Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T16:45:11.185Z Has data issue: false hasContentIssue false
  • English
  • Français

On subgradient duality with strong and weak convex functions

Published online by Cambridge University Press:  09 April 2009

V. Jeyakumar
V. Jeyakumar
Affiliation:
Department of Mathematics, University of Melbourne Parkville, Victoria 3052, Australia
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.

A duality theorem of Wolfe for nonlinear differentiable programs is extended to nondifferentiable programs with strong and weak convex functions, by replacing gradients by local subgradient. A converse duality theorem is also proved.

MSC classification

Secondary: 90C25: Convex programming 90C30: Nonlinear programming 90C48: Programming in abstract spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 2 , April 1986 , pp. 143 - 152
Copyright
Copyright Australian Mathematical Society 1986

References

1Clarke, F. H., A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165174.CrossRefGoogle Scholar
2Craven, B. D., Mathematical programming and control theory, (Chapman and Hall, London, 1978).CrossRefGoogle Scholar
3Craven, B. D., Non-differentiable Pontryagin principle, Research Report No. 20, (1977), Department of Mathematics, University of Melbourne.Google Scholar
4Craven, B. D. and Mond, B., Lagrange conditions for quasidifferentiable optimization, pp. 177191, IX Intern. Symp. on Math. Prog.Google Scholar
(Budapest, , in Surveys of Mathematical Programming, Vol. 1, Prekopa, A. (ed.), North-Holland, Amsterdam, (1976)).Google Scholar
5Craven, B. D. and Mond, B., On converse duality in nonlinear programming, Operations Res. 19 (1971), 10751078.Google Scholar
6Holmes, R. B., Geometric functional analysis and its applications (Springer-Verlag, 1975).CrossRefGoogle Scholar
7Rockafellar, R. T., Convex analysis (Princeton University Press, Princeton, 1972).Google Scholar
8Schechter, M., A subgradient duality theorem, J. Math. Anal. Appl. 61 (1977), 850855.CrossRefGoogle Scholar
9Schechter, M., More on subgradient duality, J. Math. Anal. Appl. 71 (1979), 251262.CrossRefGoogle Scholar
10Vial, J. P., Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983), 231259.CrossRefGoogle Scholar
11Vial, J. P., Strong convexity of sets and functions, J. Math. Econom. 9 (1982), 187205.CrossRefGoogle Scholar
12Wolfe, P., A duality theorem for nonlinear programming, Quart. Appl. Math. 19 (1961), 239244.Google Scholar
13Zowe, J. and Kureyusz, S., Regularity and stability for the mathematical programming problems in Banach spaces, Appl. Math. Optim. 5 (1979), 4962.Google Scholar