Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity

Home , Duality (optimization)

Applied Mathematics, 2011, 2, 1236-1242 doi:10.4236/am.2011.210172 Published Online October 2011 (http://www.SciRP.org/journal/am)

Mohamed Abd El-Hady Kassem Mathematical Department, Faculty of Science, Tanta University, Tanta, Egypt E-mail: [email protected] Received April 8, 2011; revised June 9, 2011; accepted June 16, 2011

Abstract

The purpose of this paper is to introduce second order (K, F)-pseudoconvex and second order strongly (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 established. Finally, a self duality theorem is given.

Keywords: Multiobjective Programming, Second-Order Symmetric Dual Models, Duality Theorems, Pseudoconvex Functions, Cones

1. Introduction rems were proved for these programs. Yang et al. [6] formulated a pair of Wolf type non-differentiable second Duality is an important concept in the study of nonlinear order symmetric primal and dual problems in mathemati- programming. Symmetric duality in nonlinear program- cal programming. The weak and strong duality theorems ming in which the dual of the dual is the primal was first were established under second order F-convexity assump- introduced by Dorn [1]. Subsequently Dantzig et al. [2] tions. Symmetric minimax mixed integer primal and dual established symmetric duality results for convex/concave problems were also investigated. Khurana [10] intro- functions with nonnegative orthant as the cone. The sym- duced cone-pseudoinvex and strongly cone-pseudoinvex metric duality result was generalized by Bazaraa and functions, and formulated a pair of Mond-Weir type Goode [3] to arbitrary cones. Kim et al. [4] formulated a symmetric dual multiobjective programs over arbitrary pair of multiobjective symmetric dual programs for pseu- cones. The duality theorems and the self-dual theorem doinvex functions and arbitrary cones. The weak, strong, were established under these functions. Yang et al. [8] converse and self duality theorems were established for proved the weak, strong and converse duality theorems that pair of dual models. under F-convexity conditions for a pair of second order The study of second order duality is significant due to symmetric dual programs. Yang et al. [7] established the computational advantage over first order duality as it various duality results for nonlinear programming with provides tighter bounds for the value of the objective cone constraints and its four dual models introduced by function when approximations are used (see Hou and Chandra and Abha [11]. Yang [5], Yang et al. [6,7], Yang et al. [8]). In this paper, we present new definitions dealing with Hou and Yang [5] introduced a pair of second order second order (K, F)-pseudoconvex and second order symmetric dual non-differentiable programs and second strongly (K, F)-pseudoconvex functions which are a gen- order F-pseudoconvex and proved the weak and strong eralized of cone-pseudoconvex and strongly cone-pseu- duality theorems for these second order symmetric dual doconvex functions. We suggest a pair of multiobjective programs under the F-pseudoconvex assumption. Suneja nonlinear second order symmetric dual programs. More- et al. [9] formulated a pair of multiobjective symmetric over, we establish the duality theorems using the above dual programs over arbitrary cones for cone-convex func- generalization of cone-pseudoconvex functions. Finally, tions. The weak, strong, converse and self-duality theo- a self-duality theorem is given by assuming the skew-

1 T 2. Notations and Definitions f xfup    uu fup int K ; 2  The following conventions for vectors in Rn will be used: and the function f is said to be second-order strongly (K, F)-pseudoconvex at uX if x,,pXRn x yxyiii ,1,2,,   n, x≦≦yxyi,1,2,, n, ii Fxu,  u fu  uu fup int K ≦,  x yxyiii 1, 2, , n but x  y . A general multiobjective nonlinear programming 1 T f xfup  uu fupK . problem can be expressed in the form: 2 f is second-order (K, F)-pseudoconcave if  f is second- (P): min f xfxfxfx ,,, 12  m order (K, F)-pseudoconvex and f is second-order strongly (K, F)-pseudoconcave if  f is second-order strongly subject to x X x Rn g x≦0, j  1,2, , k j  (K, F)-pseudoconvex. Remark 1. If p = 0 and F fu xu,  fu where f :and:RRnm gRR nk. xu,    where  :XX Rnn,,XR the second-order Definition 1. A point x  X is said to be an efficient   strongly (K, F)-pseudoconvex functions and second-order (or a Pareto optimal) solution of problem (P) if there (K, F)-pseudoconvex functions reduce to strongly K- exists no other x  X such that f xfx , pseudoinvex functions and K-pseudoinvex functions de- f xf≦ x, im 1, 2,, but f xfx .  ii     fined by Khurana [10]. Recall the following three definitions aiming to give Remark 2. Every second-order strongly (K, F)-pseu- the desired definition (i.e., Definition 5). doconvex function is second-order (K, F)-pseudoconvex Definition 2. [5,7,8] A functional but converse is not necessarily true as can be seen from nn F : XXR  RXR   is sublinear in its third the following example. component if, for all x,,uXX Example 1. Let n 1) Fxua,; a≦ Fxua ,; Fx,;ua aa ,  R; 12 1 2 12  x  and Kxyxy,4≦≦  ,0 x, 2) F xu,; a F xua ,; a Rn ,  R,  ≧0. 2 For notational convenience, we write fx x2  xe,,1x p 

Fxu, aFxua  ,; . and

m 3 Let K be a closed convex pointed cone in R with Fxu,  AAxu . int K   and f : RRn m be a differentiable function. It can be seen that f  x is second-order (K, F)-pseu- Definition 3. [4,10,12] The polar cone K * of K is doconvex at u 0 but f  x is not second-order strongly defined as (K, F)-pseudoconvex at u 0 because for x 1

* mT F fu   fup int K K zRxz≧0.  xK xu,  u uu and

Definition 4. [5] The function f is said to be second- 1 T n f xfup  fupK . order F-pseudoconvex at uX if x, pXR , 2 uu ≧ The following example show that a function which is Ffufupxu,  u uu  0 second-order strongly (K, F)-pseudoconvex but not sec- 1 T fx fu puu fup. ond-order F-pseudoconvex where K is a closed convex 2 cone. f is second-order F-pseudoconcave if  f is second-order Example 2. Let F-pseudoconvex. Kxyyxyxx ,,, ≧≧ ≧0, Now, we are in position to give our definitions of second-order (K, F)-pseudoconvex functions and second- fx x22 3, xx , p  1 order strongly (K, F)-pseudoconvex functions. and Definition 5. The function f is said to be second-order n 3 (K, F)-pseudoconvex at uX if x,,pXR Fxu,  AAxu .

Then f  x is second-order strongly (K, F)-pseudoco- GY:  Y Rl  R YR l  satisfying: nvex at u 0. However, f x is not F-pseudoconvex   F aauT CaC, (5) at u 0, because for x 3, xu,  11 Gb  byT CbC. (6) Ffufupxu,  u uu  0 vy,22 but Furthermore, assume that either 1) f (.,v ) is second-order (K, F)-pseudoconvex at u 1 T and f (.,v ) is second-order (K, G)-pseudoconcave at y; f1 xfup  uu fup. 2 or We formulate the following multiobjective nonlinear 2) f (.,v ) is second-order strongly (K, F)-pseudoco- symmetric dual problems: nvex at u and f (.,v) is second-order strongly (K, G)- pseudoconcave at y. 1 TT (SP): minf xy , pyy  f xy , p Then 2 1 TT TT* fuv,,ruu  fuvr subject toyyy f x,, y  f x y p  C2 , 2   (1) 1 TT f xy,,pyy  f xy pint K 2 TTT yfxyfxypyy,,y ≧0, (2) Proof: Suppose the contrary, i.e., *  K , xC1 . 1 TT fuv,,ruu  fuvr 2 1 TT (7) (SD): maxf uv , ruu  f uv , r 1 2 TT f xy,,intpyy  f xy p K 2 subject toTT f u,, v f u v r C* , (3) uuu  1 Since (,x yp , , ) is a feasible solution for the problem (SP) and (,,uv,)r is a feasible solution for the ufuvfuvrTT(,T ,≦0, (4) uu uproblem (SD), we have:

* By the dual constraint (3), the vector  K , vC, TT * 2 afovfu ,  ,vr belongs to C ,  uuu    1 where f : RRnl Rm is a thrice differentiable func- and so by (5) we get from (4) tion of x and y. C and C are closed convex cones- with 1 2 FfuvfuvrTT,, nonempty interiors in Rn and Rl respectively. For exam- xu,  u  uu n ple, the nonnegative orthant xRx ≧0 is a convex ≧≧ufuvfuvrTT,,  T 0. * * uuu  cone). C1 and C2 are positive polar cones of C1 and m C2, respectively. K is a closed convex pointed cone in R This gives such that int K   and K * is its positive polar cone. TT T T Fxu, ufuv,,  uu fuvr int K. (*)  y  f xy, and  yy  f xy,  are the gradient  T and the Hessian matrix of  f xy,  with respect to y, In a similar fashion, respectively. Similarly,  T f xy, and  T f xy, are GfxyfxypTT,,   ≧0 u  uu  vy,  y  yy the gradient and the Hessian matrix of T f  xy,  with respect to u, respectively. for the vector Observe that if pr0, then (SP) and (SD) be-  TT bfxyfx yyy,,  yp comes (P) and (D) given by Khurana [8], respectively.   * in C2 , and so 3. Symmetric Duality TT Gfxyfxypvy, y,,  yy intK. (**)  Now, we establish the symmetric duality theorems for (1) Since the function f (.,v ) is second-order (K, F)- the problems (SP) and (SD) as follows. pseudoconvex at u, relation (*) implies to Theorem 1. (Weak duality). Let (,x yp , , ) be feasible 1 TT solution for the problem (SP) and (,,uv ,) r be feasible f xv,, f  uv ruu  f uv ,in r t K. (8) 2 solution for the problem (SD). Suppose there exist sub-  linear functionals F : XXRn R X  Rn and Similarly from (1) and (6), where

m k TT* interiors in R and R , respectively, we will make use the bfxyfxyp yyy,, C2 , following proposition which gives generalized form of we get Fritz-John optimality conditions established by suneja et al. [9] for a point to be a weak minimum point of the GfxTT,,yfx yp intK. vy,   y yy following multiobjective nonlinear programming prob- (**) lem:

Also, since the function fx(,.) is second-order (K, G)- (MONLP): K min fx  fx12,,, f x fm  x pseudoconcave at y (i.e.,  fx(,.) is second-order (K, G)- n pseudoconvex at y), we have subject to x X x R  G x

1 TT  g12xgx,  ,..., gxk   Q f xv,, f  xy pyy  f xy ,in p t K. (9) 2   Definition 6. [6,8] A point x  X is said to be a Adding (8) and (9), we get weak minimum point of (MONLP) if for every x  X , f xfxint K. 1 TT     fuv,, ruu  f uvr Proposition. [9]. If x  X is a weak minimum point 2 of (MONLP), then there exist   K ,   Q not 1 TT fxy,) pyy  f xyp ,  int, K both zero such that 2 f xGxxxxTT  ≧0, C this contradicts (7). Hence, the result follows for (1).  (2) From (*) and since the function fv(., ) is sec-  T Gx   0. ond-order strongly (K, F)-pseudoconvex at u, we get Theorem 2. (Strong duality). Let x,,,yp be a 1 TT weak minimum point for the problem (SP): fix    fxv,, fuv  ruu  f uvr ,  K. (10) 2 and rr in the problem (SD). Assume that 1) the matrix T f xy, is nonsingular, Also, from (**) and since the function fx(,.) is sec-  yy    ond-order strongly (K, G)-pseudoconcave at y (i.e., 2) the set yifxyi ,,1,2,,  m is linearly in-  fx(,.) is second-order strongly (K, G)-pseudoconvex dependent, at y), we get 3)  TTfxy,, fxyp0, yyy    1 TT  f xy,, f xv pyy  f xy , p  K. (11) then  xy,,, p r 0 is feasible solution for the 2  problem (SD) and the objective values of the problems Adding (10) and (11), we get (SP) and (SD) are equal. Furthermore, under the assumptions of Theorem 1, 1 TT fuv,, ruu  f uvr xy,,, p r 0 is a weak maximum point of the 2   problem (SD). 1 TT fxy,, pyy  f xyp  K,Proof: Since  x,,,yp  is a weak minimum point 2 for the problem (SP), by the Fritz-John conditions of the    this contradicts (7). Hence, the result follows for (2). above proposition, there exist   K ,  CC22,  Therefore, the proof is completed.   0 , ,,  0, such that for each x C1 ,   K , For the closed convex cones K and Q with nonempty p≧0 ,

 T  TTT 1 T xxyxyyfxy,, y f xy  y p f xyp ,  x  x  2   T  T T TT1  yyyyyyfxy,,)  y  p   f xy   y  p  f xyp,  y  y  2  (12) T T 1 yfxyyp yyy,,    fxyp  2 yp T  T fxy, pp ≧0 yy  TTT  yyfxy,,y  fxyp0 (13) 

Substituting x  xC , yyC and pp in yfxyfxypTT,, T 0 (14) 1 2 yy y the inequality (12), we have

T T 1  ≧ * yfxyypyy,,   y fxyp 0  K 2

T T 1 yfxyyp yy,,   y fxyp 0, 2 this can be written in the following form

T TT1 TT yfxyfxyppfxypyyyyy  ,,      , . 0 (15) 2 Subtract (14) from (13), we have yfxyfxypT  TT,,  0, yyy    then Equation (15) becomes using condition (1), we get

TT   y  p . (17) pfxypyy  , 0 (16) We claim that   0 . Indeed, if   0 , then (17) Similarly, if x  x , y  y , and    in Equation implies (12), we get    y . (18) T T yp  yy   fxy,0  Therefore, equality (12) becomes

TTf xy,,0 T f xyP y y≦ y Rp  yy y 

TTfxy,,0  T fxyP , yy y and from the condition (3), we have   (23)   0 (19) Using (21), (22) and (23) in (12), we have Therefore, (18) becomes T ≧  x f xy,0 x x, K   0 (20) (24) x f xy,0 x  x≧  x  C1 Hence, ,, 0, which contradicts the assumption ,, 0. Therefore,   0 and Equation (16) As C1 is closed convex cone, x xCxC11  , TT  take the form pfxypyy  ,0 and since hence from (24) and   K , we get T TT ≧  yy  f xy, is nonsingular (condition (1)) we get x x  fxy ,0 x C1 and by using (21), we p  0 (21) get xTTfxy,,0T fxyp≧ xC, So, Equation (17) becomes xx  x  1    y (22) this implies that Substituting from Equations (21), (22) and x  x in TT xxxf  xy,,  f xy p C1 . the inequality (12), we get Similarly, by letting x  0 in (24) we have T p y f xy,0 y  y≧  y R  TTT xfxyfxypxx,,x≦0. T     y fxy,0.  Thus  xy,,, p 0 is feasible solution for the prob- And since  y f xy,  is linearly independent (con- lem (SD) and the values of the objective function for the dition (2)), we get problems (SP) and (SD) are same at xy,,, p 0.

We will now show that uv,, , r 0 is a weak TTT ≧ ufvufvurvv,,v 0, maximum point for the problem (SD).  Suppose not, then there exists a feasible solution   K ,.vC2 uv,, , r 0 such that Therefore, this dual problem (SD)’ is formally identi- 1 TT cal to the primal problem (SP), that is, the objective and fuv, ruu  f uvr, 2 constraint functions of the problems (SP) and (SD)” are 1 TT identical. Hence, this problem is self dual. Consequently, f xy,, pyy  f xy p int K, 2 the feasibility point  xy,,, p r 0 for the primal problem (SP) implies the feasibility point  yx,, , which contradicts the weak duality theorem. pr  0 for the dual problem (SD) and vice versa. Theorem 3. (Converse duality). Let uv,, , r be a Theorem 4. (Self duality). Under the assumptions of weak maximum point for the problem (SD). Fix    , the weak duality theorem and the point xy,,, p 0 pp in the problem (SP). Assume that is a weak minimum point for the problem (SP), we as- T 1) the matrix uu  f uv,  is nonsingular, sume that 1) the primal problem (SP) is self dual, 2) the set uif uv,, i 1,2,, m is linearly in- T dependent, 2) the matrix  yy  f  xy,  is nonsingular, 3) TTfuv,, fuvr0, f xy,,1,2,, i m uuu  3) the set yi   is linearly in- then uv,, , r 0 is feasible solution for the problem dependent, 4)  TTfxy,, fxyp0, (SP) and the objective values of the problems (SP) and yyy   (SD) are equal. then  xy,,, p 0 is a weak minimum point and a Furthermore, under the assumptions of Theorem 1, weak maximum point, respectively for both the problems xy,,, p 0 is a weak minimum point for the prob- (SP) and (SD) and the common optimal value is zero. lem (SP). Proof: From the strong duality theorem xy,,, p 0 The proof follows on the same lines of Theorem 2. is a weak maximum point for the problem (SD) and the optimal values of the problems (SP) and (SD) are identi- 4. Self Duality cal. By using the self duality, we have xy,,, p 0 is feasible for both problems (SP) and (SD) and using the A nonlinear programming problem is said to be self-dual theorems 1-3, we get that it is optimal for both the prob- if, when the dual is recast in the form of the primal, the lems (SP) and (SD). new problem so obtained is the same as the primal prob- To show that the common optimal value is zero, since lem. f is skew symmetric, we have Now we establish the self-duality of the problem (SP). So, we assume that nl , f xy,, f yx (i.e., fxy ,,  fxy,   f is skew-symmetric) and CC , pr . 12 yy f xy,,xx f xy. The dual problem (SD) may be rewritten as a miniza- Hence, tion form: 1 TT 1 TT fxy,, pyy  f xyp (SD)’: minf uv , ruu  f uv , r 2 2 1 TT TT f yx,, pxx  f yx p subject touuu f u,, v f u v r C1 , 2

TT T 1 TT ufuvfuvr,, ≦0, f xy,,  p yy  f xy p,  uu   2  and so  K ,.vC2 1 TT fxy,, pyy  f xyp Since f uv,, f vu, f uv,, f vu 2   uv   and uu f uv,vv f  vu,, the above dual problem 1 TT  fyx,, pxx  f yxp 0. (SD)’ reduces to 2

1 TT (SD)”: minf vu , rvv  f vu , r 2 5. Conclusions

subject toTT f v,, u  f v u r  C , vvv 1 A pair of symmetric dual programs has been formulated

Copyright © 2011 SciRes. AM 1242 M. ABD EL-H. KASSEM by considering the optimization with respect to an arbi- Programming with F-Convexity,” European Journal of trary cone under the assumptions of second order (K, F)- Operational Research, Vol. 144, 2003, pp. 554-559. pseudoconvex and second order strongly (K, F)-pseu- doi:10.1016/S0377-2217(02)00156-X doconvex functions. The results may be further general- [7] X. M. Yang, X. Q. Yang and K. L. Teo, “Converse Dual- ized by relaxing the condition of cone-pseudoconvex ity in Nonlinear Programming with Cone Constraints,” European Journal of Operational Research, Vol. 170, functions to cone-pseudobonvex functions. 2006, pp. 350-354. doi:10.1016/j.ejor.2004.05.028

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