G. L. YU ET AL.

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925

[3] T. Weir and V. Jeyakumar, “A Class of Nonconvex Func-

tions and Mathematical Programming,” Bulletin of Aus-

tralian Mathematical Society, Vol. 38, No. 2, 1988, pp.

177-189. doi:10.1017/S0004972700027441

2) Assume that is generalized invex with respect to S

and

and

is generalized -preinvex on

with respect to

C S

and

. If (3.3) holds, then

is a

strong efficient solution of .

VOP [4] T. Weir and B. Mond, “Preinvex Functions in Multiple-

Objective Optimization,” Journal of Mathematical Analy-

sis and Applications, Vol. 136, No. 1, 1988, pp. 29-38.

doi:10.1016/0022-247X(88)90113-8

Proof: (1) We proceed by contradiction. Assuming

that the condition (3.3) is not satisfied, then there exists

ˆ

S such that

ˆ

;,

f

Yx xxC

. [5] L. Batista Dos Santos, R. Osuna-ómez, M. A.Rojas-Me-

dar and A. Rufián-Lizana, “Preinvex Functions and Weak

Efficient Solutions for Some Vectorial Optimization

Problem in Banach Spaces,” Computers and Mathematics

with Applications, Vol. 48, No. 5-6, 2004, pp. 885-895.

doi:10.1016/j.camwa.2003.05.013

Thus, there exists small enough such that

ˆ0t

ˆ

ˆˆ

txxS

.

and [6] A. J. V. Brandāo, M. A. Rojas-Medar and G. N. Silva,

“Optimality Conditions for Pareto Nonsmooth Noncon-

vex Programming in Banach Spaces,” Journal of Opti-

mization Theory and Applications, Vol. 103, No. 1, 1999,

pp. 65-73. doi:10.1023/A:1021769232224

ˆ

ˆˆ

xt xxfxC

,

which contradicts to that

S

is a strong efficient

solution of . It is easy to obtain inequality (3.4)

from the definition of lower Dini directional derivative

for

VOP

.

[7] S. K. Mishra, G. Giorgi and S. Y. Wang, “Duality in

Vector Optimization in Banach Spaces with Generalized

Convexity,” Journal of Global Optimization, Vol. 29, pp.

2004, pp. 415-424. doi:10.1016/j.camwa.2007.05.002

2) Suppose that the condition (3.3) is satisfied, that is

;, ,

f

Yx xxCxS

. [8] G. L. Yu and S. Y. Liu, “Some Vector Optimization

Problems in Banach Spaces with Generalized Convex-

ity,” Computers and Mathematics with Applications, Vol.

54, No. 11-12, 2007, pp. 1403-1410.

doi:10.1016/j.camwa.2007.05.002

By Proposition 2.1,

;, ,

f

xfxkYxxx CCxS

Hence [9] J. H. Qiu, “Cone-Directed Contingent Derivatives and

Generalized Preinvex Set-Valued Optimization,” Acta

Mathematica Scientia, Vol. 27, No. 1, 2007, pp. 211-218.

doi:10.1016/S0252-9602(07)60019-8

,

xfxCxS.

That is,

is a strong efficient solution of problem

.

)(VOP [10] M. I. Henig, “Proper Efficiency with Respect to Cones,”

Journal of Optimization T h eory and Application s, Vol. 36,

No. 3, 1982, pp. 387-407. doi:10.1007/BF00934353

4. References [11] J. M. Borwein and D. Zhuang, “Super Efficiency in Vec-

tor Optimization,” Transactions of the American Mathe-

matical Society, Vol. 338, No. 1, 1993, pp. 105-122.

doi:10.2307/2154446

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