Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T09:10:49.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonsmooth invexity

Published online by Cambridge University Press:  17 April 2009

Thomas W. Reiland
Thomas W. Reiland
Affiliation:
Department of Statistics and Graduate Program in Operations Research, Box 8203, North Carolina State University, Raleigh, NC 27695-8203, United States of America
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 concept of invexity is extended to nondifferentiable functions. Characterisations of nonsmooth invexity are derived as well as results in unconstrained and constrained optimisation and duality. The principal analytic tool is the generalised gradient of Clarke for Lipschitz functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 3 , December 1990 , pp. 437 - 446
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Ben-Israel, A. and Mond, B., ‘What is invexity?’, J. Austral. Math. Soc. Ser. B 28 (1986), 1–9.CrossRefGoogle Scholar
[2]Clarke, F.H., ‘Generalized gradients of Lipschitz functionala’, Adv. in Math. 40 (1981), 52–67.CrossRefGoogle Scholar
[3]Clarke, F.H., Optimization and nonsmooth analysis (Wiley, New York, 1983).Google Scholar
[4]Craven, B.D., ‘Duality for generalized convex fractional programs’, in Generalized concavity in optimization and economics, Editors Schaible, S. and Ziemba, T., pp. 473–490 (Academic Press, 1981).Google Scholar
[5]Craven, B.D., ‘Invex functions and constrained local minima’, Bull. Austral Math. Soc. 24 (1981), 357–366.CrossRefGoogle Scholar
[6]Craven, B.D., ‘Nondifferentiable optimization by smooth approximations’, Optimization 17 (1986), 3–17.CrossRefGoogle Scholar
[7]Craven, B.D. and Glover, B.M., ‘Invex functions and duality’, J. Austral. Math. Soc. Series A 39 (1985), 1–20.CrossRefGoogle Scholar
[8]Egudo, R.R. and Mond, B., ‘Duality with generalized convexity’, J. Austral. Math. Soc. Ser. B 28 (1986), 10–21.CrossRefGoogle Scholar
[9]Hanson, M.A., ‘On sufficiency of the Kuhn-Tucker conditions’, J. Math. Anal. Appl. 80 (1981), 545–550.CrossRefGoogle Scholar
[10]Hanson, M.A. and Mond, B., ‘Further generalizations of convexity in mathematical programming’, J. Inform. Optim. Sci (1982), 25–32.Google Scholar
[11]Hanson, M.A. and Mond, B., ‘Necessary and sufficient conditions in constrained optimization’, Math. Programming 37 (1987), 51–58.CrossRefGoogle Scholar
[12]Hiriart-Urruty, J.B., ‘On optimality conditions in nondifferentiable programming’, Math. Prog. 14 (1978), 73–86.CrossRefGoogle Scholar
[13]Hiriart-Urruty, J.B., ‘Refinements of necessary optimality conditions in nondifferentiable programming I’, Appl. Math. Optim. 5 (1979), 63–82.CrossRefGoogle Scholar
[14]Jeyakumar, V., ‘Strong and weak invexity in mathematical programming’, Methods Oper. Res. 55 (1985), 109–125.Google Scholar
[15]Martin, D.H., ‘The essence of invexity’, J. Optim. Theory Appl. 47 (1985), 65–76.CrossRefGoogle Scholar
[16]Mifflin, R., ‘Semismooth and semiconvex functions in constrained optimization’, SIAM J. Control Optim. 15 (1977), 959–972.CrossRefGoogle Scholar
[17]Mond, B. and Hanson, M.A., ‘On duality with generalized convexity’, Mathematische Operationsforschung und Statistik Series Optimization 15 (1984), 313–317.CrossRefGoogle Scholar
[18]Mond, B. and Weir, T., ‘Generalized concavity and duality’, in Generalized Concavity in Optimization and Economics, Editors Schaible, S. and Ziemba, T., pp. 473–490 (Academic Press, 1981).Google Scholar
[19]Reiland, T.W., ‘A geometric approach to nonsmooth optimization with sample applications’, Nonlinear Anal. 11 (1987), 1169–1184.CrossRefGoogle Scholar
[20]Stern, E.M., Singular Integrals and Differentiability Properties of Functions: Princeton Mathematics Series 30 (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar