Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-23T14:52:58.537Z Has data issue: false hasContentIssue false
  • English
  • Français

The Farkas lemma of Shimizu, Aiyoshi and Katayama

Published online by Cambridge University Press:  17 April 2009

Charles Swartz
Charles Swartz
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003, USA.
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.

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 3 , June 1985 , pp. 445 - 450
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Clarke, F., Optimization and nonsmooth analysis (John Wiley & Sons, New York and London, 1983).Google Scholar
[2]Farkas, J., “Über die Theorie der einfachen Ungleichen”, J. Reine Angew. Math. 124 (1902), 1–24.Google Scholar
[3]Glover, B.M., “A generalized Farkas lemma with applications to quasidifferential programming”, Z. Oper. Res. Ser. A 26 (1982), 125–141.Google Scholar
[4]Holmes, Richard B., Geometric functional analysis and its applications (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[5]Horváth, John, Topological vector spaces and distributions, Volume I (Addison-Wesley, Reading, Massachusetts; Palo Alto; London; 1966).Google Scholar
[6]Pshenichnyi, B.N., Necessary conditions for an extremum (Marcel Dekker, New York, 1971).Google Scholar
[7]Shimizu, K., Aiyoshi, E. and Katayama, R., “Generalized Farkas' theorem and optimization of infinitely constrained problems”, J. Optim. Theory Appl. 40 (1983), 451–462.CrossRefGoogle Scholar
[8]Swartz, Charles, “A general Farkas lemma”, J. Optim. Theory Appl. (to appear).Google Scholar
[9]Zalinescu, C., “A generalization of the Farkas lemma and applications to convex programming”, J. Math. Anal. Appl. 66 (1978), 651–678.CrossRefGoogle Scholar
[10]Zalinescu, C., “On an abstract control problem”, Numer. Funct. Anal. Optim. 6 (1980), 531–542.CrossRefGoogle Scholar