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Applied Mathematics, 2011, 2, 452-460 doi:10.4236/am.2011.24057 Published Online April 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Efficiency and Duality in Nondiffer entiable Multiobjective Programming Involving Directional Derivative Izhar Ahmad Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Saudi Arabia, India E-mail: izharmaths@hotmail.com; drizhar@ kf up m.edu.sa Received December 24, 2010; revised February 24, 2011; accepted February 26, 2011 Abstract In this paper, we introduce a new class of generalized dI-univexity in which each component of the objective and constraint functions is directionally differentiable in its own direction di for a nondifferentiable multiob- jective programming problem. Based upon these generalized functions, sufficient optimality conditions are established for a feasible point to be efficient and properly efficient under the generalised dI-univexity re- quirements. Moreover, weak, strong and strict converse duality theorems are also derived for Mond-Weir type dual programs. Keywords: Multiobjective Programming, Nondifferentiable Programming, Generalized dI-Univexity, Sufficiency, Duality 1. Introduction The field of multiobjective programming, also known as vector programming, has grown remarkably in different directions in the setting of optimality conditions and dual- ity theory. It has been enriched by the applications of var- ious types of generalizations of convexity theory, with and without differentiability assumptions, and in the frame- work of continuous time programming, fractional prog- ramming, inverse vector optimization, saddle point theory, symmetric duality and vector variational inequalities etc. Hanson [1] introduced a class of functions by genera- lizing the difference vector xx in the definition of a convex function to any vector function , xx . These functions were named invex by Craven [2] and -con- vex by Kaul and Kaur [3]. Hanson and Mond [4] defined two new classes of functions called Type I and Type II functions, which were further generalized to pseudo Type I and quasi Type I functions by Rueda and Hanson [5]. Zhao [6] established optimality conditions and dual- ity in nonsmooth scalar programming problems assum- ing Clarke [7] generalized subgradients under Type I functions. Kaul et al. [8] extended the concept of type I and its generalizations for a multiobjective programming prob- lem. They investigated optimality conditions and derived Wolfe type and Mond-Weir type duality results. Suneja and Srivastava [9] introduced generalized d-type I func- tions in terms of directional derivative for a multiobjec- tive programming problem and discussed Wolfe type and Mond-Weir type duality results. In [10], Kuk and Tanino derived optimality conditions and duality theorems for non-smooth multiobjective programming problems in- volving generalized Type I vector valued functions. Gu- lati and Agarwal [11] discussed sufficiency and duality results for nonsmooth multiobjective problems under (,,, F d -type I functions. Agarwal et al. [12] estab- lished sufficient conditions and duality theorems for nonsmooth multiobjective problems under V-type I func- tions. Recently, Jayswal et al. [13] obtained some opti- mality conditions and duality results fo r nonsmooth mul- tiobjective problems involving generalized ,,,F dV -univexity. Antczak [14] studied d-invexity is one of the genera- lization of invex function, which is introduced by [15]. In [14], Antczak established, under weaker assumptions than Ye, the Fritz John type and Karush-Kuhn-Tucker type necessary optimality conditions for weak Pareto optimality and duality results which have been stated in terms of the right differentials of functions involved in the considered multiobjective programming problem. Some authors [16-18] proved that the Karush-Kuhn- Tucker type necessary conditions [14] are sufficient un- der various generalized d-invex functions. Antczak [19] I. AHMAD Copyright © 2011 SciRes. AM 453 corrrected the Karush-Kuhn-Tucker necessary conditions in [14] and discussed the sufficiency and duality under drtype I functions. Recently, Silmani and Radjef [20] introduced generalzed dI-invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction and es- tablished the necessary and sufficien t conditions for effi- cient and properly efficient solutions. The duality results for a Mond-Weir type dual are also derived in [20]. They also observed that the Karush-Kuhn-Tucker sufficient conditions discussed in [16-18] are not applicable. More recently, Agarwal et al. [21] introduced a new class of generalized ,d type I for a non-smooth multiobjective programming problem and discussed op- timality conditions and duality results. In this paper, we introduce I dV-univexity and ge- neralized I dV-univexity in which each component of the objective an d constraint functions of a multiobj ective programming problem is semidirectionally differentiable in its own direction d i. Various Karush-Kuhn-Tucker suf- ficient optimality cond itions for efficient and properly ef- ficient solutions to the problem are established involv ing new classes of semidirectionally differentiable generali- zed type I functions. Moreover, usual duality theorems are discussed for a Mond-Weir type dual involving afo- resaid assumptions. The results in this paper exten d many earlier work appeared in the literature [9,10,12,14-16, 19]. 2. Preliminaries and Definitions The following conventio ns for equalities and inequalities will be used. If 11 =,,, =,,n nn xx yyxy , then ==,=1,, ii x yi nxy ; <<,=1,, ii x yi nxy ; ,=1, ,; ii x yi n xy and xy xyxy, We also note q (resp. q or q ) the set of vectors q y with 0y (resp. 0y or >0y). Definition 1 [22]. Let D be a nonempty subset of n , :n DD and let 0 x be an arbitrary point of D.The set D is said to be invex at 0 x with respect to , if for each x D, 00 ,,0,1.xxxD D is said to be an invex set with respect to , if D is invex at each 0 x D with respect to the same . Definition 2 [23]. Let n D be an invex set with respect to :n DD .A function :fD is called pre-invex on D with respect to , if for all 0 , x xD, 00 0 1,,0,1.fxfx fxxx Definition 3 [14]. Let n D be an invex set with respect to :n DD . A m-dimensional vector valued function :m D is pre-invex with respect to , if each of its components is pre-invex on D with respect to the same function . Definition 4 [7]. Let D be a nonempty open set in n . A function :fD is said to be locally Lipschitz at 0 x D , if there exist a neighborhood 0 x of 0 x and a constant >0K such that 0 , ,, f yfx Kyxxyx where . denotes the Euclidean norm. We say that f is locally Lipschitz on D if its locally Lipschitz at any point of D. Definition 5 [7]. If :n f DR is locally Lip- schitz at 0 x D , the Clarke generalized directional de- rivative of f at 0 x in the direction n d, denoted by 000 0 ;= sup. lim yx t f ytd fy fxd t And the usual one-sided directional derivative of f at 0 x in the direction d is defined by 00 00 ;= , lim f xdfx fxd whenever this limit exists. Obviou sly, 000 ;; f xdf xd . We say that f is directionally differentiable at 0 x if its directional derivative 0; f xd exists finite for all n d. Definition 6 [15]. Let : N fD be a function de- fined on a nonempty open set n D and directional- ly differentiable at 0. x D f is called d-invex at 0 x on D with respect to , if there exists a vector function :, n DD such that for any x D, 0000 , ,;,, for all=1,,, ii i fy fxKyx xyxf xf xfxxx iN where 00 ;, i f xxx denotes the directional derivative of i f at 0 x in the direction 000 000 0 ,: ;, , =. lim i ii xxf xxx f xxxfx If Inequalities (1) are satisfied at any point 0 x D , then f is said to be d-i nvex on D with respect to . Definition 7 [20]. Let D be a nonempty set in n and :n DD a function. We say that :fD is a semi-directionally I. AHMAD Copyright © 2011 SciRes. AM 454 differentiable at 0 x D,if there exist a nonempty subset n S such that 0; f xd exists finite for all dS We say that f is a semi-directionally differentiable at 0 x D in the direction 0 , x x , if its direc- tional derivative 00 ;,fx xx exists finite for all x D. Definition 8 [20]. Let : N fD be a function de- fined on a nonempty open set n D and for all =1,,, i iNf is semi-directionally differentiable at 0 x D in the direction :n iDD . f is called dI-invex at 0 x on D with respect to =1, , iiN if for any x D, 000 ;,,for all ,2,,, ii ii fx fxfxxxiN where 00 ;, ii f xxx denotes the directional deriva- tive of i f at 0 x in the direction 00 0 000 0 ,:; , ,. lim iii ii i xxf xxx fxxx fx If Inequalities (2) are satisfied at any point 0 x D , then f is said to be I d-invex on D with respect to =1, , iiN Consider the following multiobjective programming problem 12 Minimize=,,,N M Pfxfxfxfx Subject to0,gx , x D where :, :, N k fD gD D is a nonempty open subset of n . Let =: 0XxDgx be the set of feasible solutions of (MP). For 0 x D, we denote by 0 J x the set 00 1,2,,:= 0,= j jkgxJJx and by 00 resp. J xJx the set 0 1, 2,,:0 j jkgx (resp. 0>0 j gx . we have 000 1, 2,, J xJxJx k and if 00 , =xXJx. We recall some optimality concepts, the most often studied in the literature, for the prob lem (MP). Definition 9. A point 0 x X is said to be a local weakly efficient solution of the problem (MP), if there exists a neighborhood 0 Nx around 0 x such that 00 for all f xfxxNxX Definition 10. A Point 0 x X is said to be a weakly efficient (an efficient) solution of the problem (MP), if there exists no x X such that 00 <.fxfx fxfx Definition 11. An efficient solution 0 x X of (MP) is said to be properly efficient, if there exists a positive real number M such that inequality 00ii jj fxfxMfx fx is verified for all 1, ,iN and x X such that 0 < ii f xfx, and for a certain 1, ,jN such that 0 >. jj f xfx Following Jeyakumar and Mond [24], Kaul et al. [8] and Slimani and Radjef [20], we give the following defi- nitions. Definition 12. , f g is I dV-univex type I at 0 x D if there exist positive real valued functions and ij defined on X D, nonnegative functions 01 andbb, also defined on 01 ,: ,:XDR RR ;:,and : nn ij RXDR XDR such that 00000 00 ,,;, iiiii bxxfxfxxx fxxx (3) and 1101 00 00 ,,;, jjjj bxxg xxxg xxx (4) for every x X and for all =1,2, ,iN and =1,2, ,jk. If the inequality in (3) is strict (wheneve r 0 x x ), we say that (MP) is of semistrictly I dV-univex type I at 0 x with respect to =1, =1, and ij iN j k . Definition 13. , f g is quasi-I dV-univex type I at 0 x D if there exist positive real valued functions and j , defined on X D, nonnegative functions 01 andbb, also defined on 0 , :, X DRR 1:RR and Nk dimensional vector functions :,=1, n i X DRi N and :,=1, n j X DRj k such that for some vectors and N k RR : 000 00 =1 00 =1 ,, 0(;(,))0 N iii i i N ii i i bxxxx fx fx fxxxx X (5) and 1010 0 =1 00 =1 ,,0 ;, 0. k jj j j k jj j j bxxxxg x g xxx xX (6) If the second inequality in (5) is strict 0 x x, we say that (MP) is of semi-strictly quasi I d-V-univex type I at X with respect to =1, =1, and ij iN j k . I. AHMAD Copyright © 2011 SciRes. AM 455 Definition 14. , f g is pseudo-I dV-univex type I at 0 x D if there exist positive real valued functions i and j , defined on X D, nonnegative functions 0 b and 1 b, also defined on X D, 0:RR , 1:RR and Nk dimensions vector functions :,=1, n i X DRi N and :,=1, n j X DRj k such that for some vectors N R and k R : 00 =1 000 00 =1 ;,0 ,,()0 N iii i N iii i i fx xx bxxxxfxfxx X (7) and 00 =1 1010 0 =1 ;, 0 ,,0 . k jj j j k jj j j gx xx bxxxxg xxX (8) Definition 15. , f g is quasi pseudo-I dV-univex type I at 0 x D if there exist positive real valued func- tions i and j , defined on X D, nonnegative fun- ctions 0 b and 1 b, also defined on X D,0:,RR 1:RR and ()Nk dimensions vector functions :, n i X DR =1,iN and :, n j X DR =1,jk such that the relation (5) and (8) are satisfied. If the second inequality in (8) is strict 0 ( x x, we say that ()VP is of quasi strictly-pseudo I d-Vtype I at 0 x with respect to =1, =1, and . ij iN j k Definition 16. , f g is pseudoquasi -I d-V-univex type I at 0 x D if there exist positive real valued func- tions i and j , defined on X D, nonnegative functions 0 b and 1 b, also defined on X D , 01 :, :RR RR and ()Nk dimensions vector functions :, =1, n i X DRiN and :, =1, n j X DRjk , such that k R the rela- tions (7) and (6) are satisfied. If the second inequality in (7) is strict 0 x x, we say that MP is of strictly- pseudo quasi I dVtype I at 0 x with respect to =1, iiN and =1, . j j k 3. Optimality Conditions In this section, we discuss some sufficient conditions for a point to be an efficient or properly efficient for (MP) under generalized I dVunivex type I assumptio ns. Theorem 3.1. Let 0 x be a feasible solution for (MP) and suppose that there exist NJ vector functions :,=1,, n i X XRiN 0 :, n j X XRjJx and scalars =1 0, =1,, =1; N ii i iN 0, j 0 jJx such that 000 0 =1( ) 0 ;,;, 0, , N ij iij j ijJx f xxxgxxx xX (9) Further, assume that one of the following conditions is satisfied: a) i) , f g is quasi strictly-pseudo I dV -univex type I at 0 x with respect to =1, 0 ,,, ij iN jJx and for some positive functions ,=1, , iiN , j 0 jJx, ii) for any uR , 0 00;uu 1<0u <0; u 00 ,>0,bxx 10 ,>0;bxx b) i) , f g is strictly-pseudo I dV-univex type I at 0 x with respect to =1, iiN , 0, , jjJx and for some positive functions , =1,, , ij iN 0,jJx ii) for any uR , 0( )>0>0;uu 1 0()0,uu 00 (, )>0,bxx 10 (, )0.bxx Then 0 x is an efficient solution for MP . Proof: Condition a). Suppose that 0 x is not an effi- cient solution of MP . Then there exists an x X such that 0, f xfx which implies that 00 =1 ,0. N iiii i xxfx fx (10) Since 00 ,>0bxx ; 0 00uu , the above inequality gives 000 00 =1 ,, 0. N iiii i bxxxxfx fx From the above inequality and Hypothesis i) of a), we have 00 =1 ;, 0. N iii i fx xx By using the Inequality (9) we deduce that 00 () 0 ;, 0, jjj jJx gx xx which implies from the condition part ii) of a) that 0 1010 0 ,,<0. jj j jJx bxxxxg x Since 10 1 ,>0; <0<0,bxxuu we get I. AHMAD Copyright © 2011 SciRes. AM 456 00 0 ,<0. jj j jJx xx gx (11) As 0 and 00 =0; j g xjJx, it follows that 00 =0, , jj g xjJx which implies that 00 0 ,=0. jj j jJx xx gx The above equation contradicts Inequality (11) and hence the conclusion of the theorem follows: Condition b): Since 00 =0, 0, , jj g xjJx and 00 ,>0, , j x xjJx we obtain 00 0 ,=0,. jj j jJx x xgx xX By Hypothesis ii) of b), we get 101 0 0 ,,0. jj jJx bxxxx From the above inequality and the Hypothesis i) of b)( in view of reverse implication in (8), if follows that 00 0 0 ;,<0, \. jj j jJx g xxx xXx By using Inequality (9), we deduce that 00 0 =1 ;,>0, \, N iii i f xxx xXx (12) which by virtue of relation (7) implies that 000 00 =1 0 ,, >0, \. N iii i i bxxxx fx fx xX x The above inequality along with Hypothesis ii) of b) gives 00 0 =1 ,>0,\. N iii i i x xfxfx xXx (13) Since (10) and (13) contradicts each other, and hence the conclusion follows: Theorem 3.2. Let 0 x be a feasible solution for (MP) and suppose that there exist NJ vector functions 0 :, =1,, :, nn ij X XRi NXXRjJx and scalars 0 =1 0, =1,, =1, 0, N iij i iN jJx such that Inequality (9) of Theorem 3.1 is satisfied. Moreover, assume that one of the following conditions is satisfied. a) i) , f g is pseudo quasi I dVunivex type I at x0 with respect to =1, 0 ,,, ij iN jJx and for some positive functions 0 ,=1, and,, ij iN jJx ii) for any uR , 10 00, 00,uuuu 00 10 ,>0, ,0;bxx bxx b) i) , f g is strictly pseudo I dVunivex type I at x0 with respect to =1, 0 ,,, ij iN jJx and for positive functions =1, iN and 0 ,, jjJx ii) for any uR 01 00 10 00; 00; ,>0, ,0. uuuu bxx bxx Then 0 x is an efficient solution for MP . Further Suppose that these exist positive real numbers , ii nm such that 0 <,<, =1, ii i nxxmiN for all feasible x.Then 0 x is a properly efficient solution for MP Proof: Condition a). Suppose that 0 x is not an effici- ent solution of MP . Then there exists an x X such that 0 f xfx which implies that 00 =1 ,<0. N iii i i xxfxfx (14) Since 0=0, 0 jj gx and 00 ,>0, j x xjJx we obtain 00 0 ,=0. jj j jJx xx gx From the above inequality and Hypothesis ii) of a), we have 10100 () 0 ,,0. jj j jJx bxxxxg x Using Hypothesis i) of a), we deduce that 00 0 () 0 ,;,0. jjj j jJx xx gxxx (15) The Inequalities (9) and (14) yield that 00 =1 ;, 0, N iii ifx xx which by Hypothesis i) of a), we obtain 000 00 =1 ,, 0, N iii i i bxxxx fxfx (16) The Inequality (16) and Hypothesis ii) of a) give 00 =1 ,0. N iii i i xxfxf x (17) Since (14) and (17) contradict each other, we conclude I. AHMAD Copyright © 2011 SciRes. AM 457 that 0 x is not an efficient solution of MP . The pro- perly efficient solution follows as in Hanson et al. [25]. For the proof of part b), we proceed as in part b) of Theorem 3.1, we get Inequality (17). Thus complete the proof. 4. Mond-Weir Type Duality Consider the following multiobjective dual to problem MP MD Maximize 12 =,,, N f yfyfy fy subject to =1 =1 ;,;, 0, Nk iiij jj ij f yxygyxy xX 0, =1,2,,, , , N k jj g yj kyDRR :, =1,2,,, :,=1,2,,. n i n j X DR iN XD Rjk Let Y be the set of feasible solutions of problem MD ; that is, 2 =1 =1 =,,, ,: ;,;, 0, ij ij Nk iij jj ij Yy f yxygyxy 0, ; ,, ; :=1,2,, ; N k jj n i g yxXyDRR XD RiN :,=1,2,,. n j X DRj k We denote by rD PY, the projection of set Y on D. Theorem 4.1. (Weak Duality). Let x and =1, =1, ,,, , ij iN j k y be feasib le solution for (MP) and (MD) respectively. Moreover, assume that one of the following conditions is satisfied: a) i) , f g is pseudo quasi I d-V-univex type I at y with respect to =1, >0, , , iiN =1, j j k and for some positive functions i , j for =1,2, ,iN and=1,2, ,jk, ii) for any uR 01 01 00; 00; ,>0, ,0 uuu u bxy bxy b) i) , f g is strictly-pseudo quasi I d-V-univex type I at y with respect to =1, , , iiN , =1, j j k and for some positive function i , j for =1,2, ,iN and=1,2, ,jk, ii) for any uR, 01 10 0>0; 00; ,0, ,>0; uuu u bxy bxy c) i) , f g is quasi strictly-pseudo I dV -univex type I at y with respect to =1, =1, ,,, ij iN j k and for some positive functions ,for= ij i 1,2,, N and =1,2, ,jk, ii) for any uR , 01 01 >0>0; >0>0; ,>0, ,>0. uuu u bxy bxy Then f xfy. Proof: Since 11 0,=1,2,,, 00,,>0 and,>0,=1,2,, jj j gy jk uubxy x yjk , we have 11 =1 ,,0. k jj j j bxyxyg y By Condition a) (in view of definition 16), it follows that =1 ,;,0. k jjjj j xyg yxy (18) Since =1, =1, ,,, , ij iN j k y is a feasible solu- tion for (MD), the first dual constraint with (18) implies that =1 ;, 0. N iii i fy xy (19) From (19) and Hypothesis i) of a), we obtain 00 =1 ,, 0. N iii i i bxyxy fxfy (20) Condition ii) of a) and Inequality (20) give =1 ,0. N iii i i xyf xf y (21) Assume that f xfy. Since >0, =1,2,,and>0 iiN , we obtain =1 ,<0, N iii i i xyf xf y (22) which contradicts (21), Therefore, the conclusion follows: The proof of part b ) and c) are ve ry similar to proof of part a), except that: for part b), the Inequality (21) beco- mes strict > and Inequality (22) becomes non strict . For part c), the Inequality (18) becomes strict <, I. AHMAD Copyright © 2011 SciRes. AM 458 it follows that the Inequalities (20) and (21) become strict >. Since 0 , then the Inequality (22) becomes non strict . In this cases, the Inequalities (21) and (22) contradicts each other always. Remark 1: If we omit the assumption >0 in the condition i) of a) or the word “strictly” in the condition b),we obtain, for this part of theorem, f xfy. Theorem 4.2. (Weak Duality). Let x and =1, =1, ,,,, ij iN j K y be feasible solutions for MP and MD respectively, Assume that 1) , f g is semi-strictly I dV-univex type I at y with respect to >0 , =1, , iiN , =1, j j k and for some positive functions =1, =1, , ij iN jk , 2) for any uR, 01 01 >0>0, 00, ,>0, ,0. uauu bxy bxy Then f xfy. Proof: Since 0, =1,2,,, jj g yjk which imp- lies that =1 ,0. k jj j j xyg y (23) By (23) and Hypothesis i) (with 1,, j bxy xy ) in Definition 12 replaced by , j x y it follows that =1 ,, 0. k jj j j gy xy (24) The first dual constraint and (24) give =1 ,, 0. N iii i fy xy (25) Dividing both sides of (3) by , i x y and taking x y, by Hypothesis i), we get 00 1 ,>,,, , =1,2,, . ii ii i bxyfxfyfyxy xy iN On Multiplying by i and taking 1 =, ii x y , we get 00 ,>,,, =1,2, , iiiiii i bxyfx fyfyxy iN Adding with respect to i, and applying (25) and Hy- pothesis ii), we have =1 ,>0. N iii i i xyf xfy (26) Assume that f xfy. Since >0and >0 i , we have =1 ,<0, N iii i i xyf xf y which contradicts (26). Theorem 4.3. (Strong Duality ).Let x0 be a weakly effi- cient solution for MP . Assume that the function g satisfies the I d-constraint qualification at 0 x with res- pect to =1, j j k . Then there exist >> and N K RR such that 0=1, ,,,iiN x , =1, jjk Y and objective functions of MP and MD have the same values at 0 x and 0=1, =1, ,,, , ij iN j k x , respectively. If, further, the weak duality between MP and MD in theorem holds with the condition a) without >0 (resp. with the condition b) or c)), then 0=1, =1, ,,, , ij iN jk x Y is a weakly efficient (resp. an efficient) solutions of MD . Proof. By the Theorem 31 [20], there exists k and 0 J x such that 0000 =1 =1 ;,;, 0, . Nk iiij jj ij f xxxgxxx xX It follows that 0=1, =1, ,,,,. ij iN jk x Y Tri- vially, the objective function values of (MP) and (MD) are equal. Suppose that 0=1, =1, ,,, , ij iN jk x Y is not a weakly efficient solution of MD . Then there exists =1, =1, ,,, , ij iN jk y Y such that 0< f xfy which violates the weak duality theorem. Hence 0=1, =1, ,,, , ij iN jK x Y is in- deed a weakly efficient solution of (MD). Theorem 4.4. (Strict Converse Duality). Let 0 x and 0=1, =1, ,,, , ij iN jk y be feasible solutions for (MP) and (MD) respectively, such that 00 =1 =1 =. NN ii ii ii f xfy (27) Moreover, assume that , f g is strictly pseudo quasi I dV type I at o y with respect to =1, =1, , ij iN j k and for and . Then 00 = x y. Proof. Since 00 =1,2,, jj g yj k , we have I. AHMAD Copyright © 2011 SciRes. AM 459 100100 0 =1 ,0. k jj j j bxyxyg y Using the second part of the hypothesis, we get 000 =1 ;, 0. k jjj j gy xy (28) The Inequality (28) and feasibility of 0=1, =1 , ,,, , ij iN j k y for (MD) give 000 =1 ;, 0, N iii i fy xy which by the first part of Hypothesis ii), we obtain 000000 00 =1 ,, >0, . N iii i i bxyxyfxfy xX The above inequality along with Hypothesis iii) gives 00 00 =1 ,>0. N iii i i xy fxfy (29) By Hypothesis i), iii) and 00 ,>0, ixy =1,2, , iN we have 00 00 =1 ,=0. N iii i i xy fxfy (30) Now (29) and (30) contradict each other. Hence the conclusion follows. 5. Conclusion and Future Developments In this paper, generalized I dV -univex functions have been introduced. The sufficient optimality conditions are discussed for a point to be an efficient or properly efficient for (MP) under the introduced functions. App- ropriate Mond-Weir type duality relations are established under these assumptions. Sufficiency and duality with generalized I dV-univex functions will be studied for nonsmooth variational and nonsmooth control problems, which will orient the future research of the author. 6. References [1] M. A. Hanson, “On Sufficiency of the Kunn-Tucker Con- ditions,” Journal of Mathematical Analysis and Applica- tions, Vol. 80, 1981, pp. 445-550. [2] B. D. Craven, “Invex Functions and Constrained Local Minima,” Bulletin of Australian Mathematical Society, Vol. 24, No. 3, 1981, pp. 357-366. doi:10.1017/S0004972700004895 [3] R. N. Kaul and K. 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