Skip to main content
SpringerLink
Log in

Local maximum points of explicitly quasiconvex functions

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This work concerns generalized convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold: first, to show that any local maximum point of an explicitly quasiconvex function is a global minimum point whenever it belongs to the intrinsic core of the function’s domain and second, to characterize strictly convex normed spaces by applying this property for a particular class of convex functions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boţ, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cambini, A., Martein, L.: Generalized Convexity and Optimization. Theory and Applications. Springer, Berlin (2009)

    MATH  Google Scholar 

  4. Greenberg, H.J., Pierskalla, W.P.: A review of quasi-convex functions. Oper. Res. 19, 1553–1570 (1971)

    Article  MATH  Google Scholar 

  5. Karamardian, S.: Strictly quasi-convex (concave) functions and duality in mathematical programming. J. Math. Anal. Appl. 20, 344–358 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  6. Martos, B.: The direct power of adjacent vertex programming methods. Manag. Sci. 12, 241–255 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ponstein, J.: Seven kinds of convexity. SIAM Rev. 9, 115–119 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Phohomsiri, P.: On the extrema of linear least-squares problems. J. Optim. Theory Appl. 127, 665–669 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

Nicolae Popovici’s research was supported by CNCS-UEFISCDI, within the project PN-II-ID-PCE-2011-3-0024. The authors wish to thank professor Valeriu Anisiu for suggesting them to investigate whether Corollary 4.1 could be used in order to characterize the class of strictly convex normed spaces, which led to Corollary 4.2. They are also grateful to the referee whose valuable comments and suggestions improved the paper.

Author information

Authors and Affiliations

  1. School of Computing and Mathematics, University of Derby, Kedleston Road, Derby, DE22 1GB, UK

    Ovidiu Bagdasar

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University of Cluj-Napoca, Kogălniceanu Street Nr. 1, 400084, Cluj-Napoca, Romania

    Nicolae Popovici

Authors
  1. Ovidiu Bagdasar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolae Popovici
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolae Popovici.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bagdasar, O., Popovici, N. Local maximum points of explicitly quasiconvex functions. Optim Lett 9, 769–777 (2015). https://doi.org/10.1007/s11590-014-0781-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-014-0781-3

Keywords

Search

Navigation