Rosenbrock function

It is noted that the Rosenbrock function given by the next equation has been used to test the performance of various algorithms including modified Newton s and conjugate gradient methods (Scales, 1986)... 

Example 2.4.1 Minimization of the Rosenbrock function by the simplex method of Nelder and Mead... 

Find the minimum of the Rosenbrock function by solving the nonlinear equations af(x)/9x = 61. ... 

Edgar and Himmelblau (1988) demonstrate the use of the method for a function of two variables. Nelder and Mead (1965) presented the method for a function of N variables as a flow diagram. They demonstrated its use by applying it to minimize Rosenbrock s function (Equation 5.22) as well as to the following functions ... 

Rosenbrock, H.H. An Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal, Oct. 1960, p. 175. 

Will Newton s method minimize Rosenbrock s function... 

List the relative advantages and disadvantages (there can be more than one) of the following methods for a two-variable optimization problem such as Rosenbrock s banana function (see Fig. P6.19)... 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

H.H. Rosenbrock, An automatic method for finding the greatest or least value of a function, Computer J. 3 (1960) 175-184. 

Both the development and the optimization of simplex methods are still continuing. Several functions have been designed to test the performance of the simplex algorithms, one example is the famous ROSENBROCK valley. Other test functions have been reported by ABERG and GUSTAVSSON [1982]. Most analytical applications of simplex optimization are found in atomic spectroscopy [SNEDDON, 1990] and chromatography [BERRIDGE, 1990],... 

An alternative, called semi-implicit methods in such texts as [351], avoids the problems, and some of the variants are L-stable (see Chap. 14 for an explanation of this term), a desirable property. This was devised by Rosenbrock in 1962 474]. There are two strong points about this set of formulae. One is that the constants in the implicit set of equations for the k s are chosen such that each can be evaluated explicitly by easy rearrangement of each equation. The other is that the method lends itself ideally to nonlinear functions, not requiring iteration, because it is, in a sense, already built-in. This is explained below. 

Rosenbrock s function is often used as a minimization test problem, because its minimum lies at the base of a banana-shaped valley and can be difficult to locate. This function is defined for even integers n as the sum... 

The contour plot of Rosenbrock s function for n - 2 is shown in Figure 14. The minimum point is (1,1), where f( ) = 0. The gradient components of this function are given by... 

An illustration of simulated annealing optimization is presented in Figures 1-5. In this 2-dimensional example the objective fimction is a modified Rosenbrock s function ... 

Figure 2. Biased random walk with a random step size — Rosenbrock s function.

Redlich-Kister, 344 refinement, 16 Reynolds-averaged Navier-Stokes/probability density-function (RANS/PDF), 33 Rosenbrock, 196 Runge-Kutta, 34 space-time, 397 conservation element and solution element (CESE), 389, 396 spectral, 41... 

Identification of the theoretical and experimental transfer functions in order to estimate the effective diffusivity De is obtained by minimizing a relative error function taken between the two transfer functions. The Rosenbrock method of optimization has been used. All the measurements have been made at room temperature and something close to normal atmospheric pressure. The only parameter that changes is the carrier gas flow rate. 

Other methods of multidimensional search without using derivatives include Rosenbrock s method (1960) and the simplex method of Spendley et al. (1962), which was later modified by Nelder and Meade (1974). Although it has the same name, this simplex method is not the same algorithm as that used for linear progranuning it is a polytope algorithm that requires only functional evaluations and requires no smoothness assumptions. 

For example, let us consider the well-known Rosenbrock s function problem ... 

It is important to realize why even the Rosenbrock method becomes inefficient when the function valleys are particularly narrow. 

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