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Generalized convexity in mathematical programming

Published online by Cambridge University Press:  17 April 2009

B. Mond
B. Mond
Affiliation:
Department of Pure Mathematics, La Trobe University, Bundoora, Victoria, Australia 3168; Department of Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa.
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Abstract

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The role of convexity in the development of mathematical programming is reviewed. Many recent generalizations of convexity and their applications to optimization theory are discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 27 , Issue 2 , April 1983 , pp. 185 - 202
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Avriel, Mordecai, Nonlinear programming, analysis and methods (Prentice-Hall, Englewood Cliffs, New Jersey, 1976).Google Scholar
[2]Avriel, Mordecai, “r-convex functions”, Math. Programming 2 (1972), 309–323.CrossRefGoogle Scholar
[3]Avriel, M., “Solution of certain nonlinear programs involving r-convex functions”, J. Optim. Theor. Appl. 11 (1973), 159–174.CrossRefGoogle Scholar
[4]Avriel, M., Diewert, W.E., Schaible, S. and Ziemba, W.T., “Introduction to concave and generalized concave functions”, Generalized concavity in optimization and economics, 21–50 (Proc. NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980. Academic Press [Harcourt Brace Jovanovich], New York and London, 1981).Google Scholar
[5]Ben-Tal, A., “On generalized means and generalized convex functions”, J. Optim. Theor. Appl. 21 (1977), 1–13.CrossRefGoogle Scholar
[6]Tal, A. Ben and Israel, A. Ben, “F-convex functions: properties and applications”, Generalized concavity in optimization and economics, 301–334 (Proc. NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980. Academic Press [Harcourt Brace Jovanovich], New York and London, 1981).Google Scholar
[7]Chandra, Suresh, “Strong pseudo-convex programming”, Indian J. Pure Appl. Math. 3 (1972), 278–282.Google Scholar
[8]Craven, B.D., Mathematical programming and control theory (Chapman and Hall, London; John Wiley & Sons, London; 1978).CrossRefGoogle Scholar
[9]Craven, B.D., “Duality for generalized convex fractional programs”, Generalized concavity in optimization and economics, 473–489 (Proc. NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980. Academic Press [Harcourt Brace Jovanovich], New York and London, 1981).Google Scholar
[10]Craven, B.D., “Invex functions and constrained local minima”, Bull. Austral. Math. Soc. 24 (1981), 357–366.CrossRefGoogle Scholar
[11]Doeringer, W., “A note on K-convex functions”, Z. Oper. Res. Ser. A–B 26 (1982), 49–55.Google Scholar
[12]Hanson, Morgan A., “On sufficiency of the Kuhn-Tucker conditions”, J. Math. Anal. Appl. 80 (1981), 545–550.CrossRefGoogle Scholar
[13]Hanson, M.A., Mond, B., “Further generalizations of convexity in mathematical programming”, J. Inform. Optim. Sci. 3 (1982), 25–32.Google Scholar
[14]Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities (Cambridge University Press, Cambridge, 1959).Google Scholar
[15]Lata, Manju, “Strong pseudo-convex programming in Banach space”, Indian J. Pure Appl. Math. 6 (1976), 45–48.Google Scholar
[16]Mangasarian, Olvi L., Nonlinear programming (McGraw-Hill, New York, London, Sydney, 1969).Google Scholar
[17]Martos, Béla, Nonlinear programming: theory and methods (North-Holland, Amsterdam, Oxford; American Elsevier, New York; 1975).Google Scholar
[18]Mond, B. and Hanson, M.A., “On duality with generalized convexity” (Pure Mathematics Research Paper, 80–3. La Trobe University, Bundoora, Australia, 1980).Google Scholar
[19]Mond, B. and Weir, T., “Generalized concavity and duality”, Generalized concavity in optimization and economics, 263–279 (Proc. NATO Advanced Study Institute, University of British Columbia, Vancouver, 1980. Academic Press [Harcourt Brace Jovanovich], New York and London, 1981).Google Scholar
[20]Nehse, Reinhard, “Strong pseudoconvex mappings in dual problems”, Math. Operationsforsch. Statist. Ser. Optim. 12 (1981), 483–491.CrossRefGoogle Scholar
[21]Ponstein, J., “Seven kinds of convexity”, SIAM Rev. 9 (1967), 115–119.CrossRefGoogle Scholar
[22]Weir, Terence, “Generalized convexity and duality in mathematical programming” (PhD thesis, La Trobe University, Bundoora, Australia, 1982).Google Scholar