Convergence of the restricted Nelder-Mead algorithm in two dimensions

The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function f of n real variables using only function values, without any derivative information. Each Nelder-Mead iteration is associated with a nondegenerate simplex defined by n+1 vertices and their function values; a typical iteration produces a new simplex by replacing the worst vertex by a new point. Despite the method's widespread use, theoretical results have been limited: for strictly convex objective functions of one variable with bounded level sets, the algorithm always converges to the minimizer; for such functions of two variables, the diameter of the simplex converges to zero, but examples constructed by McKinnon show that the algorithm may converge to a nonminimizing point. This paper considers the restricted Nelder-Mead algorithm, a variant that does not allow expansion steps. In two dimensions we show that, for any nondegenerate starting simplex and any twice-continuously differentiable function with positive definite Hessian and bounded level sets, the algorithm always converges to the minimizer. The proof is based on treating the method as a discrete dynamical system, and relies on several techniques that are non-standard in convergence proofs for unconstrained optimization.Comment: 27 page

Biological effects of prolonged expoure of small animals to unusual gaseous environments semiannual report, 1 sep. 1964 - 28 feb. 1965

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The impact of water on free-falling bodies

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Using visualization for visualization : an ecological interface design approach to inputting data

Visualization is experiencing growing use by a diverse community, with continuing improvements in the availability and usability of systems. In spite of these developments the problem of how first to get the data in has received scant attention: the established approach of pre-defined readers and programming aids has changed little in the last two decades. This paper proposes a novel way of inputting data for scientific visualization that employs rapid interaction and visual feedback in order to understand how the data is stored. The approach draws on ideas from the discipline of ecological interface design to extract and control important parameters describing the data, at the same time harnessing our innate human ability to recognize patterns. Crucially, the emphasis is on file format discovery rather than file format description, so the method can therefore still work when nothing is known initially of how the file was originally written, as is often the case with legacy binary data. © 2013 Elsevier Ltd

Reaction of the chick to one atmosphere of oxygen

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Embryo development and chick growth in a helium - oxygen atmosphere

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