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Convergence Properties of the NelderMead Simplex Method in Low Dimensions
 SIAM Journal of Optimization
, 1998
"... Abstract. 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 pr ..."
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Cited by 446 (3 self)
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Abstract. 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. Key words. direct search methods, Nelder–Mead simplex methods, nonderivative optimization AMS subject classifications. 49D30, 65K05
Direct search methods: Once scorned, now respectable
 PITMAN RESEARCH NOTES IN MATHEMATICS SERIES
, 1996
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New Anisotropic Covariance Models and Estimation of Anisotropic Parameters Based On the . . .
"... Many heterogeneous media and environmental processes are statistically anisotropic, that is, their moments have directional dependence. The term range anisotropy denotes processes that have variograms characterized by directiondependent correlation lengths and directionally independent sill. We dis ..."
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Cited by 11 (7 self)
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Many heterogeneous media and environmental processes are statistically anisotropic, that is, their moments have directional dependence. The term range anisotropy denotes processes that have variograms characterized by directiondependent correlation lengths and directionally independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models are separable and reduce to a product of onedimensional functions along the principal axes. We present a Class (A) model for multiscale processes and suggest applications in subsurface hydrology. This model is based on a truncated power law with short and longrange cutoffs. We also present a family of Class (B) models generated by superellipsoidal functions that are based on nonEuclidean distance metrics. We propose a new method for determining the orientation of the principal axes and the degree of anisotropy (i.e., the ratios of the correlation lengths). This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic, and Class (B) models to onedimensional functions. Our method is based on an explicit relation between the secondrank slope tensor (SRST), which can be estimated from the data, and the secondrank covariance tensor. The method is conceptually simple and numerically efficient. It is more accurate for regular (ongrid) data distributions, but it can also be used for irregular (offgrid) spatial distributions. We illustrate its implementation with numerical simulations.
Convergence of the restricted NelderMead algorithm in two dimensions, in preparation
, 1997
"... The Nelder–Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalarvalued function f of n real variables using only function values, without any derivative information. Each Nelder–Mead iteration is associated with a nond ..."
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Cited by 1 (1 self)
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The Nelder–Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalarvalued 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 twicecontinuously 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 nonstandard in convergence proofs for unconstrained optimization. 1
An examination of the sign and volatility switching ARCH models under alternative distributional assumptions
, 2000
"... This paper relaxes the assumption of conditional normal innovations used by Fornari and Mele (1997) in modelling the asymmetric reaction of the conditional volatility to the arrival of news. We compare the performance of the Sign and VolatilitySwitching ARCH model of Fornari and Mele (1997) and ..."
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Cited by 1 (0 self)
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This paper relaxes the assumption of conditional normal innovations used by Fornari and Mele (1997) in modelling the asymmetric reaction of the conditional volatility to the arrival of news. We compare the performance of the Sign and VolatilitySwitching ARCH model of Fornari and Mele (1997) and the GJR model of Glosten et al. (1993) under the assumption that the innovations follow the Generalised Student's t distribution. Moreover, we hedge against the possibility of misspecication by basing the inferences on the robust variancecovariance matrix suggested by White (1982). The results suggest that using more exible distributional assumptions on the  nancial data can have a signicant impact on the inferences drawn. 1 Introduction There is growing evidence that the response of current volatility to past shocks is asymmetric with negative shocks having more impact on current volatility than positive shocks (see Engle and Ng (1993) and Fornari and Mele (1997)). One explan...
Monograph on Structural Health Monitoring, Inst. of Smart Structures and Systems, Bangalore/India Structural Health Management of Ageing Aircraft and Other Infrastructure
"... Large, complex and costly engineering structures are mainly made to last for a long period. This is specifically known for civil engineering buildings but also for heavy machinery, trains, ships or aircraft. Any of these items are used intensively. Long endurance combined with intensive usage leads ..."
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Large, complex and costly engineering structures are mainly made to last for a long period. This is specifically known for civil engineering buildings but also for heavy machinery, trains, ships or aircraft. Any of these items are used intensively. Long endurance combined with intensive usage leads to deterioration. Of course an engineering structure is designed to withstand deterioration for a certain
INVERSE PROBLEMS PII: S02665611(03)531604
, 2003
"... Determining the Gaussian probability distribution of the bestfit ellipsoid of revolution for a polymer chain from planar projections ..."
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Determining the Gaussian probability distribution of the bestfit ellipsoid of revolution for a polymer chain from planar projections
Assessment of temperature, trace species, and ozone in chemistry
"... [1] Simulations of the stratosphere from thirteen coupled chemistryclimate models (CCMs) are evaluated to provide guidance for the interpretation of ozone predictions made by the same CCMs. The focus of the evaluation is on how well the fields and processes that are important for determining the oz ..."
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[1] Simulations of the stratosphere from thirteen coupled chemistryclimate models (CCMs) are evaluated to provide guidance for the interpretation of ozone predictions made by the same CCMs. The focus of the evaluation is on how well the fields and processes that are important for determining the ozone distribution are represented in the simulations of the recent past. The core period of the evaluation is from 1980 to 1999 but longterm trends are compared for an extended period (1960–2004). Comparisons of polar highlatitude temperatures show that most CCMs have only small biases in the Northern Hemisphere in winter and spring, but still have cold biases in the Southern Hemisphere spring below 10 hPa. Most CCMs display the correct stratospheric response of polar temperatures to wave forcing in the Northern, but not in the Southern Hemisphere. Global longterm stratospheric temperature trends are in reasonable agreement with satellite and radiosonde observations. Comparisons of simulations of methane, mean age of air, and propagation of the annual cycle in water vapor show a wide spread in the results, indicating differences in transport. However, for around half the models there is reasonable agreement with observations. In these models the mean age of
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"... Assessment of temperature, trace species, and ozone in chemistryclimate model simulations of the recent past ABCDEFB ..."
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Assessment of temperature, trace species, and ozone in chemistryclimate model simulations of the recent past ABCDEFB