), ( µt )t∈T ∈ R . (9) 0 Let D {(u, λ , µ ) (u, λ , µ ) satisfies (7) − (9)} de-note +
++
T +
the set of all feasible solutions of (SIVD). Theorem4.1. (Weak duality) Assume that for feasible solutions x ∈ X 0 and (u, λ , µ ) ∈ D 0 for (SIVP) and (SIVD), respectively, assumption A0 holds, and for λ ∈ Λ + , ( µt )t∈T ∈ R+T , we have
+
∑
t∈T ( u ) t ≠ t0
and together with µt0 > 0 ;
0 ; b0 ( x, u )
(v) ρ 0 + ∑ µt ρ ≧ 0 . t∈T
+
0, b1 ( x, u ) > 0 ;
* t
Since λ ∈ Λ + , it follows that λ T f ( x ) ≦ λ T f (u ) According to (iv), we have b0 ( x, u )φ0 [(λ T f )( x ) − (λ T f )(u )] ≦ 0 Then (i) yields F ( x, u; α ( x, u )ξ ) + ρ 0 d 2 ( x, u ) ≦ 0, ∀ξ ∈ ∂ K0 (λ T f )(u ) For x ∈ X , we have 0
(10)
∀ξ ∈ ∂ K0 (λ T f )(u )
(13)
µt ρ t*d 2 ( x 0 , u ) = 0
Adding (10), (11), (12) and (13), then by the sublinearity of F and (v), we obtain F ( x 0 , u; α ( x 0 , u )(ξ + ∑ µtζ t )) t∈T
< − ( ρ 0 + ∑ µt ρ ) d ( x 0 , u ) ≦ 0 * t
2
(SIV- D), respectively, assumption A0 for λ ∈ Λ ++ , ( µt )t∈T ∈ R+T , we have
holds, and
(i) λ T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − pseudoconvex on X 0 with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex on X 0 with respect to b1 and φ1 ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≦≦ 0 ⇒ φ1 ( a )
0;
b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ;
(iv) ρ 0 + ∑ µt ρ t* ≧ 0 .
Then we can obtain f ( x ) f (u ) . Proof: Suppose that the result does not hold, then there exists x ∈ X 0 , such that f ( x ) ≦ f (u ) . Since λ ∈ Λ ++ , it follows that λ T f ( x ) < λ T f (u ) By (iii), we obtain b0 ( x, u )φ0 [(λ T f )( x ) − (λ T f )(u )] < 0 Then (i) yields
By (iii), we obtain 0 g t0 ( x= ) < 0 g t0 (u ), t0 ∈ T (u ) From (iv), we get b1 ( x 0 , u )φ1[ g t0 ( x 0 ) − g t0 (u )] < 0 Then by (ii), we have
F ( x 0 , u; α ( x 0 , u )ζ t0 ) + ρ t*0 d 2 ( x 0 , u ) < 0, ∀ζ t0 ∈ ∂ t0 g t0 (u ) K
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t∈T \ T ( u )
µt ζ t )
t∈T
F ( x 0 , u; α ( x 0 , u )ξ ) + ρ 0 d 2 ( x 0 , u ) ≦ 0,
Since µt0 > 0 , it follows that
∑
t∈T
Then we can obtain f ( x ) f (u ) . Proof: Suppose that the result does not hold, then there exists x ∈ X 0 , such that f ( x ) ≦ f (u ) .
0
∑
t∈T \ T ( u )
But which contradicts (7). Hence the result follows. Theorem4.2. (Weak duality) Assume that for feasible solutions x ∈ X 0 and (u, λ , µ ) ∈ D 0 for (SIVP) and
(iv) a < 0 ⇒ φ1 ( a ) < 0 and φ1 (0) = 0;
a ≦≦≧ 0 ⇒ φ0 ( a )
(12)
µt ρ t*d 2 ( x 0 , u ) ≦ 0
F ( x 0 , u; α ( x 0 , u )
0
(iii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for t0 ∈ T (u ) ,
µt ζ t )
Let µt = 0 , for t ∈ T \ T (u ) , so we have
(i) λ f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
( F , α , ρ t* , d ) − convex on X 0 with respect to b1 and φ1 ;
∑
t∈T ( u ) t ≠ t0
T
convex on X with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
(11)
F ( x, u; α ( x, u )ξ ) + ρ 0 d 2 ( x, u ) < 0, ∀ξ ∈ ∂ K0 (λ T f )(u )
(14)
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By µt ≧ 0, t ∈ T (u ) and (11), we obtain
Suppose that ( x * , λ * , µ * ) is not an efficient solution of
g t ( x ) ≦≦ 0 g t (u ) Then from (iii) and (ii), we have F ( x, u; α ( x, u ) µt ζ t ) + µt ρ t*d 2 ( x, u ) ≦ 0,
(SIVD), then there exists ( x, λ , µ ) , such that
f ( x * ) ≦ f (u ) By the weak duality theorem 4.1, we get f ( x ) f (u )
∀ζ t ∈ ∂ Kt g t (u ), t ∈ T (u ) Let µt = 0 , for t ∈ T \ T (u ) , so we have
∑ F ( x , u; α ( x , u ) µ ζ ) + ∑ µ ρ d t
t∈T
t
t
t∈T
* t
2
( x, u ) ≦ 0,
∀ζ t ∈ ∂ Kt g t (u )
(15)
Adding (14) and (15), then by the sublinearity of F and (iv), we obtain F ( x, u; α ( x, u )(ξ + ∑ µtζ t )) t∈T
< − ( ρ 0 + ∑ µt ρ ) d 2 ( x , u ) ≦ 0 t∈T
* t
But which contradicts (7). Hence the result follows. Theorem4.3. (Strong duality) Suppose that x * is a weakly efficient solution of (SIVP), assumption A0 holds, and for λ ∈ Λ + ,( µt )t∈T ∈ R+T , such that (i) λ f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − T
convex at u ∈ D with respect to b0 and φ0 ; 0
(ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − convex at u ∈ D 0 with respect to b1 and φ1 ; (iii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for t0 ∈ T (u ) , and together with µt0 > 0 ; (iv) a < 0 ⇒ φ1 ( a ) < 0 and φ1 (0) = 0;
a ≦≦≧ 0 ⇒ φ0 ( a )
(v) ρ 0 + ∑ µt ρ ≧ 0 ; t∈T
0 ; b0 ( x, u )
0, b1 ( x, u ) > 0 ;
* t
(vi) The Kuhn-Tuker constraint qualification is satisfied. Then there exists λ * ∈ Λ + , such that ( x * , λ * , µ * ) is an efficient solution of (SIVD). Furthermore, the two problems (SIVP) and (SIVD) have the same objective values. Proof: Since x * is a weakly efficient solution of (SIVP), and the problem satisfies the Kuhn-Tuker ,s constraint qualification. So there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+T , such that 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) t∈T * * g ( x ) 0, t T µ = ∈ t t * * (λ , µ ) ≥ 0 Hence ( x * , λ * , µ * ) is a feasible solution of (SIVD). It is clear that the two problems (SIVP) and (SIVD) have the same objective values.
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Then we have a contradiction. Hence ( x * , λ * , µ * ) is an efficient solution of (SIVD). Theorem4.4. (Strong duality) Suppose that x * is an efficient solution of (SIVP), assumption A0 holds, and for λ ∈ Λ ++ ,( µt )t∈T ∈ R+T , such that (i) λ T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − pe- seudoconvex on X 0 with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex on X 0 with respect to b1 and φ1 ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≧ 0 ⇒ φ1 ( a ) ≧ 0 ; b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ; (iv) ρ 0 + ∑ µt ρ t* ≧ 0 ; t∈T
(v) The Kuhn-Tuker constraint qualification is satisfied at x * . Then there exists λ * ∈ Λ ++ , such that ( x * , λ * , µ * ) is an efficient solution of (SIVD). Furthermore, the two problems (SIVP) and (SIVD) have the same objective values. Proof: The proof follows along similar lines as the proof of theorem 4.3 and hence is omitted. Theorem 4.5. (Reverse duality) Suppose that x * ∈ X 0 , (u* , λ * , µ * ) ∈ D 0 , f ( x * ) = f (u* ) , assumption A0 holds, and for λ * ∈ Λ + , ( µt* )t∈T ∈ R+T , we have (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − convex at u* with respect to b0 and φ0 ; (ii) For t ∈ T (u* ) , g t is generalized uniform K t −
( F , α , ρ t* , d ) − convex at u* with respect to b1 and φ1 ; (iii) 0 ∈ ∂ K0 (λ *T f )(u* ) + ∑ µt*∂ Kt g t (u* ) ; t∈T
(iv) µt* g t (u*= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ= 0; a ≦ 0 ⇒ φ1 ( a ) ≦ 0 ; 0 (0) b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ; (vi) ρ 0 + ∑ µt* ρ t* ≧ 0 . t∈T
(vii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for all
t0 ∈ T (u* ) , and corresponding µt0 > 0 . Then x * is a weakly efficient solution of (SIVP), and (u* , λ * , µ * ) is an efficient solution of (SIVD).
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Proof: Form theorem 3.1, we known that u* is also a µt* F ( x* , u* ; α ( x* , u* )ζ t ) + ∑ µt* ρ t*d 2 ( x* , u* ) ∑ t∈T t∈T weakly efficient solution of (SIVP). It is clear that x * is * * * * * = F ( x , u ; α ( x , u ) µ ζ ) + µt* ρ t*d 2 ( x* , u* ) ≦ 0 also a weak efficient solution of (SIVP). If it is not true, ∑ ∑ t t t∈T t∈T there exists x ∈ X 0 , such that From (iv), we obtain there exists ξ 0 ∈ ∂ K0 (λ *T f )(u* ) , f ( x ) < f ( x* ) = f (u* ) * ζ t0 ∈ ∂ Kt gt (u* )(t ∈ T ) , such that We have a contradiction. Under the conditions of theorem 4.1, we known the ξ 0 + ∑ µt*ζ t0 = 0 * t∈T weak duality holds at u . By the sublinearity of F , we obtain Suppose that ( x * , λ * , µ * ) is not an efficient solution of = 0 F ( x * , u* ; α ( x * , u* )(ξ 0 + ∑ µt*ζ t0 )) (SIVD). Then there exists u , such that t∈T * * * * 0 * * F ( x , u ; α ( x , u ) ξ ) + F ( x * , u* ; α ( x * , u* )∑ µt*ζ t0 ) ≦ f (u ) ≥ f (u ) = f (x ) This contradicts weak duality. Hence ( x * , λ * , µ * ) is an efficient solution of (SIVD). Theorem 4.6. (Strict reverse duality) Suppose that x * ∈ X 0 , (u* , λ * , µ * ) ∈ D 0 , assumption A0 holds, and for
λ * ∈ Λ + , ( µt* )t∈T ∈ R+T , we have (ii)
λ f
is
strictly
generalized
uniform
K 0 − ( F , α , ρ 0 , d ) − pseudoconvex at u with respect to b0 and φ0 ; t ∈ T (u* ), g t
is
generalized
uniform
K t − ( F , α , ρ , d ) − quasiconvex at u with respect to b1 and φ1 ; *
* t
(iv) 0 ∈ ∂ K0 (λ *T f )(u* ) + ∑ µt*∂ Kt g t (u* ) ; t∈T
(v) µt* g t (u*= ) 0, t ∈ T ; (vi) φ0 ( a ) > 0 ⇒ a > 0; a ≦ 0 ⇒ φ1 ( a ) ≦ 0 ;
b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ;
(vii) ρ 0 + ∑ µ ρ t* ≧ 0 . t∈T
* t
Then x * = u* . Proof: We assume that x * ≠ u* . By hypothesis (ii) and (iii), we have *
*
*
2
*
⇒ b0 ( x * , u* )φ0 [λ *T f ( x * ) − λ *T f (u* )] > 0 b1 ( x , u )φ1[ g t ( x ) − g t (u )] ≦ 0 *
*
*
(16)
⇒ F ( x , u ; α ( x , u )ζ t ) + ρ d ( x , u ) ≦ 0, *
*
*
*
* t
2
*
*
∀ζ t ∈ ∂ Kt g t (u* ), t ∈ T (u* ) As t ∈ T (u* ) , we have
g t ( x * ) ≦ 0 = g t (u* ) Using (vi), we get b1 ( x * , u* )φ1[ g t ( x * ) − g t (u* )] ≦ 0 From (v), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
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This contradicts hypothesis (i). Hence, x * = u* . V. CONCLUSION Throughout this paper, we have defined a new generalized convex function, extending many well-known classes of generalized convex functions. Furthermore, we have achieved some sufficient optimality conditions for a class of multiobjective semi-infinite 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.
This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 08JK237).
*
∀ξ ∈ ∂ K0 (λ *T f )(u* ) *
Then from (vi) and (16), we have λ *T f ( x* ) > λ *T f (u* )
ACKNOWLEDGMENT
F ( x , u ; α ( x , u )ξ ) + ρ 0 d ( x , u ) ≧ 0, *
t∈T
t∈T
*
(iii)For
≧- F ( x * , u* ; α ( x * , u* )∑ µt*ζ t0 ) ≧ ∑ µt* ρ t*d 2 ( x * , u* ) ≧- ρ 0 d 2 ( x * , u* )
(i) λ *T f ( x * ) = λ *T f (u* ) *T
t∈T
which with (vii) yields F ( x * , u* ; α ( x * , u* )ξ 0 )
REFERENCES [1] Canovas M. J., Lopez M. A., Mordukhovich B. S., Parra J., “Variational analysis in semi-infinite and finite programming, I: stability of linear inepuality systems of feasible solutions”, SIAMJ. Optim., vol. 20, no. 5, pp. 1504-1526, 2009. [2] Shapiro A., “Semi-infinite programming, duality, discretization and optimality condition”, Optimization, vol. 58, no. 2, pp. 133-161, 2009. [3] N. Kanzi, S. Nobakhtian, “Nonsmooth semi-infinite programming problems with mixed constraints”, J. Math. Anal. Appl., vol. 351, no. 1, pp. 170-181, 2009. [4] N. Kanzi, S. Nobakhtian, “Optimality conditions for nonsmooth semi-infinite programming”, Optimization, vol. 59, no. 5, pp. 717-727, 2010. [5] Qingxiang Zhang, “Optimality conditions and duality for semi-infinite programming involving B-arcwise connected
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functions”, J. Glob. Optim, vol. 45, no. 4, pp. 615-629, 2009. N. Kanzi, “Necessary optimality conditions for nonsmooth semi-infinite programming problems”, J. Glob. Optim., vol. 49, no. 4, pp. 713-725, 2011. V.Preda, “On efficiency and duality for multiobjective programs”, J. Math. Anal. Appl., vol. 42, no. 3, pp. 234240, 1992. Kuk H., Lee G.M., Kim D.S., “Nonsmooth multiobjective programs with-invexity”, Ind. J. Pure. Appl. Math., vol. 29, no. 3, pp. 405-412, 1998. 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. 3, pp. 611-619, 2001. 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. 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. Anurag, Jayswal, “On sufficiency and duality in multiobjective programming problem under generalized-type I univexity”, J. Glob. Optim., vol. 46, no. 2, pp. 207-216, 2010. Vasile P., Loan S.M., Miruna B. and Andreea M.S., “Generalized V-univexity type-I for multiobjective programming with n-set functions”, J. Glob. Optim., vol. 44, no. 1, pp. 131-148, 2009.
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[14] Kim DS, Bae KD, “Optimality conditions and duality for a class of nondifferentiable multiobjective programming problems”, Taiwanese J. Math., vol. 13, no. 2, pp. 789-804, 2009. [15] Bae KD, Kang YM, Kim DS, “Efficiency and generalized convex duality for nondifferentiable multiobjective programs”, J. Inequ. Appl., vol. 71, no. 2, pp. 429-440, 2010. [16] Kim DS, Lee HJ, “Optimality conditions and duality in nonsmooth multiobjective programs”, J. Inequal. Appl., vol. 2010, no. 1, pp. 1-13, 2010. [17] Marco Castellani, “Nonsmooth invex functions and sufficient optimality conditions”, J. Math. Anal. Appl., vol. 225, pp. 319-332, no. 2, 2001. [18] E.Polak, “On the mathematical functions of nondifferentiable optimization in Engineering design”, Siam. Rev., vol. 29, pp. 13-28, no. 1, 1987.
Xiaoyan Gao, 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 multi-objective programming, etc.
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Optimality and Duality for Non-Smooth Multiple Objective 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 non-smooth multiobjective semi-infinite programming problem. Based on the concepts of local cone approximation, K − directional derivative and K − subdifferential, a new generalization of convexity, namely generalized uniform K − ( F , α , ρ , d ) − convexity, 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 Mond-Weir type multiobjective dual programs, using generalized invexity on the functions involved. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases for the results described in the paper. Index Terms—local cone approximation, K − directional derivative, K − subdifferential, semi-infinite programming, sufficient optimality condition, duality
I. INTRODUCTION In recent years, there has been considerable interest in so-called semi-infinite programming problem--the optimization of an objective function in finitely many variables over a feasible region defined by an infinite number of constraints, since this model arises in a large number of applications in different fields of mathematics, economics and engineering. We can see in [1, 2]. To date, many authors investigated the optimality conditions and duality results for semi-infinite programming problems. In particular, Kanzi and Nobakhtian [3] established some alternative theorems and several necessary optimality conditions of Fritz-John and Karush-Kuhn-Tucher type for nonsmooth semi-infinite programming problem. In [4], they also established necessary and sufficient optimality conditions under various constraints qualifications for nonsmooth semi-infinite programming problem using Clarke subdifferential. We also refer [5, 6] to understand different aspects of semi-infinite programming. On the other hand, the concept of convexity and generalized convexity plays a central role in mathematical economics, management science, and optimization theory. Therefore, the research on convexity and generalized © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.2.413-420
convexity is one of the most important aspects in mathematical programming. Recently, various generalized convexity notations have been introduced. In particular, the concept of generalized ( F , ρ ) − convexity, introduced by Preda [7] was in turn an extension of the convexity and was used by several authors to obtain relevant results. In [8, 9], 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 [10, 11]. In [12, 13], the sufficient optimality conditions and duality results were obtained under the generalized convex functions. For details, the readers are advised to consult [14, 15, 16]. In this paper, motivated by the above work, we first define a kind of generalize convex functions about the local cone approximation, K − directional derivative and K − subdifferential. Then, the sufficient optimality conditions are obtained for a class of multiobjective semiinfinite 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 be a nonempty set of R n . The epigragh of a real-valued function f : X → R is the following subset of X ×R: epi= f {( x, r ) ∈ X × R f ( x ) ≤ r} Definition2.1. Let K (⋅, ⋅) be a local cone approximation. Then, f k ( x; ⋅) : X × X → R {+∞} is said to be K-direc tional deriveative at x , where f K ( x; y ) = inf{ξ ∈ R ( y , ξ ) ∈ K ( epi f , ( x, f ( x )))} Definition2.2. [17] f : X → R is said to be K-subdi fferentiable, if there exists convex compact set ∂ K f ( x ) , such that = f K ( x; y ) max ξ , y , ∀y ∈ R n K ξ ∈∂ f ( x )
Where ∂ f ( x )= {x ∈ X * y , x * ≤ f K ( x; y ), ∀y ∈ R n } is K
*
K -subdifferential of f at x .
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Definition2.3.[17] A functional F : X × X × R n → R ( X ⊂ R n ) is said to be sublinear about the third variable, if for ∀( x1 , x2 ) ∈ X × X , it satisfies (i) F ( x1 , x2 ; α1 + α 2 ) ≤ F ( x1 , x2 ; α1 ) + F ( x1 , x2 ; α 2 ),
∀α1 , α 2 ∈ R n . (ii) F= ( x1 , x2 ; rα ) rF ( x1 , x2 ; α1 ), ∀r ∈ R+ , α ∈ R n . We suppose that X ⊂ R n is nonempty; f : X → R is local Lipschitz function; F : X × X × R n → R is sublinear; φ : R → R; b : X × X × [0,1] → R+ , lim+ b( x, x 0 ; λ ) = b( x, x 0 ); λ →0
α : X × X → R+ \ {0}; d : X × X → R is a pseudometric n
Assumption A0 Let the local cone approximation K be one among the tangent cone, arrival directional cone, Clarke tangent cone, and feasible directional cone. Lemma2.1.[18] (i)The f K ( x, ⋅) is positively homogeneous and subadditive function. (ii) f K ( x, ⋅) is convex function. Lemma2.2. [18] 0 ∈ ∂ K f ( x ) ⇔ f K ( x; y ) ≥ 0, ∀y ∈ R n . Theorem2.1. If x is a local minimum of f ( x ) on X , and satisfies the assumption A0 , then there exists 0 ∈
∂ K f ( x) . The result can be obtained easily.
on R n , ρ ∈ R . Definition2.4. f is said to be generalized uniform
III. SUFFICIENT OPTIMALITY CONDITIONS
K − ( F , α , ρ , d ) − convex at x 0 ∈ X , if for all x ∈ X , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≥ F ( x, x 0 , α ( x, x 0 )ξ )
In this paper, we consider the following multiobjective semi-infinite programming problem: ( SIVP ) min imize f ( x ) = ( f1 ( x ), f 2 ( x ),, f p ( x )) ,
+ ρ d 2 ( x, x 0 ), ∀ξ ∈ ∂ K f ( x0 ) Definition2.5. f is said to be strictly generalized uniform K − ( F , α , ρ , d ) − convex at x 0 ∈ X , if for all x ∈ X ,
x ≠ x0 , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )]
> F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ), ∀ξ ∈ ∂ K f ( x0 )
subject to g t ( x ) ≦ 0 , t ∈ T , x ∈ X . Where X ⊂ R n is a nonempty open subset, f : x →
R n , g t : x → R, t ∈ T and T is an infinite compact index set. We put X 0 = {x ∈ X g t ( x ) ≦ 0, t ∈ T } for the feasible set of problem (SIVP). = f i : x → R(i 1, 2,, p )and g t : x → R(t ∈ T ) are local Lipschitz and K-subdifferentiable at x ∈ X . Then we define p
Definition2.6. f is said to be generalized uniform K − ( F , α , ρ , d ) − pseudoconvex at x 0 ∈ X , if for all
Λ + ={ λ =(λ1 , λ2 ,, λ p )T λi ≧ 0, i = 1, 2, p, ∑ λi = 1},
x ∈ X , there exists the local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] < 0
Λ ++ ={ λ =(λ1 , λ2 ,, λ p )T λi > 0, i = 1, 2, p, ∑ λi = 1},
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) < 0, ∀ξ ∈ ∂ K f ( x0 ) Definition2.7. f is said to be strictly generalized uniform K − ( F , α , ρ , d ) − pseudoconvex at x 0 ∈ X , if for all x ∈ X , x ≠ x0 , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≤ 0 ⇒
F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) < 0, ∀ξ ∈ ∂ K f ( x0 ) Definition2.8. f is said to be generalized uniform K − ( F , α , ρ , d ) − quasiconvex at x 0 ∈ X , if for all x ∈ X , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] ≤ 0
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) ≤ 0, ∀ξ ∈ ∂ K f ( x0 )
i =1 p
i =1
T ( x ) =∈ {t T g t ( x ) = 0}, R
(T ) +
= {µ : T → R+ t ∈ T }.
Where T ( x ) is active constraint set; the set R+(T ) denotes that for all t ∈ T , µt ≧ 0 , and only finitely many are strictly positive. The following notation conventions are used in this paper: For x, y ∈ R n , x = ( x1 ,, xn )T , y = ( y1 ,, yn )T , where the superscript T denotes the transpose of a vector, (1) x < y ⇔ xi < yi , i = 1,, n ; (2) x ≤ y ⇔ xi ≦ yi , i = 1,, n , and at least one
xi0 < yi0 holds for some i0 ; (3) x ≦≦ y ⇔ xi yi , i = 1,, n ; (4) x ≧≧ y ⇔ xi yi , i = 1,, n . Throughout this paper, we denote
Definition2.9. f is said to be generalized uniform K − ( F , α , ρ , d ) − weakly quasiconvex at x 0 ∈ X , if for
λ T f ( x ) = ∑ λi f i ( x )
all x ∈ X , there exists local cone approximation K , such that b( x, x 0 )φ [ f ( x ) − f ( x 0 )] < 0
Definition3.1. A point x * ∈ X 0 is said to be an efficient solution of (SIVP), if there exists no x ∈ X 0 such that f ( x ) ≤ f ( x* )
⇒ F ( x, x 0 , α ( x, x 0 )ξ ) + ρ d 2 ( x, x 0 ) ≤ 0, ∀ξ ∈ ∂ K f ( x0 )
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p
i =1
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Definition3.2. A point x * ∈ X 0 is said to be a weakly µt* F ( x, x* ; α ( x, x* )ζ t ) + ∑ µt* ρ t*d 2 ( x, x* ) ∑ 0 t T t∈T ∈ efficient solution of (SIVP), if there exists no x ∈ X (2) * * * = ∑ F ( x, x ; α ( x, x ) µt ζ t ) + ∑ µt* ρ t*d 2 ( x, x * ) ≦ 0 such that t∈T t∈T f ( x ) < f ( x* ) Adding (1) and (2), then by the sublinearity of F and Definition3.3. It is said that x * satisfies the (vi), we have Kuhn-Tuker constraint qualification for (SIVP), if there exists λ * ∈ Λ + ( Λ ++ ) and ( µt* )t∈T ∈ R+(T ) , such that
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t ))
0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) t∈T * * g ( x ) 0, t T µ = ∈ t t * * (λ , µ ) ≥ 0 Theorem3.1. Let x * ∈ X . Assume that for all x ∈ X , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
convex at x with respect to b0 and φ0 ; *
(ii) For any t ∈ T ( x ), g t is generalized uniform K t −
( F , α , ρ , d ) − convex at x with respect to b1 and φ1 ; *
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
a ≦≦≧ 0 ⇒ φ1 ( a )
0;
(vi) ρ 0 + ∑ µ ρ ≧ 0 . t∈T
* t
t∈T
Then x * is a weakly efficient solution of (SIVP). Proof: Suppose that x * is not weakly efficient solution of (SIVP). Then there exists x ∈ X 0 , such that
0;
Then x * is a weakly efficient solution of (SIVP). Proof: Suppose that x * is not weakly efficient solution
Since λ * ∈ Λ + , it follows that
λ f ( x) < λ f ( x )
λ *T f ( x ) < λ *T f ( x* )
*
Condition (v) implies b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] < 0 Then from (i), we get
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, ∀ξ ∈ ∂ K0 (λ *T f )( x * )
By (v), we obtain b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] < 0 Then (i) implies (1)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) By (v), we obtain b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0, ∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * ) From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
* t
f ( x ) < f ( x* )
Since λ * ∈ Λ + , it follows that *T
* t
of (SIVP). Then there exists x ∈ X 0 , such that
f ( x ) < f ( x* )
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0; b0 ( x, x * ) > 0, b1 ( x, x * )
(vi) ρ 0 + ∑ µ ρ ≧ 0 .
* t
*T
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; t∈T
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
0; b0 ( x, x * ) > 0, b1 ( x, x * )
pseudoconvex at x * with respect to b0 and φ0 ;
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
t∈T
a ≦≦≧ 0 ⇒ φ1 ( a )
t∈T
Finally, we have a contradiction. Hence x * is a weakly efficient solution of (SIVP). Theorem3.2. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − (ii) For any t ∈ T ( x * ), g t is generalized uniform K t −
*
* t
t∈T
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, ∀ξ ∈ ∂ K0 (λ *T f )( x * )
(3)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) From (v), we get b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0,
∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * ) From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
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JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
∑ µ F ( x, x ;α ( x, x )ζ ) + ∑ µ ρ d t∈T
* t
*
*
t
t∈T
* t
* t
2
∑ µ F ( x, x ;α ( x, x )ζ ) + ∑ µ ρ d
( x, x * ) (4)
t∈T
* t
*
*
t
* t
t∈T
* t
2
( x, x * )
( x, x ) ≦ 0 ∑ F ( x, x ; α ( x, x ) µ ζ ) + ∑ µ ρ d ∑ F ( x, x ;α ( x, x )µ ζ ) + ∑ µ ρ d= *
*
* t
t∈T
t
* t
t∈T
* t
2
*
*
* t
t∈T
Adding (3) and (4), then by the sublinearity of F and (vi), we get
t
t∈T
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t ))
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
< −( ρ 0 + ∑ µt* ρ t* )d 2 ( x, x * ) ≦ 0
t∈T
(ii) For any t ∈ T ( x * ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; (iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
b0 ( x, x ) ≧≧ 0, b1 ( x, x )
0 ⇒ φ1 ( a )
0;
0;
(ii) For any t ∈ T ( x * ), g t is strictly generalized uniform K t − ( F , α , ρ t* , d ) − pseudoconvex at x * with respect to b1 and φ1 ;
Then x is an efficient solution of (SIVP). Proof: Suppose that x * is not an efficient solution of (SIVP). Then there exists x ∈ X , such that 0
f ( x ) ≤ f ( x* ) Since λ * ∈ Λ + , it follows that
λ *T f ( x ) ≦ λ *T f ( x* ) By (v), we obtain b0 ( x, x * )φ0 [λ *T f ( x ) − λ *T f ( x * )] ≦ 0 Then (i) implies
F ( x, x * ; α ( x, x* )ξ ) + ρ 0 d 2 ( x, x* ) < 0, *
(5)
As t ∈ T ( x * ) , we have
gt ( x ) ≦ 0 = gt ( x* ) Using (v), we get b1 ( x, x * )φ1[ g t ( x ) − g t ( x * )] ≦ 0 Then (ii) yields F ( x, x * ; α ( x, x* )ζ t ) + ρ t*d 2 ( x, x * ) ≦ 0, ∀ζ t ∈ ∂ Kt g t ( x * ), t ∈ T ( x * )
(vi) ρ 0 + ∑ µt* ρ t* ≧ 0 . t∈T
*
Then x is a weakly efficient solution of (SIVP). Remark3.1. If the assumption λ * ∈ Λ + in the above theorem 3.1 and theorem 3.2 is replaced by the assumption λ * ∈ Λ ++ , we get the stronger conclusion that x * is an efficient solution of (SIVP). The proof follows on the same line. Theorem3.5. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , the following conditions. (i) f i ( x )(i = 1, 2,, p )
is
*
))
, satisfying
generalized
uniform
α , ρ i , d ) − convex at x with respect to b0
Ki − ( F ,
*
and φ0 ; (ii) For any t ∈ T ( x * ), g t is generalized uniform K t* −
( F , α , ρ t* , d ) − quasiconvex at x * with respect to b1 and φ1 ; p
(iii) 0 ∈ ∑ λi*∂ Ki f i ( x * ) + i =1
∑
µt*∂ K g t ( x* ) ; * t
t∈T ( x* )
(iv) a < 0 ⇒ φ0 ( a ) < 0 and φ0 (0) = 0;
a ≦≦≧ 0 ⇒ φ1 ( a )
From (iv), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
*
there exists λ * ∈ Λ ++ and ( µt* )t∈T ( x* ) ∈ R+(T ( x
f )( x )
p
(v)
∑λ ρ i =1
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0;
b0 ( x, x ) > 0, b1 ( x, x ) ≧ 0 ; *
t∈T
∀ξ ∈ ∂ (λ
weakly quasiconvex at x * with respect to b0 and φ0 ;
t∈T
*
*T
(6)
Finally, we have a contradiction. Hence x * is an efficient solution of (SIVP). Similarly, we can derive the following theorem easily. Theorem3.4. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the following conditions. (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
(iv) µt* g t ( x *= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0; a ≦≦ 0 ⇒ φ1 ( a )
(vi) ρ 0 + ∑ µt* ρ t* ≧ 0 .
K0
( x, x * ) ≦ 0
(iii) 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) ;
t∈T
*
*
2
t∈T
, d ) − pseudoconvex at x * with respect to b0 and φ0 ;
*
* t
t∈T
Finally, we have a contradiction. Hence x * is a weakly efficient solution of (SIVP). Theorem3.3. Let x * ∈ X 0 . Assume that for all x ∈ X 0 , there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+(T ) , satisfying the follow- ing conditions. (i) λ *T f is strictly generalized uniform K 0 − ( F , α , ρ 0
(iv) µ g t ( x = ) 0, t ∈ T ; (v) a ≦≦≦≦ 0 ⇒ φ0 ( a ) 0; a
* t
Adding (5) and (6), then by the sublinearity of F and (vi), we obtain
F ( x, x * ; α ( x, x* )(ξ + ∑ µt*ζ t )) t∈T
* t
*
* i
i
+
∑
t∈T ( x* )
0; b0 ( x, x * ) > 0, b1 ( x, x * )
µt* ρ t* ≧ 0 .
0;
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
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F ( x 0 , u; α ( x 0 , u ) µt0 ζ t0 ) + µt0 ρ t*0 d 2 ( x 0 , u ) < 0,
Then x * is an efficient solution of (SIVP).
∀ζ t0 ∈ ∂ t0 g t0 (u ) K
IV. DUALITY THEOREMS In this section, we consider the following Mond-Weir type dual model for (SIVP): (SIVD) max f (u )
s.t. 0 ∈ ∂ K0 (λ T f )(u ) + ∑ µt ∂ Kt g t (u );
(7)
t∈T
µt gt (u ) ≧ 0, t ∈ T ;
(8)
By µt ≧ 0, t ∈ T (u ), t ≠ t0 and (4.2), we have
g t ( x 0 ) ≦≦ 0 g t (u ) Then from (iv) and (ii), for any ζ t ∈ ∂ Kt g t (u ) , we have
F ( x 0 , u; α ( x 0 , u ) µt ζ t ) + µt ρ t*d 2 ( x 0 , u ) ≦ 0, t ∈ T (u ), t ≠ t0 Now, together with t ∈ T (u ), t ≠ t0 implies
F ( x 0 , u; α ( x 0 , u )
λ ∈ Λ ( Λ ), ( µt )t∈T ∈ R . (9) 0 Let D {(u, λ , µ ) (u, λ , µ ) satisfies (7) − (9)} de-note +
++
T +
the set of all feasible solutions of (SIVD). Theorem4.1. (Weak duality) Assume that for feasible solutions x ∈ X 0 and (u, λ , µ ) ∈ D 0 for (SIVP) and (SIVD), respectively, assumption A0 holds, and for λ ∈ Λ + , ( µt )t∈T ∈ R+T , we have
+
∑
t∈T ( u ) t ≠ t0
and together with µt0 > 0 ;
0 ; b0 ( x, u )
(v) ρ 0 + ∑ µt ρ ≧ 0 . t∈T
+
0, b1 ( x, u ) > 0 ;
* t
Since λ ∈ Λ + , it follows that λ T f ( x ) ≦ λ T f (u ) According to (iv), we have b0 ( x, u )φ0 [(λ T f )( x ) − (λ T f )(u )] ≦ 0 Then (i) yields F ( x, u; α ( x, u )ξ ) + ρ 0 d 2 ( x, u ) ≦ 0, ∀ξ ∈ ∂ K0 (λ T f )(u ) For x ∈ X , we have 0
(10)
∀ξ ∈ ∂ K0 (λ T f )(u )
(13)
µt ρ t*d 2 ( x 0 , u ) = 0
Adding (10), (11), (12) and (13), then by the sublinearity of F and (v), we obtain F ( x 0 , u; α ( x 0 , u )(ξ + ∑ µtζ t )) t∈T
< − ( ρ 0 + ∑ µt ρ ) d ( x 0 , u ) ≦ 0 * t
2
(SIV- D), respectively, assumption A0 for λ ∈ Λ ++ , ( µt )t∈T ∈ R+T , we have
holds, and
(i) λ T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − pseudoconvex on X 0 with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex on X 0 with respect to b1 and φ1 ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≦≦ 0 ⇒ φ1 ( a )
0;
b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ;
(iv) ρ 0 + ∑ µt ρ t* ≧ 0 .
Then we can obtain f ( x ) f (u ) . Proof: Suppose that the result does not hold, then there exists x ∈ X 0 , such that f ( x ) ≦ f (u ) . Since λ ∈ Λ ++ , it follows that λ T f ( x ) < λ T f (u ) By (iii), we obtain b0 ( x, u )φ0 [(λ T f )( x ) − (λ T f )(u )] < 0 Then (i) yields
By (iii), we obtain 0 g t0 ( x= ) < 0 g t0 (u ), t0 ∈ T (u ) From (iv), we get b1 ( x 0 , u )φ1[ g t0 ( x 0 ) − g t0 (u )] < 0 Then by (ii), we have
F ( x 0 , u; α ( x 0 , u )ζ t0 ) + ρ t*0 d 2 ( x 0 , u ) < 0, ∀ζ t0 ∈ ∂ t0 g t0 (u ) K
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t∈T \ T ( u )
µt ζ t )
t∈T
F ( x 0 , u; α ( x 0 , u )ξ ) + ρ 0 d 2 ( x 0 , u ) ≦ 0,
Since µt0 > 0 , it follows that
∑
t∈T
Then we can obtain f ( x ) f (u ) . Proof: Suppose that the result does not hold, then there exists x ∈ X 0 , such that f ( x ) ≦ f (u ) .
0
∑
t∈T \ T ( u )
But which contradicts (7). Hence the result follows. Theorem4.2. (Weak duality) Assume that for feasible solutions x ∈ X 0 and (u, λ , µ ) ∈ D 0 for (SIVP) and
(iv) a < 0 ⇒ φ1 ( a ) < 0 and φ1 (0) = 0;
a ≦≦≧ 0 ⇒ φ0 ( a )
(12)
µt ρ t*d 2 ( x 0 , u ) ≦ 0
F ( x 0 , u; α ( x 0 , u )
0
(iii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for t0 ∈ T (u ) ,
µt ζ t )
Let µt = 0 , for t ∈ T \ T (u ) , so we have
(i) λ f is generalized uniform K 0 − ( F , α , ρ 0 , d ) −
( F , α , ρ t* , d ) − convex on X 0 with respect to b1 and φ1 ;
∑
t∈T ( u ) t ≠ t0
T
convex on X with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
(11)
F ( x, u; α ( x, u )ξ ) + ρ 0 d 2 ( x, u ) < 0, ∀ξ ∈ ∂ K0 (λ T f )(u )
(14)
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By µt ≧ 0, t ∈ T (u ) and (11), we obtain
Suppose that ( x * , λ * , µ * ) is not an efficient solution of
g t ( x ) ≦≦ 0 g t (u ) Then from (iii) and (ii), we have F ( x, u; α ( x, u ) µt ζ t ) + µt ρ t*d 2 ( x, u ) ≦ 0,
(SIVD), then there exists ( x, λ , µ ) , such that
f ( x * ) ≦ f (u ) By the weak duality theorem 4.1, we get f ( x ) f (u )
∀ζ t ∈ ∂ Kt g t (u ), t ∈ T (u ) Let µt = 0 , for t ∈ T \ T (u ) , so we have
∑ F ( x , u; α ( x , u ) µ ζ ) + ∑ µ ρ d t
t∈T
t
t
t∈T
* t
2
( x, u ) ≦ 0,
∀ζ t ∈ ∂ Kt g t (u )
(15)
Adding (14) and (15), then by the sublinearity of F and (iv), we obtain F ( x, u; α ( x, u )(ξ + ∑ µtζ t )) t∈T
< − ( ρ 0 + ∑ µt ρ ) d 2 ( x , u ) ≦ 0 t∈T
* t
But which contradicts (7). Hence the result follows. Theorem4.3. (Strong duality) Suppose that x * is a weakly efficient solution of (SIVP), assumption A0 holds, and for λ ∈ Λ + ,( µt )t∈T ∈ R+T , such that (i) λ f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − T
convex at u ∈ D with respect to b0 and φ0 ; 0
(ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − convex at u ∈ D 0 with respect to b1 and φ1 ; (iii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for t0 ∈ T (u ) , and together with µt0 > 0 ; (iv) a < 0 ⇒ φ1 ( a ) < 0 and φ1 (0) = 0;
a ≦≦≧ 0 ⇒ φ0 ( a )
(v) ρ 0 + ∑ µt ρ ≧ 0 ; t∈T
0 ; b0 ( x, u )
0, b1 ( x, u ) > 0 ;
* t
(vi) The Kuhn-Tuker constraint qualification is satisfied. Then there exists λ * ∈ Λ + , such that ( x * , λ * , µ * ) is an efficient solution of (SIVD). Furthermore, the two problems (SIVP) and (SIVD) have the same objective values. Proof: Since x * is a weakly efficient solution of (SIVP), and the problem satisfies the Kuhn-Tuker ,s constraint qualification. So there exists λ * ∈ Λ + and ( µt* )t∈T ∈ R+T , such that 0 ∈ ∂ K0 (λ *T f )( x * ) + ∑ µt*∂ Kt g t ( x * ) t∈T * * g ( x ) 0, t T µ = ∈ t t * * (λ , µ ) ≥ 0 Hence ( x * , λ * , µ * ) is a feasible solution of (SIVD). It is clear that the two problems (SIVP) and (SIVD) have the same objective values.
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Then we have a contradiction. Hence ( x * , λ * , µ * ) is an efficient solution of (SIVD). Theorem4.4. (Strong duality) Suppose that x * is an efficient solution of (SIVP), assumption A0 holds, and for λ ∈ Λ ++ ,( µt )t∈T ∈ R+T , such that (i) λ T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − pe- seudoconvex on X 0 with respect to b0 and φ0 ; (ii) For all t ∈ T (u ), g t is generalized uniform K t −
( F , α , ρ t* , d ) − quasiconvex on X 0 with respect to b1 and φ1 ; (iii) a < 0 ⇒ φ0 ( a ) < 0 ; a ≧ 0 ⇒ φ1 ( a ) ≧ 0 ; b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ; (iv) ρ 0 + ∑ µt ρ t* ≧ 0 ; t∈T
(v) The Kuhn-Tuker constraint qualification is satisfied at x * . Then there exists λ * ∈ Λ ++ , such that ( x * , λ * , µ * ) is an efficient solution of (SIVD). Furthermore, the two problems (SIVP) and (SIVD) have the same objective values. Proof: The proof follows along similar lines as the proof of theorem 4.3 and hence is omitted. Theorem 4.5. (Reverse duality) Suppose that x * ∈ X 0 , (u* , λ * , µ * ) ∈ D 0 , f ( x * ) = f (u* ) , assumption A0 holds, and for λ * ∈ Λ + , ( µt* )t∈T ∈ R+T , we have (i) λ *T f is generalized uniform K 0 − ( F , α , ρ 0 , d ) − convex at u* with respect to b0 and φ0 ; (ii) For t ∈ T (u* ) , g t is generalized uniform K t −
( F , α , ρ t* , d ) − convex at u* with respect to b1 and φ1 ; (iii) 0 ∈ ∂ K0 (λ *T f )(u* ) + ∑ µt*∂ Kt g t (u* ) ; t∈T
(iv) µt* g t (u*= ) 0, t ∈ T ; (v) a < 0 ⇒ φ0 ( a ) < 0 and φ= 0; a ≦ 0 ⇒ φ1 ( a ) ≦ 0 ; 0 (0) b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ; (vi) ρ 0 + ∑ µt* ρ t* ≧ 0 . t∈T
(vii) The generalized Slater ,s condition is satisfied, that is, there exists x 0 ∈ X 0 such that g t0 ( x 0 ) < 0 , for all
t0 ∈ T (u* ) , and corresponding µt0 > 0 . Then x * is a weakly efficient solution of (SIVP), and (u* , λ * , µ * ) is an efficient solution of (SIVD).
JOURNAL OF NETWORKS, VOL. 8, NO. 2, FEBRUARY 2013
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Proof: Form theorem 3.1, we known that u* is also a µt* F ( x* , u* ; α ( x* , u* )ζ t ) + ∑ µt* ρ t*d 2 ( x* , u* ) ∑ t∈T t∈T weakly efficient solution of (SIVP). It is clear that x * is * * * * * = F ( x , u ; α ( x , u ) µ ζ ) + µt* ρ t*d 2 ( x* , u* ) ≦ 0 also a weak efficient solution of (SIVP). If it is not true, ∑ ∑ t t t∈T t∈T there exists x ∈ X 0 , such that From (iv), we obtain there exists ξ 0 ∈ ∂ K0 (λ *T f )(u* ) , f ( x ) < f ( x* ) = f (u* ) * ζ t0 ∈ ∂ Kt gt (u* )(t ∈ T ) , such that We have a contradiction. Under the conditions of theorem 4.1, we known the ξ 0 + ∑ µt*ζ t0 = 0 * t∈T weak duality holds at u . By the sublinearity of F , we obtain Suppose that ( x * , λ * , µ * ) is not an efficient solution of = 0 F ( x * , u* ; α ( x * , u* )(ξ 0 + ∑ µt*ζ t0 )) (SIVD). Then there exists u , such that t∈T * * * * 0 * * F ( x , u ; α ( x , u ) ξ ) + F ( x * , u* ; α ( x * , u* )∑ µt*ζ t0 ) ≦ f (u ) ≥ f (u ) = f (x ) This contradicts weak duality. Hence ( x * , λ * , µ * ) is an efficient solution of (SIVD). Theorem 4.6. (Strict reverse duality) Suppose that x * ∈ X 0 , (u* , λ * , µ * ) ∈ D 0 , assumption A0 holds, and for
λ * ∈ Λ + , ( µt* )t∈T ∈ R+T , we have (ii)
λ f
is
strictly
generalized
uniform
K 0 − ( F , α , ρ 0 , d ) − pseudoconvex at u with respect to b0 and φ0 ; t ∈ T (u* ), g t
is
generalized
uniform
K t − ( F , α , ρ , d ) − quasiconvex at u with respect to b1 and φ1 ; *
* t
(iv) 0 ∈ ∂ K0 (λ *T f )(u* ) + ∑ µt*∂ Kt g t (u* ) ; t∈T
(v) µt* g t (u*= ) 0, t ∈ T ; (vi) φ0 ( a ) > 0 ⇒ a > 0; a ≦ 0 ⇒ φ1 ( a ) ≦ 0 ;
b0 ( x, u ) > 0, b1 ( x, u ) ≧ 0 ;
(vii) ρ 0 + ∑ µ ρ t* ≧ 0 . t∈T
* t
Then x * = u* . Proof: We assume that x * ≠ u* . By hypothesis (ii) and (iii), we have *
*
*
2
*
⇒ b0 ( x * , u* )φ0 [λ *T f ( x * ) − λ *T f (u* )] > 0 b1 ( x , u )φ1[ g t ( x ) − g t (u )] ≦ 0 *
*
*
(16)
⇒ F ( x , u ; α ( x , u )ζ t ) + ρ d ( x , u ) ≦ 0, *
*
*
*
* t
2
*
*
∀ζ t ∈ ∂ Kt g t (u* ), t ∈ T (u* ) As t ∈ T (u* ) , we have
g t ( x * ) ≦ 0 = g t (u* ) Using (vi), we get b1 ( x * , u* )φ1[ g t ( x * ) − g t (u* )] ≦ 0 From (v), we have µt* ≧ 0 , as t ∈ T ( x * ) ; we let µt* = 0 , as t ∈ T \ T ( x * ) . So it follows that
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This contradicts hypothesis (i). Hence, x * = u* . V. CONCLUSION Throughout this paper, we have defined a new generalized convex function, extending many well-known classes of generalized convex functions. Furthermore, we have achieved some sufficient optimality conditions for a class of multiobjective semi-infinite 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.
This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 08JK237).
*
∀ξ ∈ ∂ K0 (λ *T f )(u* ) *
Then from (vi) and (16), we have λ *T f ( x* ) > λ *T f (u* )
ACKNOWLEDGMENT
F ( x , u ; α ( x , u )ξ ) + ρ 0 d ( x , u ) ≧ 0, *
t∈T
t∈T
*
(iii)For
≧- F ( x * , u* ; α ( x * , u* )∑ µt*ζ t0 ) ≧ ∑ µt* ρ t*d 2 ( x * , u* ) ≧- ρ 0 d 2 ( x * , u* )
(i) λ *T f ( x * ) = λ *T f (u* ) *T
t∈T
which with (vii) yields F ( x * , u* ; α ( x * , u* )ξ 0 )
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Xiaoyan Gao, 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 multi-objective programming, etc.
