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Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity

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Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity Applied Mathematics, 2011, 2, 1236-1242 doi:10.4236/am.2011.210172 Published Online October 2011 (http://www.SciRP.org/journal/am) Multiobjective Nonlinear Symmetric Duality Involving Generalized Pseudoconvexity 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 es- tablished. 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- Copyright © 2011 SciRes. AM M. ABD EL-H. KASSEM 1237 symmetric of the functions. FfufupKxu, u uu int 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,,pXRn 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,,uXX Example 1. Let n 1) Fxua,; 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,,1x 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 func- tion. 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 sec- ond-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 . Copyright © 2011 SciRes. AM 1238 M. ABD EL-H. KASSEM 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 aauT 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 fuvr subject toyyy f x,, y f x y p C2 , 2 (1) 1 TT f xy,,pyy f xy pint K 2 TTT yfxyfxypyy,,y ≧0, (2) Proof: Suppose the contrary, i.e., * K , xC1 . 1 TT fuv,,ruu fuvr 2 1 TT (7) (SD): maxf uv , ruu f uv , r 1 2 TT f xy,,intpyy f xy p K 2 subject toTT f u,, v f u v r C* , (3) uuu 1 Since (,x yp , , ) is a feasible solution for the prob- lem (SP) and (,,uv,)r is a feasible solution for the ufuvfuvrTT(,T ,≦0, (4) uu uproblem (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 FfuvfuvrTT,, 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, ufuv,, 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 GfxyfxypTT,, ≧0 u uu vy, y yy the gradient and the Hessian matrix of T f xy, with respect to u, respectively.
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