Applied Mathematics, 2011, 2, 12361242 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 ElHady Kassem Mathematical Department, Faculty of Science, Tanta Universit y, Tanta, Egypt Email: 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 conepseudoconvex and strongly conepseudoconvex 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, SecondOrder 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 [1]. Subsequently Dantzig et al. [2] 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 [3] to arbitrary con es. Kim et al. [4] 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 [5], Yang et al. [6,7], Yang et al. [8]). Hou and Yang [5] introduced a pair of second order symmetric dual nondifferentiable programs and second order Fpseudoconvex and proved the weak and strong duality theorems for these second order symmetric dual programs under the Fpseudoconvex assumption. Suneja et al. [9] formulated a pair of multiobjective symmetric dual programs over arbitrary cones for coneconvex func tions. The weak, strong, converse and selfduality theo rems were proved for these programs. Yang et al. [6] formulated a pair of Wolf type nondif ferentiable second order symmetric primal and dual problems in mathemati cal programming. The weak and strong duality theorems were established under second order Fconvexity assump tions. Symmetric minimax mixed integer primal and dual problems were also investigated. Khurana [10] intro duced conepseudoinvex and strongly conepseudoinvex functions, and formulated a pair of MondWeir type symmetric dual multiobjective programs over arbitrary cones. The duality theorems and the selfdual theorem were established under these functions. Yang et al. [8] proved the weak, strong and converse duality theorems under Fconvexity conditions for a pair of second order symmetric dual programs. Yang et al. [7] established various duality results for nonlinear programming with cone constraints and its four dual models introduced by Chandra and Abha [11]. 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 conepseudoconvex and strongly conepseu 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 conepseudoconvex functions. Finally, a selfduality theorem is given by assuming the skew
M. ABD ELH. KASSEM 1237 , symmetric of the functions. 2. Notations and Definitions The following conventions for vectors in Rn will be used: ,1,2,, ii yxyi n ,1,2,, , ii yxyi≦≦ n 1, 2,, ii yxyi n ≦, but y. A general multiobjective nonlinear programming problem can be expressed in the form: (P): 12 min,,, m xfxfxfx 0,1,2, , nj ubject toxXxRgxjk ≦ where :and: nm nk . RR gRR Definition 1. A point X is said to be an efficient (or a Pareto optimal) solution of problem (P) if there exists no other X such that , xfx , ii xf≦x1, but 2, ,im . xfx Recall the following three definitions aiming to give the desired definition (i.e., Definition 5). Definition 2. [5,7,8] A functional :n n XXRRXR is sublinear in its third component if, for all ,, uX ,;,; X 1) 1212 12 ,;, n xua aFxua≦FxuaaaR ; and 2) ,;,; , n xuaF xuaaRR 0, ≧. For notational convenience, we write ,,; xu aFxua. Let K be a closed convex pointed cone in with m R int K and :nm RR be a differentiable func tion. Definition 3. [4,10,12] The polar cone * of K is defined as *0. mT zRxz xK ≧ Definition 4. [5] The function f is said to be second order Fpseudoconvex at if uX ,n pXR , ,0 1. 2 xu uuu Tuu Ffu fup fx fupfup ≧ f is secondorder Fpseudoconcave if is secondorder Fpseudoconvex. f Now, we are in position to give our definitions of sec ondorder (K, F)pseudoconvex functions and second order strongly (K, F)pseudoconvex functions. Definition 5. The function f is said to be secondorder (K, F)pseudoconvex at if uX ,, n pXR ,int 1int ; 2 xu uuu Tuu Ffu fupK xfup fupK and the function is said to be secondorder strongly (K, F)pseudoconvex at uX if ,, n pXR ,int 1. 2 xu uuu Tuu fufupK xfup fupK f is secondorder (K, F)pseudoconcave if is second order (K, F)pseudoconvex and f is secondorder strongly (K, F)pseudoconcave if is secondorder strongly (K, F)pseudoconvex. Remark 1. If p = 0 and ,, xu fuxu fu where ,, nn X R:XX R the secondorder strongly (K, F)pseudoconvex functions and secondorder (K, F)pseudoconvex functions reduce to strongly K pseudoinvex functions and Kpseudoinvex functions de fined by Khurana [10 ] . Remark 2. Every secondorder strongly (K, F)pseu doconvex function is secondorder (K, F)pseudoconvex but converse is not necessarily true as can be seen from the following example. Example 1. Let 2 ,4 ,0 2 ,,1 x x Kxyxy x fxx xep , ≦≦ and 3 ,. xu AAxu It can be seen that x is secondorder (K, F)pseu doconvex at u 0 but x is not secondorder strongly (K, F)pseudoconvex at u 0 because for x 1 ,int xu uuu fufup K and 1. 2 Tuu xfup fupK The following example show that a function which is secondorder strongly (K, F)pseudoconvex but not sec ondorder Fpseudoconvex where K is a closed convex cone. Example 2. Let 22 ,,, 3, ,1 Kxyyxyxx fxxxxp ≧≧ ≧0, and 3 ,. xu AAxu Copyright © 2011 SciRes. AM
1238 M. ABD ELH. KASSEM Then x is secondorder strongly (K, F)pseudoco nvex at u 0. However, x is not Fpseudoconvex at u 0, because for x 3, ,0 xu uuu Ffu fup but 11. 2 Tuu xfupfup We formulate the following multiobjective nonlinear symmetric dual problems: (SP): 1 min ,, 2 TT yy xypfxy p * 2 ,, TT yyy , ubjecttofx yfx ypC (1) ,, TTT yyy yfxy fxyp ≧0, (2) *1 , xC . (SD): 1 max ,, 2 TT uu uvrf uvr * 1 ,, TT uuu , ubjecttofu vfu vrC 0, (3) (, , TT T uuu ufuv fuvr ≦ (4) *2 , vC m , where :nl RR 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 0 n xRx≧ 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 * 1 C* 2 C int K and * is its positive polar cone. , T y xy and yy , T xy , T are the gradient and the Hessian matrix of xy with respect to y, respectively. Similarly, , T u xy and , T uu xy are the gradient and the Hessian matrix of , T xy with respect to u, respectively. Observe that if then (SP) and (SD) be comes (P) and (D) given by Khurana [8], 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 (,, ,) yp be feasible solution for the problem (SP) and (,, ,)uvr be feasible solution for the problem (SD). Suppose there exist sub linear functionals :n XXRR n R and :l GY YRR l YR satisfying: , T1 , xu aau 1 CaC (5) , T by 2 .CbC vy Gb (., ) 2 (6) Furthermore, assume that either 1) v (., ) is secondorder (K, F)pseudoconvex at u and v (., ) is secondorder (K, G)pseudoconcave at y; or 2) v(., is secondorder strongly (K, F)pseudoco nvex at u and ) v is secondorder strongly (K, G) pseudoconcave at y. Then 1 ,, 21 2 TT uu TT yy r f ,, int fuv uvr xyfxy pK p Proof: Suppose th e contrary, i.e., 1 ,, 21 ,,int 2 fuv uvr TT uu TT yy r f xyfxy pK p (7) Since (,, ,) yp 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 , TT uuu afovfu ,vr belongs to , and so by (5) we get from (4) * 1 C ,,, ,, TT xu uuu TT T uuu Ffuvfuvr ufuv fuvr ≧≧ 0. This gives ,,, TT xu uuuint. fuvfuvrK (*) In a similar fashion, ,,, TT vy yyy Gfxyfxyp ≧0 for the vector ,, TT yyy bfxyfx yp in and so * 2,C ,,, TT vy yyy Gfxyfxyp int K. (**) (1) Since the function (., ) v is secondorder (K, F) pseudoconvex at u, relation (*) implies to 1 ,, ,in 2 TT uu t xvfuvrfuvrK . (8) Similarly from (1) and (6), where Copyright © 2011 SciRes. AM
M. ABD ELH. KASSEM Copyright © 2011 SciRes. AM 1239 C * 2 ,, TT yyy bfxyfxyp , we get ,,, TT vy yyy Gfxyfxyp int.K (**) interiors in Rm and Rk, respectively, we will make use the following proposition which gives generalized form of FritzJohn optimality conditions established by suneja et al. [9] for a point to be a weak minimum point of the following multiobjective nonlinear programming prob lem: (MONLP): 12 min,,, m fxfxf xfx Also, since the function is secondorder (K, G) pseudo conc ave at y (i.e., is secondorder (K, G) pseudoconvex at y), we have (,.)fx (,.)fx 12 , ,..., n k subject toxXxRGx xgxgx Q 1 ,, ,in 2 TT yy t xvfxypfxy pK . (9) Definition 6. [6,8] A point X is said to be a weak minimum point of (MONLP) if for every X , int xfx K. Adding (8) and (9 ), we get 1 ,, 21 ,),int, 2 TT uu TT yy fuvrf uvr fxypfxypK Proposition. [9]. If X is a weak minimum point of (MONLP), then there exist , Q not both zero such that 0, TT xGxxxx C ≧ this contradicts (7). Hence, the result follows for (1). 0. TGx (2) From (*) and since the function is sec ondorder strongly (K, F)ps eudoconvex at u, we get (., )fv 1 ,,, 2 TT uu fxvfuvrf uvrK . (10) Theorem 2. (Strong duality). Let ,,, yp be a weak minimum point for the problem (SP): fix and rr in the problem (SD). Assume that 1) the matrix , T yy xy is nonsingular, Also, from (**) and since the function is sec ondorder strongly (K, G)pseudoconcave at y (i.e., is secondorder strongly (K, G)pseudoconvex at y), we get (,.)fx (,.)fx2) the set ,,1,2,, yi fxyi m is linearly in dependent, 3) ,, TT yyy fxy fxyp 0 , 1 ,, , 2 TT yy xyfxvpfxy 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 ,, 2 1 ,, 2 TT uu TT yy fuvrf uvr fxypfxyp K , Furthermore, under the assumptions of Theorem 1, ,,, 0xyp r is a weak maximum point of the problem (SD). Proof: Since ,,, yp is a weak minimum point for the problem (SP), by the FritzJohn conditions of the above proposition , there exist , , 22 CC 0 , ,, 0 , such that for each 1 C, , , 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 ,, , 2 1 ,,) 2 1 ,, 2 , T T TT T xxy xyy T TTTT yyyyyy T T yyy TT yy fxyyf xyypfxypxx fxyypf xyypf xypyy yfxyyp fxyp yp fxy , 0pp ≧ (12) ,, TT T yyy fxy fxyp 0 (13)
M. ABD ELH. KASSEM Copyright © 2011 SciRes. AM 1240 ,, TT T yyy yfxy fxyp 0 (14) Substituting 1 xC , 2 yyC and pp in the inequality (12), we have * 1 ,, 2 1 ,, 2 T T yyy T T yyy yfxyyp fxyp yfxyyp fxyp ≧0 0, K this can be written in the following form 1 ,, ,0 2 TTT TT yyy yy yfxyfxyppfxyp . (15) Subtract (14) from (13), we have ,, TTT yyy yfxy fxyp 0, then Equation (15) becom e s , TT yy pfxyp 0 (16) Similarly, if x, y, and in Equation (12), we ge t ,0 TT yy yp fxy using condition (1), we get p . (17) We claim that 0 . Indeed, if 0 , then (17) implies . (18) Therefore, equality (12) becomes ,,0 ,,0 TT T yyy TT T yyy , p xyfxyPy yyR fxy 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 ,0 TT pfxyp yy and since , T yy xy is nonsingular (condition (1)) we get 0p (21) So, Equation (17) becomes (22) Substituting from Equations (21), (22) and x in the inequality (12), we get ,0 ,0. T y T y xyy yyR fxy ≧ And since , y (23) Using (21), (22) and (23) in (12), we have 1 ,0, ,0 Tx x xyx xK xyx xx C ≧ ≧ (24) As 1 is closed convex cone, C11 , xCxC hence from (24) and , we get 1 xy is linearly independent (con dition (2)), we get ,0 TT x fxy C x ≧ and by using (21), we get 1 ,,0 TT T xxx, fxyfxyp xC ≧ this implies that 1 ,, TT xxx. xyfxyp C Similarly, by letting 0x in (24) we have ,,0 TT T xxx xfxyfxyp ≦. 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 ELH. KASSEM 1241 We 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 , 2 1 ,, 2 TT uu TT yy fuvrf uvr, int, xypf 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 , T uu uv is nonsingular, 2) the set ,,1,2,, ui uv im is linearly in dependent, 3) ,, TT uuu fuvfuvr 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 selfdual 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 selfduality of the problem (SP). So, we assume that , nl ,, xyf yx CC (i.e., f is skewsymmetric) and 12 , . pr The dual problem (SD) may be rewritten as a miniza tion form: (SD)’: 1 min ,, 2 TT uu uvrf uvr 1 ,, TT uuu , ubjecttofuvfuvrC ,, TT T uu ufuvfuvr ≦0, 2 ,. vC Since ,,, uvf vu ,, uv uvf vu , and uu , vv uvf vu, the above dual problem (SD)’ reduces to (SD)”: 1 min ,, 2 TT vv vurfvu r 1 ,, TT vvv ,, TT T vvv ufvu fvur ≧0, 2 ,. vC 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 , T yy xy is nonsingular, 3) the set ,,1,2,, yi xy im is linearly in dependent, 4) ,, TT yyy fxy 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 13, 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 xx fxy fxy, . xyf xy Hence, 1 ,, 21 ,, 21 ,, 2 TT yy TT xx TT yy fxypf xyp , yxpfyx p xypf xyp and so 1 ,, 21 ,, 2 TT yy TT xx fxypf xyp fyxpf yxp 0. 5. Conclusions , ubjecttofv ufv urC A pair of symmetric dual programs has been formulated Copyright © 2011 SciRes. AM
M. ABD ELH. 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 conepseudoconvex functions to conepseudobonvex functions. 6. References [1] W. S. Dorn, “A Symmetric Dual Theorem for Quadratic Programs,” Journal of the Operations Research Society of Japan, Vol. 2, 1960, pp. 9397. [2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, “Symmet ric Dual Nonlinear Programs,” Pacific Journal of Mathe matics, Vol. 15, No. 3, 1965, pp. 809812. [3] M. S. Bazaraa and J. J. Goode, “On Symmetric Duality in Nonlinear Programming,” Operation Research, Vol. 21, No. 1, 1973, pp. 19. doi:10.1287/opre.21.1.1 [4] D. Sang Kim, Y. B. Yun and W. J. Lee, “Multiobjective Symmetric Duality with Cone Constraints,” European Journal of Operational Research, Vol. 107, No. 3, 1998, pp. 686691. doi:10.1016/S03772217(97)003226 [5] S. H. Hou and X. M. Yang, “On SecondOrder Symmet ric Duality in NonDifferentiable Programming,” Journal of Mathematical Analysis and Applications, Vol. 255, 2001, pp. 491498. doi:10.1006/jmaa.2000.7242 [6] X M. Yang, X. Q. Yang and K. L. Teo, “NonDifferen tiable Second Order Symmetric Duality in Mathematical Programming with FConvexity,” European Journal of Operational Research, Vol. 144, 2003, pp. 554559. doi:10.1016/S03772217(02)00156X [7] X. M. Yang, X. Q. Yang and K. L. Teo, “Converse Dual ity in Nonlinear Programming with Cone Constraints,” European Journal of Operational Research, Vol. 170, 2006, pp. 350354. doi:10.1016/j.ejor.2004.05.028 [8] X. M. Yang, X. Q. Yang, K. L. Teo and S. H. Hou, “Mul tiobjective Second Order Symmetric with FConvexity,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 585591. doi:10.1016/j.ejor.2004.01.028 [9] K. Suneja, S. Aggarwal and S. Davar, “Multiobjective Symmetric Duality involving Cones,” European Journal of Operational Research, Vol. 141, No. 3, 2002, pp. 471 479. doi:10.1016/S03772217(01)002582 [10] S. Khurana, “Symmetric Duality in Multiobjective Pro gramming involving Generalized ConeInvex Functions,” European Journal of Operational Research, Vol. 165, No. 3, 2005, pp. 592597. doi:10.1016/j.ejor.2003.03.004 [11] S. Chandra and A. Abha, “A Note on PseudoInvex and Duality in Nonlinear Programming,” European Journal of Operational Research, Vol. 122, No. 1, 2000, pp. 161 165. doi:10.1016/S03772217(99)000764 [12] M. Kassem, “HigherOrder Symmetric Duality in Vector Optimization Problem involving Generalized ConeInvex Functions,” Applied Mathematics and Computation, Vol. 209, No. 2, 2009, pp. 405409. doi:10.1016/j.amc.2008.12.063
