American Journal of Operations Research, 2013, 3, 413420 http://dx.doi.org/10.4236/ajor.2013.34039 Published Online July 2013 (http://www.scirp.org/journal/ajor) Optimality in Multivalued Optimization Surjeet Kaur Suneja, Megha Sharma Department of Mathematics, University of Delhi, Delhi, India Email: surjeetsuneja@gmail.com Received March 2, 2013; revised April 3, 2013; accepted April 11, 2013 Copy ri ght © 2 013 Surjeet Kaur Suneja, Megha Sharm. This is an open access article distributed under the Creative Commons Attribu tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we apply the directional derivative technique to characterize Dmultifunction, quasi Dmultifunction and use them to obtain εoptimality for set valued vector optimization problem with multivalued maps. We introduce the notions of local and partialεminimum (weak) point and study εoptimality, εLagrangian multiplier theorem and εduality results. Keywords: DMultifunction; PartialεMinimum Point; εOptimality; εDuality 1. Introduction The theory of efficiency plays an important role in vari ous knowledge fields. It is proposed as a new frontier in mathematical physics and engineering in context of pri orities concerning the alternative energies, the climate exchange and education. Pareto efficiency or Pareto op timality is a central theory in economics with broad ap plications in game theory, social sciences, management sciences, various industries etc. In set valued vector op timization problems, it is important to know when the set of efficient points is nonempty to establish its main properties (existence, connectedness and compactness) and to extend the concepts to set valued vector optimiza tion in infinite dimensional ordered vector spaces. The notion of proper efficiency was first introduced by Kuhn and Tucker [1] in their well known paper on nonlinear programming and many other notions have been pro posed since then. Some of the well known notions are Geoffrion proper efficiency [2], Borwein proper effi ciency [3], Benson proper efficiency [4] and super effi ciency [5]. Chinaie and Zafarani [6] introduced the con cepts of feeble multifunction minimum (weak) point, multifunction minimum (weak) point and obtained optimality conditions for set valued vector optimiza tion problem having multivalued objective and con straints. While it is theoretically possible to identify the com plete set of solutions, finding an exact descrip tion of this set often turns out to be practically impossible or com putationally too expensive. In practical situations we often stop the calculations at values that are sufficiently close to the optimal solutions, that is, we use algorithms that find approximate of the Pareto optimal set. Stability aspect in set valued vector optimization deals with the study of behaviour of the solution set under perturbations of the data. One of the approaches in this regard is the convergence of sequence of εsolutions to a solution of the original problem. These facts justify the need of study of approximate efficiency which is equivalent to εoptimality for set valued vector optimization problems. Some of the researchers who contributed in this area are Hamel [7], Rong and Wu [8]. Chinaie and Zafarani [9] introduced the concepts of εfeeble multifunction minimum (weak) point and ob tained optimality conditions for set valued vector opti mization problem having multivalued objective and con str aints. In this paper, we have given the notio n s o f (l o ca l ) partialεminimum point and (local) partialεweak minimum point, for set valued vector optimization prob lem and used them to study εoptimality, εLagrangian multiplier theorem and εduality results. This paper is organized as follows: In Section 2 we have given the preliminaries and results related to quasi Dmultifunction. In Section 3 we apply the directional derivative technique used by Yang [10] to characterize εoptimality conditions for set valued vector optimization problem in terms of εfeeble multifunction minimum point given by Chinaie and Zafarani [9]. In Section 4, we introduce (local) partialεminimum point and (local) partialεweak minimum point, and show that it is dif ferent from εfeeble mutifunction minimum point. Also, we prove that every local partialεminimum (weak) point is a partialεminimum (weak) point if the objective C opyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM 414 function of set valued vector optimization problem is strict quasi Dmultifunction and constraint function is quasi Dmultifunction and show that this result is not true in the case of local εfeeble multifunction minimum point. In Section 5, we obtain εLagrangian multiplier theorem in terms of partialεweak minimum point and in Section 6, we establish εweak duality and εstrong duality for dual problem of set valued vector optimization prob lem. 2. Preliminaries and Definitions Let X be locally convex topological vector space, Y, Z be real locally convex Hausdorff topological vector spaces; let be pointed closed convex cones with ,DYEZ andint.DEint for all yD D i D 0 .yD\ Let Y be the dual space of Y, the positive dual cone D+ of D is given by : 0,DfYfy . The set of strictly positive functions in is denoted by , that is :0, for all i DfYfy For a set Y, we write cone: 0, aaA ti DD int D . If D is convex cone in Y, then in and equality holds if [1]. A partial order 12D yy in Y is defined by iff, , for all . 21 yyD12 Through out this paper, we denote and ,yy:int o DD Y :0. o DD : Let UY : , be a multifunction defined on a non empty subset U of X with values in Y, which is partially ordered by cone D. Now, for a multifunction UY, denote by domF and imF the domain and the image of F, respectively. In other words : ,XFxdomFx im . xX FXFx , The set gr :,:dom xX xy x xF FyF x x : is called the graph of F. Definition 2.1: [10,11] Let U be convex subset of X. Let UY be a multifunction: 1) F is said to be a Dmultifunction on U if, for all 12 , xU and 0,1 ,t we have 1212 11;tF xtF xFtxtxD 12 , 2) F is said to be a quasi Dmultifunction on U if, for all xU 0,1 ,t we have: and 1212 1;Fx DFxD FtxtxD 12 ,, 3) F is said to be a strictly quasi Dmultifunction on U iff, for all xU 12 and0,1 ,xx twe have: 1212 1. o Fx DFxD FtxtxD : Yang [10] gave the following definitions: Definition 2.2: A function XY is said to be a continuous selection of F if f is continuous and , xFx. for all X Denote by CS (F) the set of all continuous selections of F. Definition 2.3: Let 00 ,::the re e xists 0,,0SxVvXxtvVt 0 be the cone of feasible directions. Then the limit set of F at in the direction , 00 00 0 0, ,is ,: :lim, ,, o nu n y F nn n tv n vSxFY xv fx tuy zzu SxV t CS F with where 00 xy 0 . in all directions The union of all limit sets of F at 0,vSxV is denoted by 00 ,,. o y F YxSxV , We need the following assumption: Assumpti on 2.1: Let yX. If 1,zFtx ty for all 0,1 ,,tyFx then there exists a continu ous selection CS F such that 1,zftx ty for all 0, 1t : . Theorem 2.1: Let U be convex subset of X and UY . If assumption 2.1 holds and F is Dmulti function, then for any , and , xUy Fx ,. y F xy YxxxD Proof: Since F is Dmultifunction therefore, for 0,1 ,,,txxU we have 11;tFxt FxF txtxD which gives that 11tFxty FtxtxD . ,wFx by assumption 2.1, there exist If Copyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM 415 CS F such that .t xD 11twtyftx That is, 1ftx t wy t xy D ,.xxxD Thus, y F wy Y Hence, –, . y F xyY xxxD : Theorem 2.2: Let U be convex subset of X, UY be quasi Dmultifunction on U and assump tion 2.1 hold then, for any x D,, ,xxU yFxyF ,.xx D implies that, y F Yx . Proof: For ,, , xUy Fx let yFx D That is, yF xD and .yFx D Since F is quasi Dmultifunction, therefore for 0,1 ,t 1, xDFxD Ftx tx D which implies that 1.tx D yFtx Then, by assumption 2.1 there is CS F such that 1 ,yftxtx D for all 0,1,t which gives that r all 0,1 ftxtxyDt t ,.xx D 1,fo . That is, y F Yx n xU DF Gx E x 3. εOptimality in Terms of Directional Derivatives In this section, we obtain εoptimality conditions for set valued vector optimization problem in terms of direc tional derivatives given by Yang [10] for εfeeble multi function minimum point given by Chinaie and Zafarani [9]. We consider the following set valued vector optimi zation proble m : VP mi s.t. UX: where is non empty set, UY:GU Z , , are multifunctions with nonempty values. The set of fea sible solutions of (VP) is denoted by V, that is :VxUGx E . Chinaie and Zafarani [9] gave the following defini tions. Definition 3.1: Let ,o VD. 1) is called a εfeeble multifunction minimum point (εf. m. m. p.) of problem (VP), if there exists, Fx, such that (3.1) ;FV xyD 2) is called a εfeeble multifunction weak mini mum point (εf. m. w. m. p.) of problem (VP), if there exists, , Fx such that –; o FV xyD (3.2) The set of V which satisfies (3.1) or (3.2) is de noted by ˆ,SFD ˆ,WSF D and respectively. When Vx is replaced by NxVx in (3.1) and (3.2), Nx being neighbourhood of , then we have local εf. m. m. p. and local εf. m. w. m. p. of problem (VP). We now give the necessary optimal conditions for lo cal εfeeble we ak minimum point [9] of (VP). Theorem 3.1: Let 0 V and 00 Fx be local εfeeble multifunction weak minimum point of problem (VP). Then, 00 ,,, . o yo o YxSxV DD Proof Suppose 0 V is local εfeeble multifunction weak minimum point of problem (VP) and f is any con tinuous selection of F such that 00 fx. Then, 000, o FVxN xyD 0 Nx is neighbourhood of x0. where If 00 ,, o y F zY xSxV , then there exists v, 0, n uSxVn uv0 n t, , such that ,0, lim nu n onn o tv n fx tu t zy , for someCS F 0nnn . xtu0 nN; then there exist such that Let 00 0 , for all nVx Nxxnn; 00 , for all n o yD nnx . Since f is any continuous selection of F such that Copyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM 416 00 fx 00 ,for allDnn , therefore: 00 , o y F kYxxx – n o fx fx ; 0 – ,f o n n fx fx t 0 orallDnn o zD . It follows that 0 . Now, we give the sufficient conditions for εfeeble multifunction minimum point of problem (VP). Theorem 3.2: Let V, 00 Fx, ,: o DFU Y :GU Z be a Dmulifunction and be quasi Dmulti function and f be continuous selection of F such that 00 . fx 0 forall,,xSxV If 00 ,, o y F Yxxx D then x0 is εfeeble multifunction minimum point of problem (VP). Proof: Let x0 be not εfeeble multifunction minimum point of problem (VP), then there exists 00 Fx such that 0o FV xyD , which implies that, there exists 0, xVyFx y .y D , such that 0 Since G is quasi Dmultifunction, therefore feasible set V is convex, 00 xtxx for 01.Vt Thus, 0, SxV which implies that D 00 , o y F Yxxx 0 ,, (3.3) Since F is Dmultifunction therefor e, 00 o y F xyY xxxD 00 ,.xxxD which gives that 0 o y F yy Y Thus, there exists 00 , o y Fxxx 0 yykD 0.ky yD 0 yyD kY such that , that is Also, .kD which implies that Hence, 00 ,, o y F Yxxx D Also, which is contradiction to given condition (3.3). 4. PartialεMinimum (Weak) Point In this section we introduce the notion of partialεmini mum point, and partialεweak min imum point . Definition 4.1: Let , VD . 1) is called a partialεminimum point (p.εm. p.) of problem (VP), if there exists, Fx , such that ; o FV xyD (4.1) 2) is called a partialεweak minimum point (p.ε w. m. p.) of problem (VP), if there exists, Fx, such that ; o FV xyD (4.2) The set of V which satisfies (4.1) or (4.2) is de noted by ˆ,PFD ˆ,WPFD and respectively. When Vx is replaced by NxVx in (4.1) and (4.2), Nx being neighbourhood of , then we have local p.ε.m. p. and local p.εw. m. p. of prob lem (VP). If ,gryF satisfies (4.1) then it is called partialεminimizer of (VP) and if satisfies (4.2) then it is called partialεweak min imizer of (VP ) . Now we show that partialεminimum point is differ ent from εfeeble multifunction minimum point. The following example illustrates that ˆˆ ,,.SFDPFD UXR 2 YR2 , , Example 4.1: Let R , 11 , 22 2 DR 2 ER :GU Z , , and be defined by 0,0, if0 ,0if 0 xxx Gx xx : and UY be defined by 0,0 ,,0if0 0,0,if 0 xx Fx xx x :0.VxRx then, Let 0, 0,0 Vy Fx . Then, .FV xyD Copyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM 417 Thus, ˆ, SFD ... But 0,0 FVx . o y D Thus, ˆ,. PFD ˆ ,,.SFD R2 YR The following example illustrates that ˆ PFD Example 4.2: Let , , UX 2 R, 5,3 2 , 0xyy 0,0yy x , ,: ,Dxyxy , ,:Ex :GU Z , and be defined by 0, ,0 Gx : 0, if0 if 0 xxx xx and UY be defined by 0,0,2.5,3.1 ,( 1, 1, Fx xx :0.VxRx ,0) if0 if 0 xx x then, Let 0 V , 0,0 . Fx Then, o DFVxy . Thus, ˆ,. PFD But 2.5, 3.1 Vx y D . Thus, ˆ,. SFD : The following lemma can be proved as in [9]. Lemma 4.1: Let UY :Z be a strictly quasi Dmultifunction and GU be a quasi Dmulti function. Then, ˆ PFD ˆ ,, WPFD. Now, we show that every local partialεminimum (weak) point is a partial(weak) point if F is srictly quasi Dmultifunction and G is quasi Dmultifunction and prove local εfeeble multifunction minimum point is not εfeeble multifunction minimum point of problem (VP) in above conditions. Theorem 4.1: Let F be strictly quasiDmultifunction and G be a quasi Dmultifunction. Then, any local par tialεminimum point of problem (VP) is a partialεmi nimum point o f pr obl em (VP). Proof: Let be local partialεminimum point of problem (VP), then there exists a neighbourhood Nx of and Fx such that . o FVN xxyD (4.3) Let if possible, be not partialεminimum point of problem (VP). Then, there exist Fx such that . o FV xyD Thus, there exists , Vx and Fx such that o yy D, which gives that . o yyDyD That is, .yFxDFxD Since F is a strict quasi Dmultifunction, therefore for each 0, 1t. We have 1, o Fx DFxDFtxtxD which implies that –. o yFxtxx D Let –. txtxx 0,1 ,t Then, for each to FxD tt Fx such that, and consequently there exists to yyD . On the other hand for 0,t 0 with small enough, –. tx xNx Since G is quasi Dmulti function, therefore, feasible set is convex and we have –,for 0,. t xxtxxV Nxxt Thus, we deduce that –, to yFV NxxyD UXR which contradicts (4.3). The following example illustrates that above result is not true for εfeeble multifunction minimum point of (VP). 2 YR2 , , Example 4.3: Let R , 2 DR , 3.5, 3.5 2 ER :GU Z , and de fined b y 0,0,if0 ,0if 0 xx x Gx xx Copyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM 418 Copyright © 2013 SciRes. AJOR and : UY defined by 22 22 11 1,1 ,,,, 22 1,1 ,, ,0 ,, xx Fxxx xxx if 1 if 10 if 0 x x x Here G is quasi Dmultifunction and F is strict quasi Dmultifunction. Then, Let 0 V , 1,1 Fx . Then .yDFVxNx Thus, is local εfeeble multifunction minimum point. 52, 52 DD , .D3, 3FV xy Thus, is not εfeeble multifunction minimum poin t of problem (VP). 5. εLagrangian Multiplier Theorem In this section, let L(Z, Y) be the set of co ntinuous lin ear operators from Z to Y, and let :,, ZYTL ZY TED Denote by (F, G) the multivalued map from X to Y Z defined by , GxFx Gx, for all x X. If , hY ,TLZ:hF XRY , we define and : TG XY as hF xhFx and TGxFxT Gx : , respectively. Lemma 5.1: [14]. Let XY be Dmultifunc tion on X. Then, one and only one of the following statements is true: 1) there exists X such that . o DFx 2) there exists 0D such that 0y for all . FX Theorem 5.1: Let o D , o EGV , Fx and let –, yG be Dmultifunction on V. If is partialεweak minimum point of problem (VP), then there exists ,ZTL such that Y is partialεweak minimum point of following problem: T VP min xV xTGx and 0 o TGxE D Proof: Since is partialεweak minimum point of problem (VP), therefore there exists, , Fx such that o FV xyD (5.1) Hence, ,, oo FV xyGVDE Since –, yG is Dmultifunction on V, there fore by Lemma 5.1, there exists ,,0,0hpD E such that for all–0, ,,. hy yps VxyFxsGx 0.h (5.2) We claim that In fact, if h = 0, then 0p and 0,ps for all Gx (5.3). Since , o GV E there exists x1V and 11 o Gx E. Hence, 10ps , which c ontradicts ( 5.3). Therefore, 0.h Fix with o dD 1hd and define as T(z) = p(z)d, f or all (5.4). :TZYzZ Clearly, ,.TLZY Using (5.2 ) and (5.4), we ge t –0hy yTs . (5.5) Since V , therefore Gx E . Let , Gx E then Gx and E . This gives that 0ps (5.6). Therefore we get that, 0. o Tspsd D Thus, we ha ve 0. o TGxE D Suppose that is not partialεweak minimum point of problem (VP)T, which gives that o FV xTGV xyD Then there exist 000 , Vxy Fx\ and 00 Gx such that 00 . o yyTs D Since 0,hD we get: 00 0hyy Ts
S. K. SUNEJA, M. SHARM 419 From (5.4), we get 00 –0,pshy y which contradicts (5.2). Hence is partialεweak minimum point of prob lem (VP)T. 6. εDuality Let us define a multivalued mapping by :,LZY Y T max = {y: there exists x V, y F(x) such that x is partialεweak minimum point of problem (VP)T}. Consider the following maximum problem: (VD) subject to T ,ZY TL .. Definition 6.1: A point ,TLZY .T is said to be a feasible point of problem (VD) if ,Ty We say that 00 is partialεweak maximizer of (VD) if there exists no feasible point ,ZY TL such that: 0 Ty . o D We now establish the following εduality results. Theorem 6.1 (εWeak duality): If V and ,TLZY is a feasible point of problem (VD), then –. o DFx T T Proof: Since , for any , T there exists , Vy Fx such that is partialεweak minimum point of (VD) corresponding to T. It follows that, ––FTG Vxy . o D (6.1) Now, we show that . o DFx y On contrary, suppose that . o DFx y Then there exists , Fxsuch that –, o yy D which implies that –. o D yy Since is a feasible point of problem (VP)T, there exist .x E ,TLZ zG It is given that therefore ,Y,Tz D which implies that –. ooo TzyTz DD D D Thus, \FTG Vxy , o D which contradicts (6.1). Therefore, we have –. o DFx T 0–, Theorem 6.2: (εStrong dual ity): Let yG be Dmultifunction on V. If ,, 00 00 VyFxx is partialεweak minimum point of problem (VP) and o GV E ,TLZY, then there exists 0 such that 00 ,Ty is partialεweak maximizer of problem (VD). Proof: Suppose x0 is partialεweak minimum point of problem (VP) and o GV E . Then, by Theorem 5.1 there exists ,ZY 0 TL such that x0 is partialεweak minimum point of problem 0 T It follows that, , corresponding to T0. VP T 0. 00 T . Thus, T0 is feasible point of (VD) and By εweak duality, we obtain 00 . o yT D Thus, 00 ,Ty is partialεweak maximizer of prob lem (VD). 7. Acknowledgements The first author is grateful to the University Grants Commission (UGC ), In dia fo r offering financial support. REFERENCES [1] H. W. Kuhn and A. W. Tucker, “Nonlinear Programming Proceedings of the 2nd Berkeley Symposium on Mathe matical Statistics and Probability,” University of Califor nia Press, Berkeley, 1951, pp. 481492. [2] A. M. Geoffrion, “Proper Efficiency and Theory of Vec tor Maximization,” Journal of Mathematical Analysis and Applications, Vol. 35, 1991, pp. 175184. [3] J. M. Borwein, “Proper Efficient Points for Maximiza tions with Respect to Cones,” SIAM Journal on Control and Optimization, Vol. 15, No. 1, 1977, pp. 5763. doi:10.1137/0315004 [4] H. P. Benson, “An Improved Definition of Proper Effi ciency for Vector Maximization with Respect to Cones,” Journal of Mathematical Analysis and Applications, Vol. 71, No. 1, 1979, pp. 232241. doi:10.1016/0022247X(79)902269 [5] J. M. Borwein and D. M. Zhauang, “Super Efficiency in Convex Vector Optimization,” Zeitschrift für Operations Research, Vol. 35, No. 3, 1991, pp. 175184. doi:10.1007/BF01415905 [6] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 142, No. 3, 2009, pp. 451467. [7] A. Hamel, “An εLagrange Multiplier Rule for a Mathe matical Programming Problem on Banach Spaces,” Opti mization, Vol. 49, No. 12, 2001, pp. 137149. doi:10.1080/02331930108844524 [8] W. D. Rong and Y. N. Wu, “εWeak Minimal Solutions of Vector Optimization Problems with SetValued Maps,” Copyright © 2013 SciRes. AJOR
S. K. SUNEJA, M. SHARM Copyright © 2013 SciRes. AJOR 420 Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2000, pp. 569579. doi:10.1023/A:1004657412928 [9] M. Chinaie and J. Zafarani, “Image Space Analysis and Scalarization of Multivalued Optimization,” Journal of Optimization Theory and Applications, Vol. 106, No. 3, 2010, pp. 111. [10] X. Q. Yang, “Directional Derivatives for SetValued Mappings and Applications,” Mathematical Methods of Operations Research, Vol. 48, No. 2, 1998, pp. 273285. doi:10.1007/s001860050028 [11] J. Benoist, J. M. Borwein and N. A. Popovici, “Charac terization of Quasiconvex Vector Valued Functions,” Proceedings of the American Mathematical Socitty, Vol. 131, 2003, pp. 11091113. [12] J. Benoist and N. Popovici, “Characterizations of Convex and Quasiconvex SetValued Maps,” Mathematical Me thods of Operations Research, Vol. 57, No. 3, 2003, pp. 427435. doi:10.1007/9783540248286 [13] J. Jahn, “Vector Optimization Theory, Applications and Extensions,” Springer, Berlin, 2004. [14] T. Illes and G. Kassay, “Theorem of Alternative and Op timality Conditions for Convexlike and General Convex like Programming,” Journal of Optimization Theory and Applications, Vol. 101, No. 2, 1999, pp. 243257. doi:10.1023/A:1021781308794
