Basin Analysis via Simulation

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


We illustrate simulated basins of attraction for four example objective functions applying the same four different numerical methods to each (coordinate-search, and steepest-descent with three different line-searches). We define basin size and basin entropy (quantifying basin complexity) and compute each for our examples. We apply the Nelder–Mead method to all four example functions in a separate section because its iterate-multisets are non-singleton, which requires more complicated illustrations. We use the same tools to investigate the practical significance of two well-known counterexamples to good convergence behavior in numerical minimization: the canoe function with coordinate-search and McKinnon’s function (McKinnon, SIAM J. Optim. 9, 148–158 (1998)) with Nelder–Mead. We use our notions of basin size and basin entropy to quantify the extent to which initial data are likely to lead to undesirable consequences.


  1. 9.
    Daza, A., Wagemakers, A., Georgeot, B., Guéry-Odelin, D., Sanjuán, M.A.F.: Basin entropy: a new tool to analyze uncertainty in dynamical systems. Sci. Rep. 6, 31416 (2016)CrossRefGoogle Scholar
  2. 10.
    Dennis Jr., J.E., Woods, D.J.: Optimization on microcomputers: the Nelder-Mead simplex algorithm. In: Wouk, A. (ed.) New Computing Environments: Microcomputers in Large-Scale Computing. SIAM, Philadelphia (1987)Google Scholar
  3. 14.
    Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45, 385–348 (2003)MathSciNetCrossRefGoogle Scholar
  4. 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
  5. 17.
    McKinnon, K.I.M.: Convergence of the Nelder-Mead simplex method to a nonstationary point. SIAM J. Optim. 9, 148–158 (1998)MathSciNetCrossRefGoogle Scholar
  6. 22.
    Sprott, J.C., Xiong, A.: Classifying and quantifying basins of attraction. Chaos 25, 083101 (2015)MathSciNetCrossRefGoogle Scholar
  7. 23.
    Torczon, V.: Multi-directional search: a direct search algorithm for parallel machines. Ph.D. thesis, Rice University, Houston, TX (1989)Google 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

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