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Proper efficiency in linear vector maximum problems with nonlinear constraints

Published online by Cambridge University Press:  09 April 2009

T. R. Gulati
Affiliation:
Department of Mathematics, University of Roorkee, Roorkee 247667, India
M. A. Islam
Affiliation:
Department of Mathematics, University of Dhaka, Dhaka-2, Bangladesh
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Abstract

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A linear vector maximum problem with nonlinear constraints is considered. A condition is derived which is necessary for an efficient solution and sufficient for a properly efficient solution of this problem. This leads to sufficient conditions for an efficient solution to be properly efficient. An example is discussed at the end.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Benson, H. P. and Morin, T. L., ‘The vector maximization problem: proper efficiency and stability’, SIAM J. Appl. Math. 32 (1977), 6472.CrossRefGoogle Scholar
[2]Bhatia, D. and Gupta, B., ‘Efficiency in certain nonlinear fractional vector maximization problems’, Indian J. Pure Appl. Math. 11 (1980), 669672.Google Scholar
[3]Choo, E. U., ‘Proper efficiency and the linear fractional vector maximum problem’, Oper. Res. 32 (1984), 216220.CrossRefGoogle Scholar
[4]Geoffrion, A. M., ‘Proper efficiency and the theory of vector maximization’, J. Math. Anal. Appl. 22 (1968), 618630.CrossRefGoogle Scholar
[5]Isermann, H., ‘Proper efficiency and the linear vector maximum problem’, Oper. Res. 22 (1974), 189191.CrossRefGoogle Scholar
[6]Kaul, R. N. and Gupta, B., ‘Efficiency and linear fractional vector maximum problem’, Z. Angew. Math. Mech. 60 (1980), 112113.CrossRefGoogle Scholar
[7]Kaul, R. N. and Gupta, B., ‘Multi-objective programming in complex space’, Z. Angew Math. Mech 61 (1981), 599601.CrossRefGoogle Scholar
[8]Klinger, A., ‘Improper solutions of the vector maximum problem’, Oper. Res. 15 (1967), 570572.CrossRefGoogle Scholar
[9]Kuhn, H. W. and Tucker, A. W., ‘Nonlinear programming’, Proceeding Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481492 (Univ. of California Press, Berkeley, California, 1951).Google Scholar
[10]Mangasarian, O. L., Nonlinear programming (McGraw-Hill Inc., New York, 1969).Google Scholar