Iteration Mappings

  • Adam B. Levy
Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)


Minimization methods can be modeled by iteration mappings that generate new iterate-multisets from current iterate-multisets. We introduce the notion of viability to indicate when an initial iterate-multiset generates a sequence of non-empty iterate-multisets through the repeated application of an iteration mapping. We review the minimization methods of coordinate-search, steepest-descent (with various line-searches), and Nelder–Mead, and we define the corresponding iteration mappings and discuss viability in each case. We prove an analogue to the classical Banach fixed-point theorem for iteration mappings.


  1. 5.
    Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)CrossRefGoogle Scholar
  2. 15.
    Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9, 112–147 (1998)MathSciNetCrossRefGoogle Scholar
  3. 19.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Adam B. Levy
    • 1
  1. 1.Bowdoin CollegeBrunswickUSA

Personalised recommendations