Lithuanian Mathematical Journal

, Volume 59, Issue 4, pp 507–518 | Cite as

Searching for and Quantifying Nonconvexity Regions of Functions*

  • Youri Davydov
  • Elina Moldavskaya
  • Ričardas ZitikisEmail author


Convexity plays a prominent role in a number of areas, but practical considerations often lead to nonconvex functions. We suggest a method for determining regions of convexity and also for assessing the lack of convexity of functions in the other regions. The method relies on a specially constructed decomposition of symmetric matrices. Illustrative examples accompany theoretical results.


convex analysis nonconvexity penalty function Hessian Weyl inequality risk assessment 


15B57 26B25 52A41 91G80 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    R. Bhatia, Matrix Analysis, Springer, New York, 1997.CrossRefGoogle Scholar
  2. 2.
    S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univ. Press, Cambridge, 2004.CrossRefGoogle Scholar
  3. 3.
    L. Chen, Y. Davydov, N. Gribkova, and R. Zitikis, Estimating the index of increase via balancing deterministic and random data, Math. Methods Stat., 27:83–102, 2018.MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Chen and R. Zitikis, Measuring and comparing student performance: A new technique for assessing directional associations, Education Sciences, 7:1–27, 2017.CrossRefGoogle Scholar
  5. 5.
    Y. Davydov and A.M. Vershik, Réarrangements convexes des marches aléatoires, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B, 34:73–95, 1998.CrossRefGoogle Scholar
  6. 6.
    Y. Davydov and R. Zitikis, Convex rearrangements of random elements, Fields Inst. Commun., 44:141–171, 2004.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Y. Davydov and R. Zitikis, Quantifying non-monotonicity of functions and the lack of positivity in signed measures, Mod. Stoch., Theory Appl., 4:219–231, 2017.MathSciNetCrossRefGoogle Scholar
  8. 8.
    C Gini, Variabilità e Mutabilità: Contributo allo Studio delle Distribuzioni e delle Relazioni Dtatistiche, Tipografia di Paolo Cuppini, Bologna, 1912.Google Scholar
  9. 9.
    C. Gini, On the measurement of concentration and variability of characters, Metron, 63:3–38, 1914. (English transl. by F. de Santis.)Google Scholar
  10. 10.
    N. Gribkova and R. Zitikis, Assessing transfer functions in control systems, J. Stat. Theory Pract., 13(2):35, 2018.Google Scholar
  11. 11.
    M.O. Lorenz, Methods of measuring the concentration of wealth, Publ. Am. Stat. Assoc., 9:209–219, 1905.Google Scholar
  12. 12.
    S.K. Mishra, Topics in Nonconvex Optimization: Theory and Applications, Springer, New York, 2011.CrossRefGoogle Scholar
  13. 13.
    K. Mosler, Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach, Springer, New York, 2002.CrossRefGoogle Scholar
  14. 14.
    H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71:441–479, 1912.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Youri Davydov
    • 1
  • Elina Moldavskaya
    • 2
  • Ričardas Zitikis
    • 3
    Email author
  1. 1.Chebyshev Laboratory, St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of Mathematics & Technion International School, Technion – Israel Institute of TechnologyHaifaIsrael
  3. 3.School of Mathematical and Statistical SciencesWestern UniversityLondonCanada

Personalised recommendations