Skip to main content
Log in

Abstract Convexity, Global Optimization and Data Classification

  • Invited Paper
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

This survey paper consisits of three parts: first of them contains some definitions, results and examples from abstract convexity. Applications of abstract convexity to numerical methods of global optimization are discussed in the second part. Applications of these methods to data classification is the subject of the third part.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bagirov A. M. and Rubinov A. M., “Global minimization of increasing positively homogeneous functions over the unit simplex,” to appear, Ann, Operations Research

  2. Bagirov A. M. and Rubinov A. M., “Cutting angle method and a local search,” submitted to Journal of Global Optimization.

  3. Bagirov A. M., Rubinov A. M. and Yearwood J., “A global classification approach to classification,” submitted paper.

  4. Hiriart-Urruti J.B. and Lemarechal C., “Convex Analysis and Minimization Algorithms,” Vol.2, Springer-Verlag, Berlin-Heidelberg (1993).

  5. Kutateladze S.S. and Rubinov A. M., “Minkowski Duality and its applications,” Nauka, 1976 (in Russian).)

    Google Scholar 

  6. Mangasarian O.L., “Mathematical progamming in data mining, Data Mining and knowledge Discovery,” 1 (1997), 183–201.

    Article  Google Scholar 

  7. Michie D., Spiegelhalter D.J. and Taylor C. C. (eds.), “Machine Learning, Neural and Statistical Classification,” Ellis Horwood Series in Artificial Intelligence, London, 1994.

    Google Scholar 

  8. Moreau J.J., “Inf-convolution, sous-addivite, convexite des fonctions numeriques,” J. Mathem. Pures. Appl. 49(1970), 109–154.

    Google Scholar 

  9. Pallaschke D. and Rolewicz S., “Foundations of Mathematical Optimization (Convex Analysis without Linearity),” Kluwer Academic Publishers, 1997

    Book  Google Scholar 

  10. Rockafellar R.T., “Convex analysis,” Princeton University Press, Princeton, New Jercy, 1970.

    Book  Google Scholar 

  11. Rubinov A., “Abstract Convexity and Global Optimization,” Kluwer Academic Publishers, 2000.

    Book  Google Scholar 

  12. Singer I., “Abstract Convex Analysis,” J. Wiley and Sons, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rubinov, A.M. Abstract Convexity, Global Optimization and Data Classification. OPSEARCH 38, 247–265 (2001). https://doi.org/10.1007/BF03398635

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03398635

Navigation