American Journal of Oper ations Research, 2011, 1, 243248 doi:10.4236/ajor.2011.14028 Published Online December 2011 (http://www.SciRP.org/journal/ajor) Copyright © 2011 SciRes. AJOR 243 Optimality for MultiObjective Programming Inv olvi ng Arcwise Connected dTypeI Functions* Guolin Yu, Min Wang Research Institute of Information and System Computation Science, The North University fo r Ethn ics, Yinchuan, China Email: guolin_yu@126.com Received April 26, 2011; revised May 27, 2011; accepted June 11, 2011 Abstract This paper deals with the optimality conditions and dual theory of multiobjective programming problems involving generalized convexity. New classes of generalized typeI functions are introduced for arcwise connected functions, and examples are given to show the existence of these functions. By utilizing the new concepts, several sufficient optimality conditions and MondWeir type duality results are proposed for nondifferentiable multiobjective programming problem. Keywords: MultiObjective Programming, Pareto Efficient Solution, Arcwise Connected dTypeI Functions, Optimality Conditions, Duality 1. Introduction Investigation on sufficiency and duality has been one of the most attraction topics in the theory of multiobjectiv e problems. It is well known that the concept of convexity and its various generalizations play an important role in deriving sufficient optimality conditions and duality re sults for multiobjective programming problems. The concept of typeI functions was first introduced by Han son and Mond [1] as a generalization of co nvexity. With and without differentiability, the typeI functions were extended to several classes of generalized typeI func tions by many researchers, and sufficient optimality cri teria and duality results are established for multiobjec tive programming (vector optimization) problems in volving these functions (see [112]). Another meaningful generalization of convex functions is the introduction of arcwise connected functions, which was given by Avriel and Zang [13]. Singh [14] and Mukherjee and Yadav [15] discussed some properties of arcwise connected sets and functions. Bhatia and Mehara [16] investigated some properties of arcwise connected functions in terms of their directional derivatives and established optimality conditions for scalarvalued nonlinear programming prblems involving these functions. Mehar and Bhatia [17] and Davar and Mehra [18] studied optimality conditions and duality results for minmax problems and fractional programming problems involving arcwise connected and generalized arcwise connected functions, respectively. In this paper, we first introduce new classes of gener alized convex typeI functions by relaxing definitions of arcwise connected function and typeI function. We pre sent some sufficient optimality cond itions and dual theo rems for nondifferential multiobjective programming problem under various generalized convex typeI func tions assumptions. This paper is divided into four sec tions. Section 2 recalls some definitions and related re sults which will be used in later sections, and introduces new classes generalized convex typeI functions. In Sec tion 3 and Section 4, the sufficient optimality conditions and MondWeir type duality results are established for nondifferential multiobjective programming problem involving these generalized convex functions, respec tively. 2. Preliminaries In this section, we first recall some concepts and results related arcwise connected functions. Let be the n dimensional Euclidean space and be the set of all real numbers. Throughout this paper, the following con vention for vectors in will be followed: n R 1 R n R *This research is supported by Zizhu Science Foundation of Beifang University of Nationalities; Natural Science Foundation for the Youth (No. 10901004). y if and only if ii y, , 1, 2,,in
G. L. YU ET AL. 244 y if and only if ii y ，, 1, 2,,in y if and only if ii y1, , , but 2, ,in y , y≮ is the negation of y n . Definition 2.1. (See [15]) A subset R is said to be an arcwise connected (AC) set, if for every X , , there exists a continuous vectorvalued functions uX ,xu :0,1 X ,xu , called an arc, such that 0 x, 1 ,xu u. Definition 2.2. (See [15]) Let be a realvalued function defined on an AC set n R. Then is said to be an arcwise connected function (CN) if, for every X, , there exists an arc uX, u such that )1 ,xu Hfxfu , for 01 Definition 2.3. (See [13,14]) Let n R be an AC set, and Let f be a realvalued function defined on . For any X X , , , the directional derivative of f with respect to uX u at 0 is defined as , 0 mli xu Hf x , provided the limit exists and is denoted by . If ,0 xu fH ,xu 0 mli x ,0 xu H exists and it is denoted by , then vector is called directional derivative of 0 ,xu H, u at 0 . Consider the following multiobjective programming problem: 0, fx(MP) min s.t. , xxX where :m XR , :p XR, X is a nonempty open AC set of . Let F denote the feasible solutions of (MP) assumed to be nonempty, that is n R ngx :0xRF. Definition 2.4. A point X is said to be a Pareto efficient solution of problem (MP), if xfx for all X. Definition 2.5. A point X is said to be a weak Pareto efficient solution of problem (MP), if xfx for all X. Along the lines of [1,5], we now define the following classes of functions. Definition 2. 6. , ij g, 1, 2,, im and 1,2, ,jp, is said to be CNdtypeI with respect to *, x H, at * X if there exist an arc *,:0,1 xx X such that for all X, * * ,0 ii i xx fxfxf H and * * ,0 jj xx gx gH Definition 2. 7. , ij g, and 1,2, ,im 1, 2,,jp , is said to be quasiCNdtypeI with re spect to *, x H at * X if there exist an arc *,:0,1 xx X such that for all X, * * ,00 ii i xx fxfxfH , and * * , 00 jj xx gx gH 0 . Definition 2. 8. , ij g, and 1,2, ,im pj ,,2,1 , is said to be pseudoCNdtypeI with re spect to at if there exist an arc xx H, *Xx * *,:0,1 xx X such that for all , Xx ** ,00 ii xx i Hfx fx , and ** ,00 jj xx gH gx 0. Definition 2. 9. ,fg ij , and 1, 2,,im pj ,,2,1 , is said to be quasipseudoCNdtypeI with respect to at if there exist an arc xx H, *Xx * *,:0,1 xx X such that for all , Xx * * ,00 ii i xx fxfxfH , and ** ,00 jj xx gH gx 0. Definition 2.10. ,fg ij , and 1, 2,,im 1, 2,,jp , is said to be pseudoquasiCNdtypeI with respect to at xx H, * * X if there exist an arc *,:0,1 xx X such that for all , Xx ** ,00 ii xx fHfx fx i , and * * , 00 jj xx gx gH 0 . To show the existence of the CNdtypeI functions Copyright © 2011 SciRes. AJOR
245 G. L. YU ET AL. we give the following exam ple: Example 2.1. Define a set 2 R as 22 12 1212 ,:1, 0,0Xxxxxx x . Then x is an AC set with respect to ,:0,1 xy X given by 12 12 22 22 ,112 1,1 xy Hxyx 2 y 12 , xx X, 12 ,0yyy X ,1. Define , :fX R: XR as 22 12 12 ,if 1 and 1 0, otherwise, xx xx fx 2 21 2 ,if 1 and 1 0, otherwise, xx x gx * f and g are not differentiable at 1,1 X . How ever, and *,0 xx fH *,0 xx gH existed, and they are given by * * , 22 12 , 0 ,if both components of 1 0, otherwise, xx xx fH xx H * * 2 2, , ,if both components of 00, otherwise, xx xx x gH H 1 where 12 , xx. It is obviously that , g is CNd typeI at . *1,1x 3. Sufficient Optimality Conditions In this section, we establish sufficient optimality condi tions for a feasible solution to be a weak Pareto effi cient solution for (MP) under the CNdtypeI and pseu doquasiCNdtype I assumptions. Theorem 3.1. Suppose that there exist a feasible solu tion xF and 1,, , 0 such that ,00 T xx fHg x X (3.1) 0 jj gx , 1, 2,,jp (3.2) If , ij g is CNdtypeI with respect to , x at , then is a weak Pareto efficient solution for (MP). Proof Suppose that is not a weak Pareto efficient solution of (MP). Then th ere is a feasible solution of (MP) such that () () ii xfx for any . 1, 2,,im By CNdtypeI assumption on, we get i f , 00 ii ixx fxfxf H for any . 1, 2,, im Thus, ,0 ixx fH 0 for any . 1,2,,im So, we have , 1 0 m ixx i fH 0. (3.3) It yields from (3.1) that , 10 p jj xx j gH 0. (3.4) from CNdtypeI assumption on , we get ,0 jjxx gx gH , for any . 1, 2,,jp Since 0 and Equation (3.2) holds, we can get , 00 jjjj xx gx gH , for any . 1, 2,,jp Therefore, , 100 p jj xx j gH , which contradicts to (3.4). Theorem 3.2. Suppose that there exist a feasible solu tion x F and 1,, , 0 such that (3.1) and (3.2) hold. If , ijj g is pseudoquasiCNdtype I with respect to , x at , then is a weak Pareto efficient solution for (MP). Proof Since (3.2) holds and xF, by pseudoquasi CNdtypeI hypothesis on j at , for all X we have *,00 jj xx gH . Thus *, 10 p jj xx j gH X. (3.5) 0 for all Let not be a weak Pareto efficient solution for (MP). Then there is a feasible solution ˆ for (MP) such that ˆ ii xfx for any . 1, 2,,im from pseudoquasiCNdtypeI hypothesis on i at , it yields ˆ ,0 ixx fH 0 for any . 1,2,,im so, ˆ , 1 0 m ixx i fH 0. (3.6) Copyright © 2011 SciRes. AJOR
G. L. YU ET AL. 246 Combing (3.5) and (3.6), we get ˆˆ ,, 11 00 p m ixx jjxx ij fH gH 0. But this is a contradiction to (3.1). The proof is com pleted. Remark 3.1. For the functions () x and () x in Example 2.1, we consider the programming problem (MP). Let 1 , we are easy to get that ˆˆ ,, 000 T xx xx HgH x X , which implies that 1,1x is an optimal solution (is also weak Pareto efficient) of (MP). 4. Duality Results Now, in relation to (MP) we consider the following Mond and Weir type dual under the CNdtypeI and generalized CNdtypeI assumptions. 12 ,, (DMP) max,,, s.t. 000 for all m TT yx yx fyfy f yfy fH gHy F (4.1) 10 p jj j gy , (4.2) 0 , , (4.3) 12 ,,, T m 0 , . (4.4) 12 ,,, T p Theorem 4.1. (Weak Duality) Let and ,,y be feasible soltuions for (MP) and (DMP), respectively. Moreover, we assume that any one of the following con ditions holds: a) , ij g is CNdtypeI with respect to , x at ; yb) , ijj g , is pseudoquasiCNdtypeI with re spect to x at y. Then xfy. (4.5) Proof Since ,,y is feasible solution for (DMP), we have ,, 000 TT yx yx fH gHxX 0 (4.6) and (4.2) holds. We proceed by contradiction. Suppose that xfy. Then, there exists an index such that k kk xfy, ii xfy for all i. k Since 0 i , `1,2 ,,im , we get kk kk xf y, ii ii xf y for all ik . Thus, 11 mm ii ii ii xf y. (4.7) by condition (a), we get ,0 iiiyx fyfxf H and ,0 jjyx gy gH 0 . Therefore, we can get , 11 10 mm m iiiiiiyx ii i fxfyf H (4.8) and , 11 0 pp jjjjyx jj gy gH . (4.9) Combing (4.2), (4.7) , (4.8) and (4.9 ), we get , 100 m ii yx i fH , and , 100 p jj yx j gH . so ,, 11 00 p m i iyxijyx ij fH gH (4.10) which contradicts (4.6). By condition (b), noticing that (4.2) holds, with the similar argument as that of Theorem 3.2, we can get , 1 00 m ii yx i fH , and , 100 p jj yx j gH . The above two inequalities imply (4.10), again a con tradiction to (4.6). This completes the proof. Theorem 4.2. Suppose that there exist feasible solu tions and ,,y for (MP) and (DMP), respec tively, such that Copyright © 2011 SciRes. AJOR
247 G. L. YU ET AL. ii xfy, . (4.11) 1, 2,,im Moreover, we assume that the hypotheses of Theorem 4.1 hold at , then is a Pareto efficient solution for (MP). Proof For any feasible solution for (MP), we get from Theorem 4.1 that xfy. Suppose that is not a Pareto efficient solution for (MP). Then, there exist a feasible solution ˆ for (MP) and an index k such that ˆ kk xfx, ˆ kk xfx for all ik . Using condition (4.11), we get ˆ kk xfy, ˆ kk xfy for all ik . This contradicts to Theorem 4.1. Theorem 4.3. (Converse Duality) Let ,,y y be a Pareto efficient solution for (DMP). Moreover, we as sume that the hypotheses of Theorem 4.1 hold at, then is a Pareto efficient solution for (MP). yProof We proceed by contradiction. Suppose that is not a Pareto efficient solution for (MP), that is, there exist and an index k such that y xF kk xfy, ii xfy for all ik . If any one of the hypotheses of Theorem 4.1 holds, it yields in light of Theorem 4.1 that (4.5) is satisfied. This leads to the similar contradiction as in the proof of Theorem 4.1. 5. References [1] M. A. Hanson and B. Mond, “Necessary and Sufficient Conditions in Constraint Optimization,” Mathematical Programming, Vol. 37, No. 1, 1987, pp. 5158. doi:10.1007/BF02591683 [2] B. Aghezzaf and M. Hachimi, “Generalized Invexity and Duality in Multiobjective Programming Problems,” Jour nal of Global Optimization, Vol. 18, No. 1, 2000, pp. 91101. doi:10.1023/A:1008321026317 [3] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Differentiable Multiobjective Programming Involving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 296, No. 2, 2004, pp. 382392. doi:10.1016/j.jmaa.2003.12.042 [4] M. Hachimi and B. Aghezzaf, “Sufficiency and Duality in Nondifferentiable Multiobjective Programming In volving Generalized Type I Functions,” Journal of Mathematical Analysis and Applications, Vol. 319, No. 1, 2006, pp. 110123. doi:10.1016/j.jmaa.2005.02.064 [5] R. N. Kaul, S. K. Suneja and M. K. Srivastava, “Optimal ity Criteria and Duality in Multiple Objective Optimiza tion Involving Generalized Invexity,” Journal of Optimi zation Theory and Applications, Vol. 80, No. 3, 1994, pp. 465482. doi:10.1007/BF02207775 [6] S. K. Mishra, G. Giorgi and S. Y. Wang, “Duality in Vec tor Optimization in Banach Spaces with Generalized Convexity,” Journal of Global Optimization, Vol. 29, No. 4, 2004, pp. 415424. doi:10.1023/B:JOGO.0000047911.03061.88 [7] S. K. Mishra and M. A. Noor, “Some Nondifferentiable MultiObjective Programming Problems,” Journal of Mathematic Analysis and Applications, Vol. 316, No. 2, 2006, pp. 472482. doi:10.1016/j.jmaa.2005.04.067 [8] S. K. Mishra, S. Y. Wang and K. K. Lai, “Multiple Ob jective Fractional Programming Involving Semilocally Type IPreinvex and Related Functions,” Journal of Mathematic Analysis and Applications, Vol. 310, No. 2, 2005, pp. 626640. [9] A. Mehra and D. Bhatia, “Optimality and Duality for Minmax Problems Involving Arcwise Connected and Generalized Arcwise Connected Functions,” Journal of Mathematical Analysis and Applications, Vol. 231, No. 2, 1999, pp. 425445. doi:10.1006/jmaa.1998.6231 [10] N. G. Rueda, M. A. Hanson and C. Singh, “Optimality and Duality with Generalized Convexity,” Journal of Op timization Theory and Applications, Vol. 86, No. 2, 1995, pp. 491500. doi:10.1007/BF02192091 [11] 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. 1112, 2007, pp. 14031410. doi:10.1016/j.camwa.2007.05.002 [12] G. L. Yu and S. Y. Liu, “Optimality for (h, )Multiob jective Programming Involving Generalized TypeI Func tions,” Journal of Global Optimization, Vol. 41, No. 1, 2008, pp. 147161. doi:10.1007/s1089800791963 [13] M. Avriel and I. Zang, “Generalized ArcwiseConnected Functions and Characterizations of LocalGlobal Mini mum Properties,” Journal of Optimization Theory and Applications, Vol. 32, No. 4, 1980, pp. 407425. doi:10.1007/BF00934030 [14] C. Singh, “Elementary Properties of Arcwise Connected Set and Functions,” Journal of Optimization Theory and Applications, Vol. 41, 1990, pp. 85103. [15] R. N. Mukherjee and S. R. Yadav, “A Note on Arcwise Connected Sets and Functions,” Bulletin of the Australian Mathematical Society, Vol. 31, No. 3, 1985, pp. 369375. doi:10.1017/S0004972700009333 [16] D. Bhatia and A. Mehra, “Optimality and Duality In volving Arcwise Connected Generalized Connected Fun ctions,” Journal of Optimization Theory and Applications, Vol. 100, No. 1, 1999, pp. 181194. doi:10.1023/A:1021725200423 [17] N. G. Rueda and M. A. Hanson, “Optimality Criteria in Mathematical Programming Involving Generalized In vexity,” Journal of Mathematical Analysis and Applica tions, Vol. 130, No. 2, 1998, pp. 375385. doi:10.1016/0022247X(88)903137 Copyright © 2011 SciRes. AJOR
G. L. YU ET AL. Copyright © 2011 SciRes. AJOR 248 [18] S. Davar and A. Mehra, “Optimality and Duality for Fractional Programming Problems Involving Arcwise Connected Functions and Their Applications,” Journal of Mathematical Analysis and Applications, Vol. 263, No. 2, 2001, pp. 666682. doi:10.1006/jmaa.2001.7651
