Computable Analysis of Linear Rearrangement Optimization

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)


Optimization problems over rearrangement classes arise in various areas such as mathematics, fluid mechanics, biology, and finance. When the generator of the rearrangement class is two-valued, they reduce to shape optimization and free boundary problems which can exhibit intriguing symmetry breaking phenomena. A robust framework is required for computable analysis of these problems. In this paper, as a first step towards such a robust framework, we provide oracle Turing machines that compute the distribution function, decreasing rearrangement, and linear rearrangement optimizers, with respect to functions that are continuous and have no significant flat zones. This assumption on the reference function is necessary, as otherwise, the aforementioned operations may not be computable. We prove that the results can be computed to within any degree of accuracy, conforming to the framework of Type-II Theory of Effectivity.


Computable analysis Rearrangements of functions Optimization 


  1. 1.
    Benjamin, T.B.: The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. In: Germain, P., Nayroles, B. (eds.) Applications of Methods of Functional Analysis to Problems in Mechanics. LNM, vol. 503, pp. 8–29. Springer, Heidelberg (1976). Scholar
  2. 2.
    Brattka, V., Yoshikawa, A.: Towards computability of elliptic boundary value problems in variational formulation. J. Complex. 22(6), 858–880 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Burton, G.R.: Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276(2), 225–253 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burton, G.R.: Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(4), 295–319 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burton, G.R., McLeod, J.B.: Maximisation and minimisation on classes of rearrangements. Proc. R. Soc. Edinb. Sect. A 119(3–4), 287–300 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Crowe, J.A., Zweibel, J.A., Rosenbloom, P.C.: Rearrangements of functions. J. Funct. Anal. 66(3), 432–438 (1986)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Elcrat, A., Nicolio, O.: An iteration for steady vortices in rearrangement classes. Nonlinear Anal. 24(3), 419–432 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Emamizadeh, B., Marras, M.: Rearrangement optimization problems with free boundary. Numer. Funct. Anal. Optim. 35(4), 404–422 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Emamizadeh, B., Zivari-Rezapour, M.: Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem. Nonlinear Anal. 74(16), 5697–5704 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Emamizadeh, B., Farjudian, A., Liu, Y.: Optimal harvesting strategy based on rearrangements of functions. Appl. Math. Comput. 320, 677–690 (2018)MathSciNetGoogle Scholar
  11. 11.
    Emamizadeh, B., Farjudian, A., Zivari-Rezapour, M.: Optimization related to some nonlocal problems of Kirchhoff type. Canad. J. Math. 68(3), 521–540 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Emamizadeh, B., Hanai, M.A.: Rearrangements in real estate investments. Numer. Funct. Anal. Optim. 30(5–6), 478–485 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kao, C.Y., Su, S.: Efficient rearrangement algorithms for shape optimization on elliptic eigenvalue problems. J. Sci. Comput. 54(2), 492–512 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ko, K.I.: Complexity Theory of Real Functions. Birkhäuser, Boston (1991)CrossRefGoogle Scholar
  15. 15.
    Selivanova, S., Selivanov, V.: Computing the solution operators of symmetric hyperbolic systems of PDE. J. Univers. Comput. Sci. 15(6), 1337–1364 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Talenti, G.: The art of rearranging. Milan J. Math. 84(1), 105–157 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Weihrauch, K.: Computable Analysis, An Introduction. Springer, Heidelberg (2000). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of Nottingham Ningbo ChinaNingboChina

Personalised recommendations