# American Institute of Mathematical Sciences

July  2011, 7(3): 523-529. doi: 10.3934/jimo.2011.7.523

## On symmetric and self duality in vector optimization problem

 1 Department of Mathematics, Chongqing Normal University, Chongqing 400047

Received  June 2010 Revised  March 2011 Published  June 2011

In this paper, we point out some errors in a recent paper of M.A.E.H.Kassen (Applied Mathematics and Computation 183(2006) 1121-1126). And a pair of the first-order symmetric dual model for vector optimization problem is proposed in this paper. Then, we prove the weak, strong and converse duality theorems for the formulated first-order symmetric dual programs under invexity conditions.
Citation: Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial and Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523
##### References:
 [1] B. D. Craven, Lagrangian conditions and quasiduality, Bull. Austral. Math. Soc., 16 (1977), 325-339. doi: 10.1017/S0004972700023431. [2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, Symmetric dual nonlinear programs, Pacific J. Math., 15 (1965), 809-812. [3] W. S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan, 2 (1960), 93-97. [4] M. A. E.-H. Kassem, Symmetric and self duality in vector optimization problem, Applied Mathematics and Computation, 183 (2006), 1121-1126. doi: 10.1016/j.amc.2006.05.131. [5] Z. A. Khan and M. A. Hanson, On ratio invexity in mathematical programming, J. Math. Anal. Appl., 205 (1997), 330-336. doi: 10.1006/jmaa.1997.5180. [6] D. S. Kim, Y. B. Yun and H. Kuk, Second-order symmetric and self-duality in multiobjective programming, Applied Mathematical Letters, 10 (1997), 17-22. doi: 10.1016/S0893-9659(97)00004-9. [7] B. Mond, A symmetric dual theorem for nonlinear programs, Quart. Appl. Math., 23 (1965), 265-269. [8] B. Mond and T. Weir, Symmetric duality for nonlinear multiobjective programming, in "Recent Developments in Mathematical Programming" (ed. Santosh Kumar), Gordon and Breach Science, London, (1991), 137-153. [9] T. Weir and B. Mond, Symmetric and self duality in multiple objective programming, Asia-Pacific J. Oper. Res., 5 (1988), 124-133. [10] X.-M. Yang and S.-H. Hou, Second-order symmetric duality in multiobjective programming, Applied Mathematical Letters, 14 (2001), 587-592. doi: 10.1016/S0893-9659(00)00198-1.

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##### References:
 [1] B. D. Craven, Lagrangian conditions and quasiduality, Bull. Austral. Math. Soc., 16 (1977), 325-339. doi: 10.1017/S0004972700023431. [2] G. B. Dantzig, E. Eisenberg and R. W. Cottle, Symmetric dual nonlinear programs, Pacific J. Math., 15 (1965), 809-812. [3] W. S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan, 2 (1960), 93-97. [4] M. A. E.-H. Kassem, Symmetric and self duality in vector optimization problem, Applied Mathematics and Computation, 183 (2006), 1121-1126. doi: 10.1016/j.amc.2006.05.131. [5] Z. A. Khan and M. A. Hanson, On ratio invexity in mathematical programming, J. Math. Anal. Appl., 205 (1997), 330-336. doi: 10.1006/jmaa.1997.5180. [6] D. S. Kim, Y. B. Yun and H. Kuk, Second-order symmetric and self-duality in multiobjective programming, Applied Mathematical Letters, 10 (1997), 17-22. doi: 10.1016/S0893-9659(97)00004-9. [7] B. Mond, A symmetric dual theorem for nonlinear programs, Quart. Appl. Math., 23 (1965), 265-269. [8] B. Mond and T. Weir, Symmetric duality for nonlinear multiobjective programming, in "Recent Developments in Mathematical Programming" (ed. Santosh Kumar), Gordon and Breach Science, London, (1991), 137-153. [9] T. Weir and B. Mond, Symmetric and self duality in multiple objective programming, Asia-Pacific J. Oper. Res., 5 (1988), 124-133. [10] X.-M. Yang and S.-H. Hou, Second-order symmetric duality in multiobjective programming, Applied Mathematical Letters, 14 (2001), 587-592. doi: 10.1016/S0893-9659(00)00198-1.
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