Convergence properties of the Nelder--Mead simplex method in low dimensions The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder--Mead algorithm. This paper presents convergence properties of the Nelder--Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder--Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder--Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.

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  1. Price, C. J.; Coope, I. D.; Byatt, D.: A convergent variant of the Nelder--Mead algorithm (2002)
  2. Sodoyer, David; Schwartz, Jean-Luc; Girin, Laurent; Klinkisch, Jacob; Jutten, Christian: Separation of audio-visual speech sources: A new approach exploiting the audio-visual coherence of speech stimuli (2002)
  3. van Ruymbeke, Evelyne; Keunings, Roland; Bailly, Christian: Determination of the molecular weight distribution of entangled linear polymers from linear viscoelasticity data (2002)
  4. Banjai, Lehel; Trefethen, Lloyd N.: Numerical solution of the omitted area problem of univalent function theory (2001)
  5. Lewis, Robert Michael; Torczon, Virginia; Trosset, Michael W.: Direct search methods: Then and now (2000)
  6. Roberts, William J. J.; Furui, Sadaoki: Maximum likelihood estimation of K-distribution parameters via the expectation-maximization algorithm (2000)
  7. Kelley, C. T.: Detection and remediation of stagnation in the Nelder-Mead algorithm using a sufficient decrease condition (1999)
  8. Tseng, Paul: Fortified-descent simplicial search method: A general approach (1999)
  9. Lagarias, Jeffrey C.; Reeds, James A.; Wright, Margaret H.; Wright, Paul E.: Convergence properties of the Nelder--Mead simplex method in low dimensions (1998)
  10. McKinnon, K. I. M.: Convergence of the Nelder--Mead simplex method to a nonstationary point (1998)
  11. Wright, M. H.: Direct search methods: Once scorned, now respectable (1996)

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