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Approximate convexity in vector optimisation

Published online by Cambridge University Press:  17 April 2009

Anjana Gupta
Affiliation:
Department of Operational Research, University of Delhi, Delhi-110007, India e-mail: [email protected]
Aparna Mehra
Affiliation:
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India e-mail: [email protected]
Davinder Bhatia
Affiliation:
Department of Operational Research, University of Delhi, Delhi 110007, India
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Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bector, C.R., Chandra, S. and Abha, , ‘On incomplete Lagrangian funtion and saddle point optimality criteria in mathematical programming’, J. Math. Anal. Appl. 251 (2000), 212.CrossRefGoogle Scholar
[2]Bector, C.R., Chandra, S. and Abha, , ‘On mixed duality in mathematical programming’, J. Math. Anal. Appl. 259 (2001), 346356.CrossRefGoogle Scholar
[3]Bector, C.R., Chandra, S. and Dutta, J., Principles of optimization theory (Narosa Publishing House, India, 2005).Google Scholar
[4]Bhatia, D. and Jain, P., ‘Generalized (F, ρ) convexity and duality for nonsmooth multiobjective programs’, Optimization 31 (1994), 153164.CrossRefGoogle Scholar
[5]Clarke, F.H., Optimization and nonsmooth analysis (Willey-Interscience, New York, 1983).Google Scholar
[6]Egudo, R., ‘Efficiency and generalized convex duality for multi-objective programs’, J. Math. Anal. Appl. 138 (1989), 8494.CrossRefGoogle Scholar
[7]Hanson, M.A. and Mond, B., ‘Further generalization of convexity in mathematical programming’, J. Inform. Optim. Sci. 4 (1982), 2532.Google Scholar
[8]Jeyakumar, V. and Mond, B., ‘On generalized convex mathematical programming’, J. Austral. Math. Soc. Ser. B 34 (1992), 4353.CrossRefGoogle Scholar
[9]Ngai, H.V., Luc, D. and Thera, M., ‘Approximate convex functions’, J. Nonlinear Convex Anal. 1 (2000), 155176.Google Scholar
[10]Preda, V., ‘On sufficiency and duality for multiobjective programs’, J. Math. Anal. Appl. 166 (1992), 365377.CrossRefGoogle Scholar
[11]Vial, J.P., ‘Strong and weak convexity of sets and functions’, Math. Oper. Res. 8 (1983), 235259.CrossRefGoogle Scholar
[12]Xu, Z., ‘Mixed type duality in multiobjective programming problems’, J. Math. Anal. Appl. 198 (1996), 621635.CrossRefGoogle Scholar