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.

show all references

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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