Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions

J. Math. Anal. Appl. 288 (2003) 365–382 www.elsevier.com/locate/jmaa

Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions Vasile Preda Mathematics Faculty, University of Bucharest, 14 Academiei Street, Bucharest 70109, Romania Received 4 January 2000 Submitted by C.R. Bector

Abstract Necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions.  2003 Elsevier Inc. All rights reserved.

1. Introduction Many optimality conditions and approaches to duality for the nonlinear multiple objective optimization problem have been of much interest in the recent past and many contributions have been made to this development, e.g., Bector et al. [2], Bitran [4], Cambini and Martein [5,6], Corley [7], Craven [8,9], Elster and Nehse [10], Geoffrion [12], Heal [15], Ivanov and Nehse [16], Jeyakumar [17–21], Singh [34], Tanino and Sawaragi [36], Weir [37], Weir and Mond [38], White [40]. Some studies differ in their approaches and/or in the sense in which the optimality concept is defined for a multiple objective programming problem. Some approaches to duality include the use of vector valued Lagrangians and Lagrangians (e.g., [9,17,18,36,37,39]), incorporating matrix Lagrange multipliers (e.g., [4,7,8,16]).

E-mail address: [email protected]. 0022-247X/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-247X(02)00460-2

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Also, it is well known (see Mangasarian [28]) that, under a convexity assumption and a regular hypothesis, there exists an equivalence between saddle-points of the Lagrangian and optima for an inequality constrained minimization problem (for discussions and extensions of this result see Heal [15], Ben-Israel and Mond [3] and Jeyakumar [17]). In [18], Jeyakumar discussed a class of nonsmooth nonconvex problems in which functions are locally Lipschitz and are satisfying some invex type conditions. Further, it is shown that duality theorems of Wolfe type [41] hold for this class of problems. Also, in Cambini and Martein [5,6], a new approach to optimality conditions in vector and scalar optimization is given. Elster and Nehse [10] considered a class of convexlike functions and obtained a saddlepoint optimality condition for mathematical programs involving such functions. Hayashi and Komiya [14] also considered Lagrangian duality for convexlike programs. BenIsrael and Mond [3] and Hanson and Mond [13] considered a class of functions called preinvex. In [20], Jeyakumar and Mond introduced new classes of generalized convex vector functions, called v-invex, and some results relative to Lagrangian sufficiency, weak duality and global optimality are given. In [9], Craven has given Lagrangian necessary conditions for optimality, of both Fritz– John and Kuhn–Tucker types for a constrained minimization problem, where the functions are locally Lipschitz and the directional derivatives are assumed to have some convexity properties as functions of direction. Further, in Craven [9], some sufficient Kuhn–Tucker conditions and a criterion for the locally solvable constraint qualification are obtained. In [19] and [21], some classes of nonsmooth programming problems are given. The concept of semilocally convex functions was introduced by Ewing [11] and was further extended to semilocally quasiconvex, semilocally pseudoconvex functions by Kaul and Kaur [22–24]. In Suneja and Gupta [35] the (strict) semilocally pseudoconvexity is defined at a point with respect to a set. A number of properties of these functions were given by Kaul and Kaur [23,24] and Suneja and Gupta [35] (see Mahajanm and Vartak [27] for the case of semilocally convex). By using these concepts for a scalar valued nonlinear programming problem in Kaul and Kaur [22–24] and Suneja and Gupta [35], some optimality conditions and duality results are obtained. These results are extended in [31] for a multiple objective programming problem. Preda and Stancu-Minasian [33] stated the Fritz–John and Karush–Kuhn–Tucker optimality conditions for weak vector minima using η-semidifferentials and functions satisfying generalized semilocally preinvex properties. These results are then used to extend the Wolfe [41] and Mond–Weir [29] duals. Also, some results of Preda [31], Preda et al. [32], and Suneja and Gupta [35] are generalized. Recently, Lyall et al. [26], some necessary and sufficient optimality conditions for a fractional programming problem with semilocally convex, semilocally quasiconvex and semilocally pseudoconvex functions are stated. Also, a dual program and duality results of weak and strong duality have been proved for the pair of primal and dual programs. In this paper we are considering necessary and sufficient optimality conditions for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions. Thus, many results of Lyall et al. [26],

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Preda [31], Preda et al. [32], Preda and Stancu-Minasian [33], and Suneja and Gupta [35] are generalized. We may remark that important results concerning the preinvex functions were recently obtained by Li and Dong [25], Antczak [1], Yang and Li [42,43], and Yang et al. [44].

2. Definitions and preliminaries For x, y ∈ Rn , by x
y we mean xi
yi for all i, x  y means xi
yi for all i and xj < yj for at least one j , 1
j
n. By x < y we mean xi < yi for all i and by x  y we mean the negation of x  y. Let X0 ⊆ Rn be a set and η : X0 × X0 → Rn be a vectorial application. We say that the set X0 is η-vex at x ∈ X0 if x + λη(x, x) ∈ X0 for any x ∈ X0 and λ ∈ [0, 1]. We say that the set X0 is η-vex if X0 is η-vex at any x ∈ X0 . We remark that if η(x, x) = x − x for any x ∈ X0 then X0 is η-vex at x iff X0 is a convex set at x. Definition 1. We say that the set X0 ⊆ Rn is an η-locally starshaped set at x, x ∈ X0 , if for any x ∈ X0 , there exists 0 < aη (x, x)
1 such that x + λη(x, x) ∈ X0 for any λ ∈ [0, aη (x, x)]. Definition 2 [31]. Let f : X0 → Rn be a function, where X0 ⊆ Rn is an η-locally starshaped set at x ∈ X0 . We say that f is: (i1 ) semilocally preinvex (slpi) at x if, corresponding to x and each x ∈ X0 , there exists a positive number dη (x, x)
aη (x, x) such that f (x +λη(x, x))
λf (x)+(1 −λ)f (x) for 0 < λ < dη (x, x); (i2 ) semilocally quasi-preinvex (slqpi) at x if, corresponding to x and each x ∈ X0 , there exists a positive number dη (x, x)
aη (x, x) such that f (x)
f (x) and 0 < λ < dη (x, x) implies f (x + λη(x, x))
f (x). Definition 3. Let f : X0 → Rn be a function, where X0 ⊆ Rn is an η-locally starshaped set at x ∈ X0 . We say that f is η-semidifferentiable at x if (df )+ (x, η(x, x)) exists for each x ∈ X0 , where
   1
(df )+ x, η(x, x) = lim f x + λη(x, x) − f (x) λ→0+ λ (the right derivative at x along the direction η(x, x)). If f is η-semidifferentiable at any x ∈ X0 , then f is said to be η-semidifferentiable on X0 . Remark. If η(x, x) = x − x, the η-semidifferentiability is the semidifferentiability notion. As is given in [26], if a function is directionally differentiable, then it is semidifferentiable but the converse is not true.

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Theorem 4. Let f : X0 → Rn be an η-semidifferentiable function at x ∈ X0 . If f is slqpi at x and f (x)
f (x) then (df )+ (x, η(x, x))
0. Definition 5 [31]. We say that f is semilocally pseudo-preinvex (slppi) at x if for any x ∈ X0 , (df )+ (x, η(x, x))  0 ⇒ f (x)  f (x). If f is slppi at any x ∈ X0 , then f is said to be slppi on X0 . Definition 6. Let X and Y be two subsets of X0 and y ∈ Y . We say that Y is η-locally starshaped at y with respect to X if for any x ∈ X there exists 0 < aη (x, y) ¯
1 such that y + λη(x, y) ∈ Y for any 0
λ
aη (x, y). Definition 7. Let be η-locally starshaped at y with respect to X and f be an η-semidifferentiable function at y. We say that f is: (i1 ) slppi at y ∈ Y with respect to X, if for any x ∈ X, (df )+ (y, η(x, y))  0 ⇒ f (x)  f (y); (i2 ) strictly semilocally pseudo-preinvex (sslppi) at y with respect to X, if for each x ∈ X, x = y, (df )+ (y, η(x, y))  0 ⇒ f (x) > f (y). We say that f is (slppi) sslppi on Y with respect to X, if f is (slppi) sslppi at any point of Y with respect to X. Definition 8 (Elster and Nehse [10]). A function f : X0 → Rk is a convexlike function if for any x, y ∈ X0 and 0
λ
1, there is z ∈ X0 such that f (z)
λf (x) + (1 − λ)f (y). Remark. The convex and the preinvex functions are convexlike functions. Lemma 9 (Hayashi and Komiya [14]). Let S be a nonempty set in Rn and ψ : S → Rk be a convexlike function. Then either ψ(x) < 0 has a solution x ∈ S or λT ψ(x)  0

for all x ∈ S,

for some λ ∈ Rk , λ  0, but both alternatives are never true. (Here the symbol T denotes the transpose of a matrix.) Using Lemma 9 from above instead of Lemma 2.9 from [33], we have that the Theorems 3.4 and 3.5 stated there are still true. Thus, in the next section we will use the following version of Theorem 3.5 from [33]. Theorem 10. Let x ∈ X be a (local) weak minimum solution for the following problem:

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 ϕ1 (x), . . . , ϕp (x)  hj (x)
0, j ∈ M, subject to x ∈ X0 ,

minimize

where ϕ = (ϕ1 , . . . , ϕp ) : X0 → Rp and h1 , . . . , hm are η-semidifferentiable at x. Also, we assume that hj (j ∈ N(x)) is a continuous function at x and (dϕ)+ (x, η(x, x)) and (d)+ (x, η(x, x)) are convexlike functions of x on X0 . If h satisfies a regularity condition at x (see [33, Definition 3.2]), then there exist λ0 ∈ Rp , u0 ∈ Rm such that

  T T λ0 (dϕ)+ x, η(x, x) + u0 (dh)+ x, η(x, x)  0 for all x ∈ X0 , T

u0 h(x) = 0, 0T

λ e = 1,

h(x)
0, λ  0, 0

u0  0,

where e = (1, . . . , 1)T ∈ Rp . 3. Necessary optimality conditions In this paper we consider the following multiobjective nonlinear fractional programming problem:   fp (x) f1 (x) (VFP) minimize ,..., g1 (x) gp (x)  hj (x)
0, j = 1, 2, . . . , m, subject to x ∈ X0 , where X0 ⊆ Rn is a nonempty set and gi (x) > 0 for all x ∈ X0 and each i = 1, . . . , p. Let f = (f1 , . . . , fp ), g = (g1 , . . . , gp ), h = (h1 , . . . , hm ). We put X = {x ∈ X0 | hj (x)
0, j = 1, 2, . . . , m} for the feasible set of problem (VFP). Definition 11. For the problem (VFP), a point x ∈ X is said to be a weak minimum if there exists no other feasible point x for which f (x)/g(x) > f (x)/g(x). For x ∈ X we put M(x) = {j ∈ M | hj (x) = 0}, h0 = (hj )j ∈M(x) and N(x) = M \ M(x), where M = {1, 2, . . . , m}. Definition 12. We say that (VFP) satisfies the generalized Slater’s constraint qualification (GSCQ) at x ∈ X if h0 is slppi at x and there exists an xˆ ∈ X such that h0 (x) ˆ < 0. Lemma 13. Let x ∈ X be a (local) weak minimum solution for (VFP). Further, we assume that hj is continuous at x for any j ∈ N(x) and that f, g, h0 are η-semidifferentiable at x. Then, the system   (df )+ (x, η(x, x)) < 0, (1) (dg)+ (x, η(x, x)) > 0,  (dh0 )+ (x, η(x, x)) < 0 has no solution x ∈ X0 .

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Proof. Let x be a (local) weak minimum solution for (VFP) and suppose there exists x ∗ ∈ X0 such that
 (2) (df )+ x, η(x ∗ , x) < 0,
 + ∗ (dg) x, η(x , x) > 0, (3)
 0 + ∗ (dh ) x, η(x , x) < 0. (4) Let ϕ(x, x ∗ , λ) = f (x + λη(x ∗ , x)) − f (x). We have ϕ(x, x ∗ , 0) = 0 and the right differential of ϕ(x, x ∗ , 0) with respect to λ at λ = 0 is given by   lim λ−1 ϕ(x, x ∗ , λ) − ϕ(x, x ∗ , 0) λ0 
  = lim λ−1 f x + λη(x ∗ , x) − f (x) λ0
 = (df )+ x, η(x ∗ , x) < 0 (using (2)). Hence, ϕ(x, x ∗ , λ) < 0 if λ is in some open interval (0, δ1 ), δ1 > 0; i.e., f (x + λη(x ∗ , x)) < f (x), λ ∈ (0, δ1 ). Similarly we get 
g x + λη(x ∗ , x) > g(x), λ ∈ (0, δ2 ), 
h0 x + λη(x ∗ , x) < h0 (x) = 0, λ ∈ (0, δ3 ), where δ2 > 0, δ3 > 0. Now, for j ∈ N(x), hj (x) < 0 and hj is continuous at x and therefore, there exists δ  > 0 such that 
hj x + λη(x ∗ , x) < 0, λ ∈ (0, δ  ) for any j ∈ N(x). Let δ = min(δ1 , δ2 , δ3 , δ  ). Then x + λη(x ∗ , x) ∈ Sδ (x) ⊆ Uδ (x),

λ ∈ (0, δ),

(5)

where Sδ (x) is a hypersphere around x and Uδ (x) is a neighbourhood of x. Now, we have 
(6) f x + λη(x ∗ , x) < f (x), 
∗ (7) g x + λη(x , x) > g(x),
 ∗ h x + λη(x , x) < 0, (8) for any λ ∈ (0, δ). By (5) and (8), we have x + λη(x ∗ , x) ∈ X0 ∩ Uδ (x), for any λ ∈ (0, δ). Using (6) and (7), for Q(x) = (f1 (x)/g1 (x), . . . , fp (x)/gp (x)) we get 
Q x + λη(x ∗ , x) < Q(x), which contradicts the assumption that x is a local weak solution of (VFP). Hence, there exists no x ∈ X0 satisfying the system (1). Thus the lemma is proved. ✷ In the next theorem we obtain an important result of Fritz–John type necessary optimality criteria.

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Theorem 14 (Fritz–John type necessary optimality criteria). Let us suppose that hj is continuous at x for j ∈ N(x), (df )+ (x, η(x, x)), (dg)+ (x, η(x, x)) and (dh0 )+ (x, η(x, x)) are convexlike functions of x on X0 . If x is a (local) weak minimum solution for (VFP), then there exist λ0 ∈ Rp , u0 ∈ Rp , v 0 ∈ Rm such that

  T T λ0 (df )+ x, η(x, x) − u0 (dg)+ x, η(x, x)
 T (9) + v 0 (dh0 )+ x, η(x, x)  0 for all x ∈ X0 , T

v 0 h(x) = 0, (λ , u , v ) = 0, 0

0

0

(10) (λ , u , v )  0. 0

0

0

(11)

Proof. If x is a (local) weak minimum solution for (VFP) then, by Lemma 13, the system (1) has no solution x ∈ X0 . But the assumption of Lemma 9 also holds and since the system (1) has no solution x ∈ X0 , we obtain that there exists λ0 ∈ Rp , u0 ∈ Rp , vj0 ∈ R (j ∈ M(x)), such that λ0  0, u0  0, vj0  0 (j ∈ M(x)), (λ0 , u0 , (vj0 )j ∈M(x) ) = 0, with  

T T λ0 (df )+ x, η(x, x) − u0 (dg)+ x, η(x, x)  +
 T
+ vj0 dh0j x, η(x, x)  0 for all x ∈ X0 .

(12)

j ∈M(x)

If we put vj0 = 0 for j ∈ N(x), by (12) we get (9). Finally, the relations (7) and (8) follow obviously and the proof is complete. ✷ The next theorem is a Karush–Kuhn–Tucker type necessary optimality criterion. In this theorem, the above defined generalized constraint qualification is very important. p p For each u = (u1 , . . . , up ) ∈ R+ , where R+ denotes the positive orthant of Rp , we consider
 (VFPu ) minimize f1 (x) − u1 g1 (x), . . . , fp (x) − up gp (x)  hj (x)
0, j ∈ M, subject to x ∈ X0 . The following lemma can be proved without difficulty: Lemma 15. If x is a (local) weak minimum for (VFP) then x is a (local) weak minimum for (VFPu0 ), where u0 = f (x)/g(x). Using this lemma we can derive a Karush–Kuhn–Tucker type necessary optimality criterion for the problem (VFP). Theorem 16 (Karush–Kuhn–Tucker type necessary optimality criterion). Let x be a (local) weak minimum solution for (VFP), let hj be continuous at x for j ∈ N(x) and let (dfi )+ (x, η(x, x)), (dgi )+ (x, η(x, x)), i ∈ P , and (dh0 )+ (x, η(x, x)) be convexlike p p functions of x on X0 . If g satisfies (GSQ) at x, then there exist λ0 ∈ R+ , u0 ∈ R+ , v 0 ∈ Rm such that

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V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382 p 




  λ0i (dfi )+ x, η(x, x) − u0i (dgi )+ x, η(x, x)


 T + v 0 (dh)+ x, η(x, x)  0

i=1 0T

v h(x) = 0,

for all x ∈ X0 ,

h(x)
0,

0T

λ e = 1,

λ  0,

u0  0,

0

v 0  0,

where e = (1, . . . , 1)T ∈ Rp . Proof. Let x be a (local) weak minimum solution for (VFP). According to Lemma 15, we have that x is a (local) weak minimum solution for (VFPu0 ), where u0 = (u01 , . . . , u0p ), u0i = fi (x)/gi (x), i ∈ P . Now, applying Theorem 10 to problem (VFPu0 ), we get that there p exist λ0 ∈ R+ , v 0 ∈ Rm such that p 




  λ0i (dfi )+ x, η(x, x) − u0i (dgi )+ x, η(x, x)


 T + v 0 (dh)+ x, η(x, x)  0

i=1

for all x ∈ X0 ,

(13)

0T

v h(x) = 0,

(14)

h(x)
0,

(15)

0T

λ e = 1, λ  0, 0

(16) u  0, 0

v  0, 0

(17)

and the theorem is proved. ✷ Remark. In the above theorem we can suppose, for any i ∈ P , that (dfi )+ (x, η(x, x)) − u0i (dgi )+ (x, η(x, x)) is convexlike on X0 , where u0i = fi (x)/gi (x), instead of considering that (dfi )+ (x, η(x, x)) and (dgi )+ (x, η(x, x)) are convexlike on X0 , for any i ∈ P .

4. Sufficient optimality criteria In this section, using the concept of (local) weak optimality, we give some sufficient optimality conditions for the (VFP) problem. Theorem 17. Let x ∈ X and f be η-semilocally convex at x, g be η-semilocally concave at x, and h be η-semilocally convex at x. Also, we assume that there exists λ0 ∈ Rp , u0 ∈ Rp and v 0 ∈ Rm such that p 

 


T λ0i (dfi )+ x, η(x, x) + v 0 (dh)+ x, η(x, x)  0

for all x ∈ X,

(18)

i=1


 (dgi )+ x, η(x, x)
0, 0T

v h(x) = 0,

∀x ∈ X, ∀i ∈ P ,

(19) (20)

V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382

h(x)
0,

(21)

0T

λ e = 1, λ  0, 0

373

(22) u  0,

v  0.

0

0

(23)

Then x is a weak minimum solution for (VFP). Proof. We proceed by contradicting. Hence there exists x˜ ∈ X such that fi (x) fi (x) ˜ < gi (x) ˜ gi (x)

for any i ∈ P .

(24)

Since f is η-semilocally convex at x, g is η-semilocally concave at x, and h is ηsemilocally convex at x, we get
 fi (x) ˜ − fi (x)  (dfi )+ x, η(x, ˜ x) , i ∈ P , (25)
 + ˜ − gi (x)
(dgi ) x, η(x, ˜ x) , i ∈ P , (26) gi (x) 
+ ˜ − hj (x)  (dhj ) x, η(x, ˜ x) , j ∈ M. (27) hj (x) p

Multiplying (25) by λ0i  0, i ∈ P , λ0 ∈ R+ , (27) by vj0  0, j ∈ M, and then summing the obtained relations, we get p 

λ0i






fi (x) ˜ − fi (x) +

m 


 vj0 hj (x) ˜ − hj (x)

j =1

i=1



p 

m

   λ0i (dfi )+ x, η(x, ˜ x) + vj0 (dhj )+ x, η(x, ˜ x)  0, j =1

i=1

where the last inequality is according to (18). Hence, p 

m
  λ0i fi (x) ˜ − fi (x) + vj0 hj (x) ˜ − v 0 h(x)  0.

(28)

j =1

i=1

Since x ∈ X, v 0  0, by (20) and (28) we get p 


 λ0i fi (x) ˜ − fi (x)  0.

(29)

i=1

Using (22), (23) and (29), we obtain that there exists i0 ∈ P such that fi0 (x) ˜  fi0 (x).

(30)

By (19) and (26) it follows gi (x) ˜
gi (x),

i ∈ P.

Now, using (30), (31) and f  0, g > 0, we obtain fi (x) ˜ fi0 (x)  0 gi0 (x) ˜ gi0 (x)

(31)

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V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382

which is in contradiction to (24). Thus, the theorem is proved and x is a weak minimum solution for (VFP). ✷ 0 p 0 p 0 m Corollary Let x ∈ p 18. Xm and0assume that there exist λ ∈ R , u ∈ R and v ∈ R such 0 that i=1 λi fi (·) + j =1 vj hj (·) is η-semilocally convex at x and (19)–(23) hold. Then x is a weak minimum solution for (VFP).

Theorem 19. Let x ∈ X and f be η-semilocally convex at x, g be η-semilocally concave at x, and h be η-semilocally convex at x. Also, we assume that there exists λ0 ∈ Rp , u0i = fi (x)/gi (x), i ∈ P , and v 0 ∈ Rm such that p 

 


λ0i (dfi )+ x, η(x, x) − u0i (dgi )+ x, η(x, x)


 T + v 0 (dh)+ x, η(x, x)  0

i=1

for all x ∈ X,

(32)

0T

v h(x) = 0,

(33)

h(x)
0,

(34)

0T

λ e = 1, λ  0, 0

(35) u  0, 0

v  0. 0

(36)

Then x is a weak minimum solution for (VFP). Proof. We proceed by contradicting. Then if x is not a weak minimum solution for (VFP), we have that there exists x˜ ∈ X such that fi (x) ˜ fi (x) < gi (x) ˜ gi (x)

for any i ∈ P ,

i.e., fi (x) ˜ < u0i gi (x) ˜

for any i ∈ P .

(37)

By the η-semilocally convexity of f and h at x and the η-semilocally concavity of g at x, we obtain
 ˜ − fi (x)  (dfi )+ x, η(x, ˜ x) , i ∈ P , fi (x)
 ˜ − gi (x)
(dgi )+ x, η(x, ˜ x) , i ∈ P , gi (x)
 ˜ − hj (x)  (dhj )+ x, η(x, ˜ x) , j ∈ M. hj (x) Using these inequalities and (36), we get p 

m

  0 0
   λ0i fi (x) ˜ − fi (x) − λi ui gi (x) ˜ − gi (x) + vj0 hj (x) ˜ − hj (x) p

i=1



i=1 p  i=1

j =1




  λ0i (dfi )+ x, η(x, ˜ x) − u0i (dgi )+ x, η(x, ˜ x)

V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382

+

m 

375


 vj0 (dhj )+ x, η(x, ˜ x)  0,

j =1

where the last inequality is according to (32). Therefore,

  p m 

 0 0 0 λi fi (x) ˜ − ui gi (x) ˜ − fi (x) − ui gi (x) + vj0 hj (x) ˜ − v 0 h(x)  0.    i=1 j =1 =0

Since

u0i

p 

= fi (x)/gi (x), i ∈ P , we obtain


 T T λ0i fi (x) ˜ − u0i gi (x) ˜ + v 0 h(x) ˜ − v 0 h(x)  0.

i=1

Now, x˜ ∈ X, (33) and (36) give p 


 λ0i fi (x) ˜ − u0i gi (x) ˜  0.

i=1 T

Since λ0i  0, λ0 e = 1, we get that there exists i0 ∈ P such that ˜ − u0i0 gi0 (x) ˜  0, fi0 (x) i.e., fi (x) ˜ fi0 (x)  0 gi0 (x) ˜ gi0 (x) which is in contradiction with (37). Hence x is a weak minimum solution for (VFP) and the proof is complete. ✷ Corollary 20. Let x ∈ X and u0i = fi (x)/gi (x). We assume that there exist λ0 ∈ Rp , v 0 ∈ p Rm such that the conditions (32)–(36) of Theorem 19 are satisfied and i=1 λ0i (fi (·) − 0 u0i gi (·)) + m j =1 vj hj (·) is η-semilocally convex at x. Then x is a weak minimum solution for (VFP). Theorem 21. Let x ∈ X, u0i = fi (x)/gi (x), i ∈ P , and λ0 ∈ Rp , v 0 ∈ Rm such that the conditions (32)–(36) of Theorem 19 hold. Moreover, we assume that for any j ∈ M, fi (·) − u0i gi (·) is η-semilocally pseudoconvex and for any j ∈ M, hj (·) is η-semilocally quasiconvex at x. Then x is a weak minimum solution for (VFP). Proof. We suppose that x is not a weak minimum solution for (VFP). Then there exists x˜ ∈ X such that fi (x) ˜ fi (x) < gi (x) ˜ gi (x)

for any i ∈ P ,

i.e., fi (x) ˜ − u0i gi (x) ˜ <0

for any i ∈ P ,

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which is equivalent to fi (x) ˜ − u0i gi (x) ˜ < fi (x) − u0i gi (x)

for any i ∈ P .

Now, by the η-semilocally pseudoconvexity of fi (·) − u0i gi (·) at x we get  

˜ x) − u0i (dgi )+ x, η(x, ˜ x) < 0 for any i ∈ P . (dfi )+ x, η(x, p

Using λ0i ∈ R+ , eT λ0 = 1 we obtain p 

 


λ0i (dfi )+ x, η(x, ˜ x) − u0i (dgi )+ x, η(x, ˜ x) < 0.

(38)

i=1

For x˜ ∈ X we have h(x) ˜
0. But for j ∈ M(x), hj (x) = 0. Hence hj (x) ˜
hj (x) for any j ∈ M(x). Now, by the η-semilocally quasiconvexity of hj at x we obtain
 (dhj )+ x, η(x, ˜ x)
0 for any j ∈ M(x). 0 But v 0 ∈ Rm + and vj = 0 for j ∈ N(x) and then we get m 


 vj0 (dhj )+ x, η(x, ˜ x)
0.

(39)

j =1

Now, by (38) and (39) we obtain p 

m    



λ0i (dfi )+ x, η(x, ˜ x) − u0i (dgi )+ x, η(x, ˜ x) + vj0 (dhj )+ x, η(x, ˜ x) < 0 j =1

i=1

which is a contradiction to (32). Hence x is a weak minimum for (VFP) and the theorem is proved. ✷ Corollary 22. Let x ∈ X, u0i = fi (x)/gi (x), i ∈ P , and λ0 ∈ Rp , v 0 ∈ Rm such that the p 0 conditions (32)–(36) hold. If i=1 λ0i (fi (·) − u0i gi (·)) + m j =1 vj hj (·) is η-semilocally pseudoconvex at x, then x is a weak minimum solution for (VFP).

5. Duality We consider, for (VFP), a general Mond–Weir dual (FMWD) as maximize

ψ(y, λ, u, v) = u − vIT0 hI0 (y)e

subject to p 





 λi (dfi )+ y, η(x, y) − ui (dgi )+ y, η(x, y) + v T (dh)+ y, η(x, y)  0

i=1

for all x ∈ X, fi (y) − ui gi (y)  0 for any i ∈ P ,

(40) (41)

V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382

vITS hIS (y)  0 (1
s
γ ), λ e = 1, T

u  0,

λ  0, u∈R , p

377

(42)

λ∈R , p

v  0,

γ

y ∈ X0 ,

(43)

where γ  1, Is ∩ It = ∅ for s = t and s=0 Is = M. (Here vIs = (vj )j ∈Is , hIs = (hj )j ∈Is .) Let W denote the set of all feasible solutions of (FMWD). Also, we define the following sets:   A = (λ, u, v) ∈ Rp × Rp × Rm | (y, λ, u, v) ∈ W for some y ∈ X0 and, for (λ, u, v) ∈ A,   B(λ, u, v) = y ∈ X0 | (y, λ, u, v) ∈ W .  We put B = (λ,u,v)∈A B(λ, u, v) and note that B ⊂ X0 . Also, we note that if (y, λ, u, v) ∈ W then (λ, u, v) ∈ A and y ∈ B(λ, u, v). Now we establish certain duality results between (VFP) and (FMWD). Assume that f , g and h are η-semidifferentiable on X. Theorem 23 (Weak duality). Assume that for all feasible solutions x ∈ X and (y, λ, u, v) ∈ W for (VFP) and (FMWD), respectively, we have: (i1 ) (i2 ) (i3 ) (i4 )

vITs hIs (y), for 1
s
γ , are slppi on B(λ, u, v); also, if any of the following holds: fi (·) − ui gi (·) + vIT0 hI0 (·) is η-sslppi at y on B(λ, u, v), for all i ∈ P ; p λ > 0 and i=1 λi (fi (·) − ui gi (·)) + vIT0 hI0 (·) is slppi on B(λ, u, v); p T i=1 λi (fi (·) − ui gi (·)) + vI0 hI0 (·) is η-sslppi on B(λ, u, v).

Then the following cannot hold: fi (x) − ui gi (x)
vIT0 hI0 (y)

for any i ∈ P ,

(44)

and fi0 (x) − ui0 gi0 (x) < vIT0 hI0 (y)

for some i0 ∈ P .

(45)

Proof. Using the feasibility of x for (VFP) and of (y, λ, u, v) for (FMWD), we obtain vITs hIs (x)
vITs hIs (y) for all s, 1
s
γ . By (46) and (i1 ) we have 
 vj (dhj )+ y, η(x, y)
0,

1
s
γ.

(46)

(47)

j ∈Is

Now we suppose to the contrary of the result of the theorem that (44) and (45) hold. Hence if (44) and (45) hold for some feasible x for (VFP) and (y, λ, u, v) feasible for (FMWD), we obtain fi (x) − ui gi (x)
vIT0 hI0 (y)

for any i ∈ P ,

(48)

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and fi0 (x) − ui0 gi0 (x)
vIT0 hI0 (y)

for some i0 ∈ P .

(49)

According to (41), (43) and the feasibility of x for (VFP), we have vIT0 hI0 (x)
0
fi (y) − ui gi (y)

for all i ∈ P .

(50)

Combining (48)–(50) we get fi (x) − ui gi (x) + vIT0 hI0 (x)
fi (y) − ui gi (y) + vIT0 hI0 (y) for all i ∈ P ,

(51)

and fi0 (x) − ui0 gi0 (x) + vIT0 hI0 (x)
fi (y) − ui gi (y) + vIT0 hI0 (y) for some i0 ∈ P .

(52)

Now, if (i2 ) holds, then by (51) and (52) we obtain

 
 vj (dhj )+ y, η(x, y) < 0 (dfi )+ y, η(x, y) − ui (dgi )+ y, η(x, y) + j ∈I0

for any i ∈ P .

(53)

By (53) and (43) we get p 




 
 λi (dfi )+ y, η(x, y) − ui (dgi )+ y, η(x, y) + vj (dhj )+ y, η(x, y) < 0. j ∈I0

i=1

Now, by (40) we obtain γ  


 vj (dhj )+ y, η(x, y) > 0

s=1 j ∈Is

which is a contradiction to (47). Thus, in the case (i2 ) the theorem is proved. Now, we suppose (i3 ). Since λi > 0 for any i, if (44) and (45) hold for some feasible x for (VFP) and (y, λ, u, v) feasible for (FMWD), we obtain p 


λi fi (x) − ui gi (x) < vIT0 hI0 (y).

(54)

i=1

Also, we have p 


λi fi (y) − ui gi (y)  0

(55)

i=1

and vIT0 hI0 (x)
0. By (54)–(56) we obtain

(56)

V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382 p 

379



  λi fi (x) − ui gi (x) + vIT0 hI0 (x) < λi fi (y) − ui gi (y) + vIT0 hI0 (y). p

i=1

i=1

Now, by (i3 ) we get p 




 λi (dfi )+ y, η(x, y) − ui (dgi )+ y, η(x, y)

i=1

+




 vj (dhj )+ y, η(x, y) < 0.

(57)

j ∈I0

This inequality and (40) imply γ  


 vj (dhj )+ y, η(x, y) > 0

s=1 j ∈Is

which is a contradiction to (47). Thus, in the case (i3 ) the theorem is proved. Finally, we suppose (i4 ). On the lines from the above case we obtain p 


 
λi fi (x) − ui gi (x) + vIT0 hI0 (x)
λi fi (y) − ui gi (y) + vIT0 hI0 (y) p

i=1

i=1

and then, by (i4 ) we obtain again (57). Now, we proceed as above and we obtain a contradiction. Thus, the theorem is proved. ✷

6. Some examples In this section we give some examples of η-locally starshaped sets and semilocally preinvex functions. Example 24. Let X0 = [−1, 0] ∪ [1/2, 1]. Then X0 is η-locally ˙ starshaped set with respect to η, where η(x, x) = x − 2x, and for aη (x, x) we consider   x if 2x − x < 0 and x ∈ [−1, 0], aη (x, x) = min 1, 2x − x   x −1 aη (x, x) = min 1, if 2x − x < 0 and x ∈ [1/2, 1], 2x − x   x+1 aη (x, x) = min 1, if 2x − x > 0 and x ∈ [−1, 0], 2x − x   2x − 1 if 2x − x > 0 and x ∈ [1/2, 1], aη (x, x) = min 1, 2(2x − x) aη (x, x) = 1

if x = 2x.

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Example 25. The set X0 = R\(−1/2, 1/2) is η-locally starshaped with respect to η, where  x − x if x > 0, x > 0 or x < 0, x < 0, η(x, x) = x − x if x > 0, x < 0 or x < 0, x > 0, and aη (x, x) = 1. Example 26. Using [30], we note that the closed interior of an astroid   X0 = (x, y) ∈ R2 | x 2/3 + y 2/3  a 2/3 , with a > 0, is an η-starshaped set, where
 η(x, y; x, y) = η1 (x, y; x, y), η2 (x, y; x, y) with

 −x −y , , 2 2   −x −y η2 (x, y; x, y) = , , 2(1 + x 2 + y 2 ) 2(1 + x 2 + y 2 ) 

η1 (x, y; x, y) =

and

 aη (x, y; x, y) = min 1,

 2 2 2 , 1 + x + y . 1 + x2 + x2

Example 27. Let X = [0, 1). Then Y = [0, 1] is an η-locally starshaped set at y = 1/2 with respect to X, where η(x, y) = x − 2y and   1 a(x, y) = min 1, . 2(1 − x) The following examples present some semilocally preinvex type functions. Example 28. Let f : X0 = R\(−1/2, 1/2) → R be defined by  x + 1 if x < 1, f (x) = 0 if x ∈ [−1, 1], 1 − x if x > 1. Then f is slpi and slqpi with respect to η, where η is defined by Example 25. Example 29. The function f : X0 = [−1, 0] ∪ [1/2, 1] → R defined by f (x) = −x 2 is slppi at x = 1, with respect to η(x, x) = x − 2x. Also, we remark that f is a nonconvex function, with (df )+ (x, η(x, x)) = 2 x (2x − x). Remark. Some remarkable results concerning invex type sets can be found in Antczak [1], and properties of invex and preinvex type functions in Li and Dong [25], Yang and Li [42, 43], and Yang et al. [44].

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381

Acknowledgment The author is very thankful to the referees for their comments and suggestions which contributed to the improvement of this paper.

References [1] T. Antczak, (p, r)-invex sets and functions, J. Math. Anal. Appl. 263 (2001) 355–379. [2] C.R. Bector, M.K. Bector, J.E. Klassen, Duality for a nonlinear programming problem, Utilitas Math. 11 (1977) 87–89. [3] A. Ben-Israel, B. Mond, What is invexity?, J. Austral. Math. Soc. Ser. B 28 (1986) 1–9. [4] G. Bitran, Duality in nonlinear multiple criteria optimization problems, J. Optim. Theory Appl. 35 (1981) 509–515. [5] A. Cambini, L. Martein, An approach to optimality conditions in vector and scalar optimization, in: Diewert, Spremann, Stehling (Eds.), Mathematical Modelling in Economics, Springer-Verlag, 1993. [6] A. Cambini, L. Martein, Generalized concavity and optimality conditions in vector and scalar optimization, in: Komlosi, et al. (Eds.), Generalized Convexity, in: Lecture Notes in Economics and Mathematical Systems, Vol. 405, Springer-Verlag, 1994. [7] B.D. Corley, Duality theory for maximization with respect to cones, J. Math. Anal. Appl. 84 (1981) 560–568. [8] B.D. Craven, Strong vector minimization and duality, Z. Angew. Math. Mech. 60 (1980) 1–5. [9] B.D. Craven, On quasidifferentiable optimization, J. Austral. Math. Soc. Ser. A 41 (1986) 64–78. [10] K.H. Elster, R. Nehse, Optimality Conditions for Some Nonconvex Problems, Springer-Verlag, New York, 1980. [11] C.M. Ewing, Sufficient conditions for global minima of suitable convex functions from variational and control theory, SIAM Rev. 19 (1977) 202–220. [12] A.M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl. 22 (1968) 618–630. [13] M.A. Hanson, B. Mond, Convex transformable programming problems and invexity, J. Inform. Optim. Sci. 8 (1987) 179–189. [14] M. Hayashi, H. Komiya, Perfect duality for convexlike programs, J. Optim. Theory Appl. 38 (1982) 179– 189. [15] G. Heal, Equivalence of saddle-points and optima for non-concave programs, Adv. Appl. Math. 5 (1984) 398–415. [16] E.H. Ivanov, R. Nehse, Some results on dual vector optimization problems, Math. Oper. Statist. Ser. Optim. 16 (1985) 505–517. [17] V. Jeyakumar, Convexlike alternative theorems and mathematical programming, Optimization 16 (1985) 505–517. [18] V. Jeyakumar, Equivalence of saddle-points and optima for and duality for a class of non-smooth non-convex problems, J. Math. Anal. Appl. 130 (1988) 334–343. [19] V. Jeyakumar, Composite nonsmooth programming with Gateaux differentiability, SIAM J. Optim. 1 (1991) 30–41. [20] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, J. Austral. Math. Soc. Ser. B 34 (1992) 43–53. [21] V. Jeyakumar, X.Q. Yang, Convex composite multi-objective nonsmooth programming, Math. Programming 59 (1993) 325–343. [22] R.N. Kaul, S. Kaur, Generalizations of convex and related functions, European J. Oper. Res. 9 (1982) 369– 377. [23] R.N. Kaul, S. Kaur, Sufficient optimality conditions using generalized convex functions, Opsearch 19 (1982) 212–224. [24] R.N. Kaul, S. Kaur, Generalized convex functions—properties, optimality and duality, Technical report SOL, 84-4, Stanford University, California, 1984.

382

V. Preda / J. Math. Anal. Appl. 288 (2003) 365–382

[25] X.F. Li, J.L. Dong, Subvexormal functions and subvex functions, J. Optim. Theory Appl. 103 (1999) 675– 704. [26] V. Lyall, S. Suneja, S. Aggarwal, Optimality and duality in fractional programming involving semilocally convex and related functions, Optimization 41 (1997) 237–255. [27] D.G. Mahajanm, M.N. Vartak, Generalizations of some duality theorems in nonlinear programming, Math. Programming 12 (1977) 293–317. [28] O.L. Mangasarian, Nonlinear Programming, McGraw–Hill, New York, 1969. [29] B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 263–280. [30] S. Mititelu, Invex sets, Math. Rep. 46 (1994) 529–532. [31] V. Preda, Optimality conditions and duality in multiple objective programming involving semilocally convex and related functions, Optimization 36 (1996) 219–230. [32] V. Preda, I.M. Stancu-Minasian, A. Batatorescu, Optimality and duality in nonlinear programming involving semilocally preinvex and related functions, J. Inform. Optim. Sci. 17 (1996). [33] V. Preda, I.M. Stancu-Minasian, Duality in multiple objective programming involving semilocally preinvex and related functions, Glas. Mat. 32 (1997) 153–165. [34] C. Singh, Duality theory in multiobjective differentiable programming, J. Inform. Optim. Sci. 9 (1988) 231– 240. [35] S.K. Suneja, S. Gupta, Duality in nonlinear programming involving semilocally convex and related functions, Optimization 28 (1993) 17–29. [36] T. Tanino, Y. Sawaragi, Duality in multiobjective programming, J. Optim. Theory Appl. 27 (1979) 509–529. [37] T. Weir, Proper efficiency and duality for vector valued optimization problems, J. Austral. Math. Soc. Ser. A 43 (1987) 21–34. [38] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136 (1988) 29–38. [39] T. Weir, B. Mond, Generalized convexity and duality in multiple objective programming, Bull. Austral. Math. Soc. 39 (1989) 287–299. [40] D.G. White, Vector maximization and Lagrange multipliers, Math. Programming 31 (1985) 192–205. [41] P. Wolfe, Duality theorems for nonlinear programming, Quart. Appl. Math. 19 (1961) 239–244. [42] X.M. Yang, D. Li, On properties of preinvex functions, J. Math. Anal. Appl. 256 (2000) 229–241. [43] X.M. Yang, D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258 (2001) 287–308. [44] X.M. Yang, X.Q. Yang, K.L. Teo, Characterizations and applications of prequasi-invex functions, J. Optim. Theory Appl. 110 (2001) 645–668.