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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 567 (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
 Eds.), Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis, 191–208
, 1996
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Convergence of the NelderMead simplex method to a nonstationary point
 SIAM J. OPTIM
, 1996
"... This paper analyses the behaviour of the NelderMead simplex method for a family of examples which cause the method to converge to a nonstationary point. All the examples use continuous functions of two variables. The family of functions contains strictly convex functions with up to three continuo ..."
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Cited by 80 (0 self)
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This paper analyses the behaviour of the NelderMead simplex method for a family of examples which cause the method to converge to a nonstationary point. All the examples use continuous functions of two variables. The family of functions contains strictly convex functions with up to three continuous derivatives. In all the examples the method repeatedly applies the inside contraction step with the best vertex remaining fixed. The simplices tend to a straight line which is orthogonal to the steepest descent direction. It is shown that this behaviour cannot occur for functions with more than three continuous derivatives. The stability of the examples is analysed.
FortifiedDescent Simplicial Search Method: A General Approach
 SIAM J. Optim
, 1995
"... We propose a new simplexbased direct search method for unconstrained minimization of a realvalued function f of n variables. As in other methods of this kind, the intent is to iteratively improve an ndimensional simplex through certain reflection/expansion/contraction steps. The method has three n ..."
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Cited by 26 (1 self)
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We propose a new simplexbased direct search method for unconstrained minimization of a realvalued function f of n variables. As in other methods of this kind, the intent is to iteratively improve an ndimensional simplex through certain reflection/expansion/contraction steps. The method has three novel features. First, a userchosen integer m k specifies the number of "good" vertices to be retained in constructing the initial trial simplicesreflected, then either expanded or contractedat iteration k. Second, a trial simplex is accepted only when it satisfies the criteria of fortified descent, which are stronger than the criterion of strict descent used in most direct search methods. Third, the number of additional function evaluations needed to check a trial reflected/expanded simplex for fortified descent can be controlled. If one of the initial trial simplices satisfies the fortified descent criteria, it is accepted as the new simplex; otherwise, the simplex is shrunk a fracti...
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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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
New Algorithmic Approaches for the Anticipatory Route Guidance Generation Problem
, 2001
"... The emergence of Intelligent Transportation Systems (ITS) has considerably changed the field of transportation modeling during the past ten years. By Intelligent Transportation Systems (ITS) we refer to transportation systems which apply emerging information technologies to alleviate congestion prob ..."
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The emergence of Intelligent Transportation Systems (ITS) has considerably changed the field of transportation modeling during the past ten years. By Intelligent Transportation Systems (ITS) we refer to transportation systems which apply emerging information technologies to alleviate congestion problems (see, for example, Transportation Research Board 1999). They combine advanced surveillance systems collecting realtime traffic data, Advanced Traffic Management Systems (ATMS) and Advanced Traveler Information Systems (ATIS) in order to achieve that objective. The complexity of these systems requires the development of sophisticated tools for their optimal explo itation. Among the many ITS applications, providing realtime travel information is a particularly important one. It consists in using available data from the surveillance system in order to generate information that will be disseminated by the ATIS (variable message signs, highway advisory radio on standard broadcast bands, onboard GPSbased computers, etc.) The messages may contain complete or partial route recommendations, qualitative or quantitative information about traffic conditions over the network. We are interested here in generating consistent anticipatory route guidance (CARG). Anticipatory guidance accounts for the probable evolution of traffic conditions over time and throughout the network. By consistent, we mean that the anticipated traffic conditions used to generate the guidance must be similar to the traffic conditions that the travelers are going to experience on the network. The problem is non trivial because, contrarily to weather forecast where the real system under consideration is not affected by information provision, the very fact of providing travel information may modify the ...
Preliminary design optimization of profile losses in multistage axial compressors based on complexmethod
, 2008
"... Abstract: This paper illustrates a numerical technique undertaken for preliminary design optimization of profile losses in multistage axial compressors. Design process has been carried out based on onedimensional row by row calculations along compressor mean line. The main objective of the optimi ..."
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Abstract: This paper illustrates a numerical technique undertaken for preliminary design optimization of profile losses in multistage axial compressors. Design process has been carried out based on onedimensional row by row calculations along compressor mean line. The main objective of the optimization process was to find the best distribution of pressure ratios along compressor stages in order to maximize the overall efficiency, which is itself a nonlinear function of governing variables. In this respect, only profile losses, identified as the most dominant ones, are focused on and minimized during the optimization process. Pressure ratios of stages have been taken as design variables. Diffusion factor and De Haller number of each blade row were considered as the main constraints during the optimization process. Numerical optimization was based on a sequential search technique, referred to as complex method. Design process in parallel to the numerical optimization is examined on a tenstage axial compressor of known general performance data. Final results showed an increase of about 2.7 per cent in the total efficiency relative to its initial value calculated during the preliminary design process.