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What is invexity?

Published online by Cambridge University Press:  17 February 2009

A. Ben-Israel  and
B. Mond
A. Ben-Israel
Affiliation:
Department of Mathematical Science, University of Delaware, Newark, DE 19716, U.S.A.
B. Mond
Affiliation:
Department of Mathematical, La Trobe University, Bundoor, Vic. 3083, Australia.
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Abstract

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Recently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.

Type
Research Article
Information
The ANZIAM Journal , Volume 28 , Issue 1 , July 1986 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Craven, B. D., “Duality for generalized convex fractional programs,” in Generalized concavity in optimization and economics, (eds. Schaible, S. and Ziemba, W. T.), (Academic Press, New York, 1981), 473–489.Google Scholar
[2]Craven, B. D., “Invex functions and constrained local minima,”, Bull. Austral. Math. Soc. 24 (1981), 357–366.CrossRefGoogle Scholar
[3]Craven, B. D., and Glover, B. M., “Invex functions and duality,” J. Austral. Math. Soc. Ser. A, 39 (1985), 1–20.CrossRefGoogle Scholar
[4]Gulenko, L. Ya., (Russian) “The relaxation method for mathematical programming problems with Hanson's functions,” Dokl. Akad. Nauk BSSR 27 (1983), 105–107.Google Scholar
[5]Hanson, M. A., “On sufficiency of the Kuhn-Tucker conditions,” J. Math. Anal. Appl. 80 (1981), 545–550.CrossRefGoogle Scholar
[6]Hanson, M. A. and Mond, B., “Necessary and sufficient conditions in constrained optimization,” FSU Statistics Report M683, Florida State University, Department of Statistics, Tallahassee, Florida, U.S.A., 1984.Google Scholar
[7]Jeyakumar, V., “Strong and weak invexity in mathematical programming”, Method. Oper. Res. 55 (1985), 109–125.Google Scholar
[8]Kaul, R. N. and Kaur, S., “Optimality criteria in nonlinear programming involving nonconvex functions,” J. Math. Anal. Applic. 105 (1985), 104–112.CrossRefGoogle Scholar
[9]Kuhn, H. W., and Tucker, A. W., “Nonlinear programming.” in Proceedings of the Second Symposium on Mathematical Statistics and Probability, (ed. Neyman, J.), (University of California Press, Berkeley, Calif., 1951), 481–492.Google Scholar
[10]Mangasarian, O. L., Nonlinear programming, (McGraw-Hill, New York, 1969).Google Scholar
[11]Martin, D. H., “The essence of invexity, J. Optim. Theory Appl. 47 (1985), 65–76.CrossRefGoogle Scholar
[12]Mond, B., and Hanson, M. A., “On duality with generalized convexity,” Math. Operationsforsch. Statist. Ser. Optim. 25 (1984), 962–967.Google Scholar
[13]Zang, I., Choo, E. U., and Avriel, M., “On functions whose stationary points are global,” J. Optim. Theory Appl. 22 (1977), 195–208.CrossRefGoogle Scholar