Skip to main content
SpringerLink
Log in

Alternative and Separation

  • Chapter
Constrained Optimization and Image Space Analysis

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 49))

  • 711 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aleksandrow P.S., “Combinatorial Topology”. Graylock Press, Albany, N.Y., 1960.

    Google Scholar 

  2. Banach S., “Sur les lignes rectifiables et les surfaces dont l’aire est finie”. Fundam. Math., Vol.7, 1925, pp.225–237.

    MATH  Google Scholar 

  3. Bazaraa M.S., “A Theorem of the Alternative with Applications to Convex Programming: Optimality, Duality, and Stability”. Jou.of Mathem. Analysis and Appls., Vol.41, 1973, pp.701–715.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bigi G. and Pappalardo M., “Regularity Conditions for the Linear Separation of Sets”. Jou.Optimiz.Th.Appls., Vol.99, No.2, 1998, pp.533–540.

    Article  MATH  MathSciNet  Google Scholar 

  5. Castellani G. and Giannessi F., “Decomposition of Mathematical Programs by means of Theorems of the Alternative for Linear and Nonlinear Systems”. Proceedings of the 9th Inter. Symposium on Mathematical Programming, Hungarian Academy of Sciences, Budapest, 1979, pp.423–439.

    Google Scholar 

  6. Craven B.D., Gwinner G. and Jeyakumar V., “Nonconvex Theorems of the Alternative and Minimization”. Optimization, Vol.18, No.2, 1987, pp. 151–163.

    MATH  MathSciNet  Google Scholar 

  7. Craven B.D. and Koliha J.J., “Generalizations of Farkas Theorem”. SIAM Jou. Mathem. Analysis, Vol.8, No.6, 1977, pp.983–997.

    Article  MATH  MathSciNet  Google Scholar 

  8. Craven B.D. and Mond B., “Transposition theorems for cone-convex functions”. SIAM Jou.Appl.Mathem., Vol.24, No.4, 1973, pp.603–612.

    Article  MATH  MathSciNet  Google Scholar 

  9. Dax A. and Sreedharan V.P., “Theorems of the Alternative and Duality”. Jou. Optimiz. Th. Appls., Vol.94, No.3, 1997, pp.561–590.

    Article  MATH  MathSciNet  Google Scholar 

  10. Dinh The Luc, “Theorems of the Alternative and their applications in Multiobjective Optimization”. Acta Math.Hungarica, Vol.45, No.3–4, 1985, pp.311–320.

    Article  MathSciNet  Google Scholar 

  11. Farkas J., “Über die Theorie der einfachen Ungleichungen”. Jou. Reine Angew. Mathem., Vol. 124, 1902, pp.1–27.

    Google Scholar 

  12. Ferrero O., “Theorems of the Alternative for Set-Valued Functions in Infinite-Dimensional Spaces”. Optimization, Vol.20, No.2, 1989, pp. 167–175.

    MATH  MathSciNet  Google Scholar 

  13. Giannessi F., “Theorems of the Alternative, Quadratic programs, and Complementarity Problems”. In [I 13], pp.151–186.

    Google Scholar 

  14. Giannessi F., “Theorems of the Alternative and Optimality Conditions”. Tech. Report No.83, Dept. of Mathem., Univ. of Pisa, Sect. of Optimization, 1982, pp.1–30. Published with the same title in Jou. Optimiz. Th. Appls., Vol.42, No. 3, 1984, pp.331–365.

    Google Scholar 

  15. Giannessi F., “Theorems of the Alternative for multifunctions with applications to Optimization. Necessary conditions”. Tech. Paper No. 131, Optimiz. Series, Dept. of Mathem., Univ. of Pisa, Pisa, Italy, 1986, pp.1–127.

    Google Scholar 

  16. Giannessi F., “Theorems of the Alternative for Multifunctions with Applications to Optimization: General Results”. Jou.Optimiz.Th.Appls., Vol. 55, No.2, 1987, pp.233256.

    MathSciNet  Google Scholar 

  17. Giannessi F., “Theorems of the Alternative and Optimization”. In [V 26], Vol.V, pp.437–444.

    Google Scholar 

  18. Golikov A.I. and Evtushenko Yu.G., “Theorems of the Alternative and their Applications in Numerical Methods”. Computational Mathematics and Mathematical Physics, Vol.43, No.3, 2003, pp.338–358.

    MATH  MathSciNet  Google Scholar 

  19. Gordan P., “Über die Auflõsungen linearer Gleichungen mit reelen Coefficienten”. Mathematishe Annalen, Vol.6, 1873, pp.23–28.

    Article  MATH  MathSciNet  Google Scholar 

  20. Hahn H., “Über lineare Gleichungen in lineare Räumen”. Jou.Mathem., Vol.157, 1927, pp.214–229.

    MATH  Google Scholar 

  21. Heinecke G. and Oettli W., “A Nonlinear Theorem of the Alternative without Regularity Assumption”. Jou.of Mathem.Analysis and Appls., Vol.146, No.2, 1990, pp.580–590.

    Article  MATH  MathSciNet  Google Scholar 

  22. Illés T, and Kassay G., “Farkas Type Theorems for Generalized Convexities”. Report No.94-23 of Tech.Univ.Delft, Fac.of Tech.Mathem.and Informatics, 1994, pp.1–12.

    Google Scholar 

  23. Jeyakumar V., “Convexlike Alternative Theorems and Mathematical Programming”. Optimization, Vol.16, No.5, 1895, pp.643–652.

    MathSciNet  Google Scholar 

  24. Jeyakumar V., “A generalization of a minimax theorem of Fan via a theorem of the alternative”. Jou. of Optimiz. Theory and Appls., Vol.48, No.3, 1986, pp.525–533.

    Article  MATH  MathSciNet  Google Scholar 

  25. Jeyakumar V., “A General Farkas Lemma and Characterization of Optimality for a Nonsmooth Program involving Convex Processes”. Jou.Optimiz.Th.Appls., Vol.55, No.3, 1987, pp.449–461.

    Article  MATH  MathSciNet  Google Scholar 

  26. Lehmann R. and Oettli W., “The Theorem of the Alternative: Key-Theorem, and the Vector Maximum Problem”. Mathematical Programming, Vol.8, 1975, pp.332–344.

    Article  MathSciNet  Google Scholar 

  27. Li Z., “A Theorem of the Alternative and its Applications to the Optimization of Set-Valued Maps”. Jou.Optimiz.Th.Appls., Vol.100, No.2, 1999, pp.365–375.

    Article  MATH  Google Scholar 

  28. MacLinden L., “Duality Theorems and Theorems of the Alternative”. Proc.of Annals of Mathem.Soc., Vol.53, 1975, pp.172–175.

    Article  Google Scholar 

  29. Mangasarian O.L., “A stable Theorem of the Alternative: an extension of the Gordan Theorem”. Linear Algebra and its Appls., Vol.41, 1981, pp.209–223.

    Article  MATH  MathSciNet  Google Scholar 

  30. Martinez-Legaz J.E. and Seeger A., “Yuan’s Alternative Theorem and the Maximization of the Minimum Eigenvalue Function”. Jou.Optimiz.Th.Appls., Vol.82, No.l, 1994, pp.159–167.

    Article  MATH  MathSciNet  Google Scholar 

  31. Mastroeni G. and Pappalardo M., “Separation and regularity in the Image Space”. In “New Trends in Mathematical Programming”, Series in Applied Optimization, Vol.13, Kluwer, Dordrecht, 1998, pp.181–190.

    Google Scholar 

  32. Mastroeni G. and Pellegrini L., “Linear separation for G-semidifferentiable problems”. Proceedings of the Conference “Convessitá e Calcolo Parallelo (Convexity and Parallel Computation)”. G. Giorgi and F. Rossi Eds., Publisher Univ.of Verona, Via dell’Artigliere, 19-Verona-Italy, 1997, pp.187–203.

    Google Scholar 

  33. Mazzoleni P., “Some generalizations of the Theorem of the Alternative for functions and multifunctions”. Proc.of the Dept.of Applied Mathem. of University of Venice, Vol.XIX, 1982, 39–51.

    Google Scholar 

  34. Motzkin T.S., “Beiträge zur Theirie der Linearen Ungleichungen”. Inaugural Diss. Basel, Jerusalem, 1936.

    Google Scholar 

  35. Nehse von R., “Some General Separation Theorems”. Mathem.Nachr., Vol.($, 1978, pp.319–327.

    Google Scholar 

  36. Oettli W., “A new version of th Hahn-Banach Theorem”. Proc.of the Int.Congress on Mathematical Programming, April 1981 (Rio de Janeiro), North-Holland, 1984, pp.289–295.

    Google Scholar 

  37. Prager W., “Optimal arrangement of the beams of a rectangular grillage”. In “Problemi attuali di Meccanica teorica e applicata (Present problems of theorical and applied Mechanics)”, Proceedings of the Int.Confenence in Memory of M.Panetti, Published by Academy of Sciences of Turin, Turin, 1977, pp.239–249.

    Google Scholar 

  38. Simons S., “Variational Inequalities via the Hahn-Banach Theorem”. Archiv der Mathematik, Vol.31, Fasc.5, 1978, pp.482–490.

    Article  MATH  MathSciNet  Google Scholar 

  39. Simons S., “Minimax and Variational Inequalities. Are they of Fixed-Point or Hahn-Banach type?”. In “Game Theory and Mathematical Economics”, O. Moeschin and D. Pallaschke Eds., North-Holland, 1981, pp.379–387.

    Google Scholar 

  40. Slater M.L., “A Note on Motzkin’s Transposition Theorem”. Econometrica, Vol.19, 1951, pp.185–186.

    Article  MATH  MathSciNet  Google Scholar 

  41. Stiemke E., “Über positive Lösungen homogener linearer Gleichungen”. Mathematische Annalen, Vol.76, 1915, pp.340–342.

    Article  MATH  MathSciNet  Google Scholar 

  42. Tricomi F.G., “Integral Equations”. Interscience, 1957.

    Google Scholar 

  43. Yang X.M., “Alternative Theorems and Optimality Conditions with weakened convexity”. OPSEARCH, Vol.29, No.2, 1992, pp.125–135.

    MATH  Google Scholar 

  44. Yang X.M., Yang X.Q. and Chen G.-Y., “Theorems of the Alternative and optimization with Set-Valued Maps”. Jou.Optimiz.Th.Appls., Vol.107, No.3, 2000, pp.627–640.

    Article  MATH  MathSciNet  Google Scholar 

  45. Zalinescu C., “A generalization of the Farkas Lemma and Applications to Convex Programming”. Jou.Mathem.Analysis Appls., Vol.66, No.3, 1978, pp.651–678.

    Article  MATH  MathSciNet  Google Scholar 

  46. Zalmai G.J., “A transposition Theorem with Applications to Constrained Optimal Control Problems”. Optimization, Vol.20, No 3, Academic-Verlag, Berlin, 1989, pp.265–279.

    Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

(2005). Alternative and Separation. In: Constrained Optimization and Image Space Analysis. Mathematical Concepts and Methods in Science and Engineering, vol 49. Springer, Boston, MA . https://doi.org/10.1007/0-387-28020-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation