Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-19T05:06:36.958Z Has data issue: false hasContentIssue false

On locally Lipschitz vector-valued invex functions

Published online by Cambridge University Press:  17 April 2009

N.D. Yen
Affiliation:
Institute of Mathematics, PO Box 631, Bo Ho 10000 Hanoi, Vietnam
P.H. Sach
Affiliation:
Institute of Mathematics, PO Box 631, Bo Ho 10000 Hanoi, Vietnam
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.

The four types of invexity for locally Lipschitz vector-valued functions recently introduced by T. W. Reiland are studied in more detail. It is shown that the class of restricted K-invex in the limit functions is too large to obtain desired optimisation theorems and the other three classes are contained in the class of functions which are invex 0 in the sense of our previous joint paper with B. D. Craven and T. D. Phuong. We also prove that the extended image of a locally Lipschitz vector-valued invex function is pseudoconvex in the sense of J. Borwein at each of its points.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Aubin, J.P. and Ekeland, I., Applied nonlinear analysis (Wiley, New York, 1984).Google Scholar
[2]Borwein, J., ‘Weak tangent cones and optimization in Banach space’, SIAM J. Control Optim. 16 (1978), 512522.CrossRefGoogle Scholar
[3]Clarke, F.H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
[4]Craven, B.D., ‘Invex functions and constrained local minima’, Bull. Austral. Math. Soc. 24 (1981), 357366.CrossRefGoogle Scholar
[5]Craven, B.D. and Glover, B.M., ‘Invex functions and duality’, J. Austral. Math. Soc. Ser. A 39 (1985), 120.CrossRefGoogle Scholar
[6]Craven, B.D., Sach, P.H., Yen, N.D. and Phuong, T.D., ‘A new class of invex multifunctions’, Papers presented at the Summer School in Erice (Sicily), June 19–30, 1991; Depart. of Math., Melbourne University, Preprint no.21 (1991), Nonsmooth Optimization: Methods and Applications (to appear).Google Scholar
[7]Hanson, M.A., ‘On sufficiency of the Kuhn-Tucker conditions’, J. Math. Anal. Appl. 80 (1981), 545550.CrossRefGoogle Scholar
[8]Mangasarian, O.L., Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar
[9]Mond, B. and Weir, T., ‘Generalized concavity and duality’, in Generalized concavity in optimization and economics, (Schaible, S. and Ziemba, W. T., Editors) (Academic Press, New York, 1981), pp. 263279.Google Scholar
[10]Reiland, T.W., ‘Nonsmooth invexity’, Bull. Austral. Math. Soc. 42 (1990), 437446.CrossRefGoogle Scholar
[11]Reiland, T.W., ‘Generalized invexity for nonsmooth vector-valued mappings’, Numer. Funct. Anal. Optim. 10 (1989), 11911202.Google Scholar
[12]Sach, P.H. and Craven, B.D., ‘Invexity in multifunction optimization’, Numer. Funct. Anal. Optim. 12 (1991), 383394.CrossRefGoogle Scholar
[13]Sach, P.H. and Craven, B.D., ‘Invex multifunctions and duality’, Numer. Funct. Anal. Optim. 12 (1991), 575591.CrossRefGoogle Scholar
[14]Tanaka, Y., ‘Note on generalized convex functions’, J. Optim. Theor. Appl. 66 (1990), 345349.CrossRefGoogle Scholar
[15]Tanaka, Y., Fukushima, M. and Ibaraki, T., ‘On generalized pseudoconvex functions’, J. Math. Anal. Appl. 144 (1989), 342355.CrossRefGoogle Scholar
[16]Wolfe, P. A., ‘A duality theorem for nonlinear programming’, Quart. Appl. Math. 19 (1961), 239244.CrossRefGoogle Scholar