Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T07:15:14.707Z Has data issue: false hasContentIssue false

Efficiency and generalised convexity in vector optimisation problems

Published online by Cambridge University Press:  17 February 2009

Pham Huu Sach
Affiliation:
Hanoi Institute of Mathematics, P.O. Box 631, Boho, Hanoi, Vietnam; e-mail: [email protected].
Gue Myung Lee
Affiliation:
Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea; e-mail: [email protected] and [email protected].
Do Sang Kim
Affiliation:
Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea; e-mail: [email protected] and [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper gives a necessary and sufficient condition for a Kuhn-Tucker point of a non-smooth vector optimisation problem subject to inequality and equality constraints to be an efficient solution. The main tool we use is an alternative theorem which is quite different to a corresponding result by Xu.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Aubin, J. P., Mathematical methods of game and economic theory (North-Holland, Amsterdam, 1979).Google Scholar
[2]Benson, H. P., “Concave minimization: theory, applications and algorithms”, in Handbook of global optimization (eds. Horst, R. and Pardalos, P. M.), (Kluwer, Dordrecht, 1995) 43148.CrossRefGoogle Scholar
[3]Blackwell, D. and Girshick, M. A., Theory of games and statistical decisions (John Wiley and Sons, New York, 1954).Google Scholar
[4]Brandas, A. J. V., Rojas-Medar, M. A. and Silva, G. N., “Invex nonsmooth alternative theorem and applications”, Optimization 48 (2000) 239253.CrossRefGoogle Scholar
[5]Clarke, F. H., Optimization and nonsmooth analysis (Wiley-Interscience, New York, 1983).Google Scholar
[6]Craven, B. D., “Nondifferentiable optimization by nonsmooth approximations”, Optimization 17 (1986) 317.CrossRefGoogle Scholar
[7]Ferguson, T. S., Mathematical statistics: a decision theoretic approach (Academic Press, New York, 1967).Google Scholar
[8]Geoffrion, A. M., “Proper efficiency and the theory of vector maximization”, J. Math. Anal. Appl. 22 (1968) 618630.CrossRefGoogle Scholar
[9]Hanson, M. A., “On sufficiency of the Kuhn-Tucker conditions”, J. Math. Anal. Appl. 80 (1981) 545550.CrossRefGoogle Scholar
[10]Hanson, M. A., “Invexity and the Kuhn-Tucker theorem”, J. Math. Anal. Appl. 236 (1999) 594604.CrossRefGoogle Scholar
[11]Hanson, M. A. and Mond, B., “Necessary and sufficient conditions in constrained optimization”, Math. Programming 37 (1987) 5158.CrossRefGoogle Scholar
[12]Mangasarian, O. L., Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
[13]Martin, D. H., “The essence of invexity”, J. Optim. Th. Appl. 47 (1985) 6576.CrossRefGoogle Scholar
[14]Osuna-Gómez, R., Beato-Moreno, A. and Rufian-Lizana, A., “Generalized convexity in multiobjective programming”, J. Math. Anal. Appl. 233 (1999) 205220.CrossRefGoogle Scholar
[15]Osuna-Gómez, R., Rufian-Lizana, A. and Ruiz-Canalez, P., “Invex functions and generalized convexity in multiobjective programming”, J. Optim. Th. Appl. 98 (1998) 651661.CrossRefGoogle Scholar
[16]Pareto, V., Manuale di economia politica, con una introduzione ulla scienza sociale (Societa Editrice Libraria, Milan, Italy, 1906).Google Scholar
[17]Phuong, T. D., Sach, P. H. and Yen, N. D., “Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function”, J. Optim. Th. Appl. 87 (1995) 579594.CrossRefGoogle Scholar
[18]Reiland, T. W., “Nonsmooth invexity”, Bull. Austral. Math. Soc. 42 (1990) 437446.CrossRefGoogle Scholar
[19]Sach, P. H., Kim, D. S. and Lee, G. M., “Generalized convexity and nonsmooth problems of vector optimization”, Preprint 2000/31, Hanoi Institute of Mathematics, (submitted).CrossRefGoogle Scholar
[20]Sach, P. H., Kim, D. S. and Lee, G. M., “Invexity as necessary optimality condition in nonsmooth programs”, Preprint 2000/30, Hanoi Institute of Mathematics, (submitted).Google Scholar
[21]Sach, P. H., Lee, G. M. and Kim, D.S., “Infine functions, nonsmooth alternative theorems and vector optimization problems”, J. Global Optim. 27 (2003) 5181.CrossRefGoogle Scholar
[22]Takayama, A., Mathematical economics (The Dryden Press, Illinois, 1974).Google Scholar
[23]Xu, Z., “Generalization of nonhomogeneous Farkas' lemma and applications”, J. Math. Anal. Appl. 186 (1994) 726734.CrossRefGoogle Scholar
[24]Zeleny, M., Multiple criteria decision making (McGraw-Hill, New York, 1982).Google Scholar