Computers and Mathematics with Applications 60 (2010) 3187–3192
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Multiobjective higher-order symmetric duality involving generalized
S.K. Guptaa, Anurag Jayswalb,∗
aDepartment of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India
bDepartment of Applied Mathematics, Birla Institute of Technology Mesra, Ranchi-835 215, India
a r t i c l ei n f o
Received 4 May 2010
Received in revised form 11 October 2010
Accepted 11 October 2010
a b s t r a c t
In this paper, a pair of Mond–Weir type multiobjective higher-order symmetric dual
higher-order K-preinvexity/K-pseudoinvexity assumptions. Symmetric minimax mixed
integer primal and dual problems are also discussed.
© 2010 Elsevier Ltd. All rights reserved.
h : Rn×Rn→ R and k : Rn×Rn→ Rm. Mond and Zhang  obtained duality results for various higher-order dual programs
under higher-order invexity assumptions while Mond and Weir  presented two pair of symmetric dual multiobjective
programming problems and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions.
Ahmad et al.  considered a general Mond–Weir type higher-order nondifferentiable multiobjective dual programs
and obtained duality relations under higher-order (F,α,ρ,d) type-I functions. Chen  studied a pair of Mond–Weir
type symmetric higher-order multiobjective nondifferentiable programs and obtained duality results under higher-order
F-convexity assumptions. Later on, Gulati and Gupta  formulated higher-order Wolfe and Mond–Weir type
symmetric dual problems with cone constraints and established usual duality theorems under higher-order η-invexity/η-
Khurana  formulated a pair of multiobjective symmetric dual programs of Mond–Weir type over arbitrary cones
in which the objective function is optimized with respect to an arbitrary closed convex cone by assuming the involved
functions to be cone-pseudoinvex and strongly cone-pseudoinvex. These results were later extended to nondifferentiable
multiobjective symmetric dual programs by Kim and Kim .
Second-order multiobjective symmetric duality over arbitrary cones has been discussed by Ahmad and Husain  for
Wolfe type programs (assuming the function involved to be second-order invex) and Gulati and Geeta  for Mond–Weir
type problems (assuming pseudoinvexity/F-convexity on the kernel function). Recently, Ahmad and Husain  studied
mixed symmetric multiobjective dual programs and obtained duality results under K-preinvexity and K-pseudoinvexity
∗Corresponding address: Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, India. Tel.: +91 542 2440628.
E-mail addresses: firstname.lastname@example.org (S.K. Gupta), email@example.com (A. Jayswal).
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
S.K. Gupta, A. Jayswal / Computers and Mathematics with Applications 60 (2010) 3187–3192
(ii) If we remove the higher-order terms h and g, then (MP) and (MD) reduce to the programs studied in Khurana .
Further, taking k = 1, it become the programs considered in Chandra and Kumar .
(iii) If hi(x,y,p) =
the second-order symmetric dual programs of Gulati et al. .
i∇yyfi(x,y)piand gi(u,v,r) =
i∇xxfi(u,v)ri, C1= Rn
+and C2= Rm
+, then the programs reduce to
The authors are thankful to the anonymous referees for their valuable suggestions, which have substantially improved
the presentation of the paper.
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