### Journal Menu >> Applied Mathematics, 2011, 2, 1236-1242 doi:10.4236/am.2011.210172 Published Online October 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity Mohamed Abd El-Hady Kassem Mathematical Department, Faculty of Science, Tanta Universit y, Tanta, Egypt E-mail: mohd60_371@hotmail.com Received April 8, 2011; revised June 9, 2011; accepted June 16, 20 1 1 Abstract The purpose of this paper is to introdu ce second order (K, F)-pseudoconvex and second order strong ly (K, F)- pseudoconvex functions which are a generalization of cone-pseudoconvex and strongly cone-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated by using the considered functions. Furthermore, the weak, strong and converse duality theorems for this pair are es-tablished. Finally, a self duality theorem is given. Keywords: Multiobjective Programming, Second-Order Symmetric Dual Models, Duality Theorems, Pseudoconvex Functions, Cones 1. Introduction Duality is an important con cept in the study of nonlinear programming. Symmetric duality in nonlinear program-ming in which the dual of the dual is the primal was first introduced by Dorn . Subsequently Dantzig et al.  established symmetric du ality results for convex/concave functions with nonneg ative orthan t as th e cone. Th e sym- metric duality result was generalized by Bazaraa and Goode  to arbitrary con es. Kim et al.  formulated a pair of multiobjective symmetr ic dual programs for pseu- doinvex functions and arbitrary cones. The weak, strong, converse and self duality theorems were established for that pair of dual models. The study of second order duality is sign ificant due to the computational ad vantage over first order duality as it provides tighter bounds for the value of the objective function when approximations are used (see Hou and Yang , Yang et al. [6,7], Yang et al. ). Hou and Yang  introduced a pair of second order symmetric dual non-differentiable programs and second order F-pseudoconvex and proved the weak and strong duality theorems for these second order symmetric dual programs under the F-pseudoconvex assumption. Suneja et al.  formulated a pair of multiobjective symmetric dual programs over arbitrary cones for cone-convex func- tions. The weak, strong, converse and self-duality theo-rems were proved for these programs. Yang et al.  formulated a pair of Wolf type non-dif ferentiable second order symmetric primal and dual problems in mathemati- cal programming. The weak and strong duality theorems were established under second order F-convexity assump- tions. Symmetric minimax mixed integer primal and dual problems were also investigated. Khurana  intro-duced cone-pseudoinvex and strongly cone-pseudoinvex functions, and formulated a pair of Mond-Weir type symmetric dual multiobjective programs over arbitrary cones. The duality theorems and the self-dual theorem were established under these functions. Yang et al.  proved the weak, strong and converse duality theorems under F-convexity conditions for a pair of second order symmetric dual programs. Yang et al.  established various duality results for nonlinear programming with cone constraints and its four dual models introduced by Chandra and Abha . In this paper, we present new definitions dealing with second order (K, F)-pseudoconvex and second order strongly (K, F)-pseudoconvex functions which are a gen-eralized of cone-pseudoconvex and strongly cone-pseu- doconvex functions. We suggest a pair of multiobjective nonlinear second order symmetric dual programs. More- over, we establish the duality theorems using the above generalization of cone-pseudoconvex functions. Finally, a self-duality theorem is given by assuming the skew- M. ABD EL-H. KASSEM 1237,symmetric of the functions. 2. Notations and Definitions The following conventions for vectors in Rn will be used: ,1,2,,iixyxyi n ,1,2,, ,iixyxyi≦≦ n1, 2,, iixyxyi n ≦, but xy. A general multiobjective nonlinear programming problem can be expressed in the form: (P):  12min,,, mfxfxfxfx 0,1,2, ,njsubject toxXxRgxjk ≦ where :and:nm nk.fRR gRR Definition 1. A point xX is said to be an efficient (or a Pareto optimal) solution of problem (P) if there exists no other xX such that  ,fxfx  ,iifxf≦x1, but 2, ,im.fxfx Recall the following three definitions aiming to give the desired definition (i.e., Definition 5). Definition 2. [5,7,8] A functional :nnFXXRRXR  is sublinear in its third component if, for all ,,xuX,;,; X 1) 1212 12,;, nFxua aFxua≦FxuaaaR; and 2) ,;,; ,nFxuaF xuaaRR 0, ≧. For notational convenience, we write ,,;xuFaFxua. Let K be a closed convex pointed cone in with mRint K and :nmfRR be a differentiable func-tion. Definition 3. [4,10,12] The polar cone *K of K is defined as *0.mTKzRxz xK ≧ Definition 4.  The function f is said to be second- order F-pseudoconvex at if uX,nxpXR ,    ,01.2xu uuuTuuFfu fupfx fupfup≧ f is second-order F-pseudoconcave if is second-order F-pseudoconvex. fNow, we are in position to give our definitions of sec-ond-order (K, F)-pseudoconvex functions and second- order strongly (K, F)-pseudoconvex functions. Definition 5. The function f is said to be second-order (K, F)-pseudoconvex at if uX,,nxpXR  ,int1int ;2xu uuuTuuFfu fupKfxfup fupK  and the function f is said to be second-order strongly (K, F)-pseudoconvex at uX if ,,nxpXR  ,int1.2xu uuuTuuFfufupKfxfup fupK   f is second-order (K, F)-pseudoconcave if f is second- order (K, F)-pseudoconvex and f is second-order strongly (K, F)-pseudoconcave if f is second-order strongly (K, F)-pseudoconvex. Remark 1. If p = 0 and ,,xuFfuxu fu  where ,,nnX R:XX R the second-order strongly (K, F)-pseudoconvex functions and second-order (K, F)-pseudoconvex functions reduce to strongly K- pseudoinvex functions and K-pseudoinvex functions de-fined by Khurana [10 ] . Remark 2. Every second-order strongly (K, F)-pseu- doconvex function is second-order (K, F)-pseudoconvex but converse is not necessarily true as can be seen from the following example. Example 1. Let 2,4 ,02,,1xxKxyxy xfxx xep,  ≦≦ and 3,.xuFAAxu It can be seen that fx is second-order (K, F)-pseu- doconvex at u 0 but fx is not second-order strongly (K, F)-pseudoconvex at u 0 because for x 1 ,intxu uuuFfufup K  and  1.2Tuufxfup fupK  The following example show that a function which is second-order strongly (K, F)-pseudoconvex but not sec-ond-order F-pseudoconvex where K is a closed convex cone. Example 2. Let 22,,,3, ,1Kxyyxyxxfxxxxp ≧≧ ≧0, and 3,.xuFAAxu Copyright © 2011 SciRes. AM 1238 M. ABD EL-H. KASSEM Then fx is second-order strongly (K, F)-pseudoco- nvex at u 0. However, fx is not F-pseudoconvex at u 0, because for x 3,  ,0xu uuuFfu fup  but  11.2Tuufxfupfup We formulate the following multiobjective nonlinear symmetric dual problems: (SP): 1min ,,2TTyyfxypfxy p *2,,TTyyy ,subjecttofx yfx ypC  (1) ,,TTTyyyyfxy fxyp≧0, (2) *1,KxC. (SD): 1max ,,2TTuufuvrf uvr *1,,TTuuu ,subjecttofu vfu vrC 0, (3) (, ,TT Tuuuufuv fuvr≦ (4) *2,KvCm, where :nlfRR R is a thrice differentiable func-tion of x and y. C1 and C2 are closed convex cones- with nonempty interiors in Rn and Rl respectively. For exam-ple, the nonnegative orthant 0nxRx≧ is a convex cone). and are positive polar cones of C1 and C2, respectively. K is a closed convex pointed cone in Rm such that *1C*2Cint K and *K is its positive polar cone. ,Tyfxy and yy,Tfxy,T are the gradient and the Hessian matrix of fxy with respect to y, respectively. Similarly, ,Tufxy and ,Tuufxy are the gradient and the Hessian matrix of ,Tfxy with respect to u, respectively. Observe that if then (SP) and (SD) be-comes (P) and (D) given by Khurana , respectively. 0,pr 3. Symmetric Duality Now, we establish the symmetric duality theorems for the problems (SP) and (SD) as follows. Theorem 1. (Weak duality). Let (,, ,)xyp be feasible solution for the problem (SP) and (,, ,)uvr be feasible solution for the problem (SD). Suppose there exist sub-linear functionals :nFXXRR nXR and :lGY YRR lYR satisfying: ,T1,xuFaau 1CaC (5) ,Tby2.CbCvyGb(., )2 (6) Furthermore, assume that either 1) fv(., ) is second-order (K, F)-pseudoconvex at u and fv(., ) is second-order (K, G)-pseudoconcave at y; or 2) fv(., is second-order strongly (K, F)-pseudoco- nvex at u and )fv is second-order strongly (K, G)- pseudoconcave at y. Then 1,,212TTuuTTyyr f,,intfuv uvrfxyfxy pK p Proof: Suppose th e contrary, i.e., 1,,21,,int2fuv uvrTTuuTTyyr ffxyfxy pK p (7) Since (,, ,)xyp is a feasible solution for the prob-lem (SP) and (,,uv ,)r is a feasible solution for the problem (SD), we h av e: By the dual constraint (3), the vector ,TTuuuafovfu,vr  belongs to , and so by (5) we get from (4) *1C,,,,,TTxu uuuTT TuuuFfuvfuvrufuv fuvr ≧≧0. This gives ,,,TTxu uuuint.FfuvfuvrK  (*) In a similar fashion, ,,,TTvy yyyGfxyfxyp ≧0 for the vector ,,TTyyybfxyfxyp  in and so *2,C,,,TTvy yyyGfxyfxyp intK. (**) (1) Since the function (., )fv is second-order (K, F)- pseudoconvex at u, relation (*) implies to  1,, ,in2TTuu tfxvfuvrfuvrK. (8) Similarly from (1) and (6), where Copyright © 2011 SciRes. AM M. ABD EL-H. KASSEM Copyright © 2011 SciRes. AM 1239C*2,,TTyyybfxyfxyp , we get ,,,TTvy yyyGfxyfxyp int.K (**) interiors in Rm and Rk, respectively, we will make use the following proposition which gives generalized form of Fritz-John optimality conditions established by suneja et al.  for a point to be a weak minimum point of the following multiobjective nonlinear programming prob-lem: (MONLP): 12min,,, mKfxfxf xfx  Also, since the function is second-order (K, G)- pseudo conc ave at y (i.e., is second-order (K, G)- pseudoconvex at y), we have (,.)fx(,.)fx 12, ,...,nksubject toxXxRGxgxgxgx Q   1,, ,in2TTyy tfxvfxypfxy pK . (9) Definition 6. [6,8] A point xX is said to be a weak minimum point of (MONLP) if for every xX, intfxfx K. Adding (8) and (9 ), we get 1,,21,),int,2TTuuTTyyfuvrf uvrfxypfxypK  Proposition. . If xX is a weak minimum point of (MONLP), then there exist K, Q not both zero such that  0,TTfxGxxxxC≧ this contradicts (7). Hence, the result follows for (1). 0.TGx (2) From (*) and since the function is sec-ond-order strongly (K, F)-ps eudoconvex at u, we get (., )fv 1,,,2TTuufxvfuvrf uvrK . (10) Theorem 2. (Strong duality). Let ,,,xyp be a weak minimum point for the problem (SP): fix  and rr in the problem (SD). Assume that 1) the matrix ,Tyyfxy is nonsingular, Also, from (**) and since the function is sec-ond-order strongly (K, G)-pseudoconcave at y (i.e., is second-order strongly (K, G)-pseudoconvex at y), we get (,.)fx(,.)fx2) the set ,,1,2,,yifxyi m is linearly in-dependent, 3) ,,TTyyyfxy fxyp0 , 1,, ,2TTyyfxyfxvpfxy pK . (11) then ,,, 0xyp r is feasible solution for the problem (SD) and the objective values of the problems (SP) and (SD) are equal. Adding (10) and (11), we get 1,,21,,2TTuuTTyyfuvrf uvrfxypfxyp K , Furthermore, under the assumptions of Theorem 1, ,,, 0xyp r is a weak maximum point of the problem (SD). Proof: Since ,,,xyp is a weak minimum point for the problem (SP), by the Fritz-John conditions of the above proposition , there exist K, , 22CC0, ,, 0, such that for each 1xC, K, , 0p≧this contradicts (7). Hence, the result follows for (2). Therefore, the proof is completed. For the closed convex cones K and Q with nonempty  1,, ,21,,)21,,2,TTTT Txxy xyyTTTTTyyyyyyTTyyyTTyyfxyyf xyypfxypxxfxyypf xyypf xypyyyfxyyp fxypyp fxy               ,0pp≧ (12) ,,TT Tyyyfxy fxyp 0 (13) M. ABD EL-H. KASSEM Copyright © 2011 SciRes. AM 1240 ,,TT Tyyyyfxy fxyp 0 (14) Substituting 1xxC, 2yyC and pp in the inequality (12), we have   *1,,21,,2TTyyyTTyyyyfxyyp fxypyfxyyp fxyp    ≧00,K this can be written in the following form 1,, ,02TTT TTyyy yyyfxyfxyppfxyp  . (15) Subtract (14) from (13), we have ,,TTTyyyyfxy fxyp  0, then Equation (15) becom e s ,TTyypfxyp0 (16) Similarly, if xx, yy, and  in Equation (12), we ge t ,0TTyyyp fxy   using condition (1), we get yp. (17) We claim that 0. Indeed, if 0, then (17) implies y. (18) Therefore, equality (12) becomes ,,0,,0TT TyyyTT Tyyy ,pfxyfxyPy yyRfxy fxyP    ≦ and from the condition (3), we have 0 (19) Therefore, (18) becomes 0 (20) Hence, , which contradicts the assump-tion . Therefore, ,, 0,, 00 and Equation (16) take the form ,0TTpfxypyy and since ,Tyyfxy is nonsingular (condition (1)) we get 0p (21) So, Equation (17) becomes y (22) Substituting from Equations (21), (22) and xx in the inequality (12), we get ,0,0.TpyTyfxyy yyRfxy ≧ And since ,y (23) Using (21), (22) and (23) in (12), we have 1,0,,0Txxfxyx xKfxyx xx C ≧≧ (24) As 1 is closed convex cone, C11,xxCxC  hence from (24) and K, we get 1fxy is linearly independent (con-dition (2)), we get ,0TTxxfxy Cx≧ and by using (21), we get 1,,0TT Txxx,xfxyfxyp xC ≧ this implies that 1,,TTxxx.fxyfxyp C  Similarly, by letting 0x in (24) we have ,,0TT Txxxxfxyfxyp≦. Thus ,,, 0xy p is feasible solution for the prob-lem (SD) and the values of the objective function for the problems (SP) and (SD) are same at ,,, 0xy p. M. ABD EL-H. KASSEM 1241We will now show that ,,, 0uv r is a weak maximum point for the problem (SD). Suppose not, then there exists a feasible solution ,,, 0uv r such that 1,21,,2TTuuTTyyfuvrf uvr,int,fxypf xypK  which contradicts the weak duality theorem. Theorem 3. (Converse duality). Let ,,,uv r be a wea k maximum point f or the prob lem (SD). Fix , pp in the problem (SP). Assume that 1) the matrix ,Tuufuv is nonsingular, 2) the set ,,1,2,,uifuv im is linearly in-dependent, 3) ,,TTuuufuvfuvr0, then ,,, 0uv r is feasible solution for the problem (SP) and the objective values of the problems (SP) and (SD) are equal. Furthermore, under the assumptions of Theorem 1, ,,, 0xy p is a weak minimum point for the prob-lem (SP). The proof follows on the same lines of Theorem 2. 4. Self Duality A nonlinear programming problem is said to be self-dual if, when the dual is recast in the form of the primal, the new problem so obtained is the same as the primal prob-lem. Now we establish the self-duality of the problem (SP). So, we assume that , nl,,fxyf yxCC (i.e., f is skew-symmetric) and 12, . prThe dual problem (SD) may be rewritten as a miniza-tion form: (SD)’: 1min ,,2TTuufuvrf uvr 1,,TTuuu ,subjecttofuvfuvrC  ,,TT Tuuufuvfuvr≦0, 2,.KvC Since  ,,,fuvf vu ,,uvfuvf vu, and uu,vvfuvf vu, the above dual problem (SD)’ reduces to (SD)”: 1min ,,2TTvvfvurfvu r 1,,TTvvv,,TT Tvvvufvu fvur≧0, 2,.KvC Therefore, this dual problem (SD)’ is formally identi-cal to the primal problem (SP), that is, the objective and constraint functions of the problems (SP) and (SD)” are identical. Hence, this problem is self dual. Consequently, the feasibility point ,,, 0xyp r for the primal problem (SP) implies the feasibility point ,, ,yx 0pr for the dual problem (SD) and vice versa. Theorem 4. (Self duality). Under the assumptions of the weak duality theorem and the point ,,, 0xy p is a weak minimum point for the problem (SP), we as-sume that 1) the primal problem (SP) is self dual, 2) the matrix ,Tyyfxy is nonsingular, 3) the set ,,1,2,,yifxy im is linearly in-dependent, 4) ,,TTyyyfxy fxyp0 , then ,,, 0xy p is a weak minimum point and a weak maximum point, respectively for both the problems (SP) and (SD) and the common optimal value is zero. Proof: From the strong duality theorem ,,, 0xy p is a weak maximum point for the problem (SD) and the optimal values of the problems (SP) and (SD) are id enti-cal. By using the self duality, we have ,,, 0xy p is feasible for both problems (SP) and (SD) and using the theorems 1-3, we get that it is optimal for both the prob-lems (SP) and (SD). To show that the common optimal value is zero, since f is skew symmetric, we have  ,,,,yy xxfxy fxy,.fxyf xy Hence, 1,,21,,21,,2TTyyTTxxTTyyfxypf xyp,fyxpfyx pfxypf xyp  and so 1,,21,,2TTyyTTxxfxypf xypfyxpf yxp0.  5. Conclusions ,subjecttofv ufv urC  A pair of symmetric dual programs has been formulated Copyright © 2011 SciRes. AM M. ABD EL-H. KASSEM Copyright © 2011 SciRes. AM 1242 by considering the optimization with respect to an arbi-trary cone under th e assumptions of second order ( K, F)- pseudoconvex and second order strongly (K, F)-pseu- doconvex functions. The results may be further general-ized by relaxing the condition of cone-pseudoconvex functions to cone-pseudobonvex functions. 6. References  W. S. 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