Rosenbrock

Rosenbrock, H.H. (1974) Computer Aided Control System Design, Academic Press, New York. 

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)... 

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 ... 

There are many variations of this method. To illustrate the procedure, a variation developed by Rosenbrock will be discussed. It is one of the best optimization methods known8,7 when there is no experimental error. The method is also very useful for determining constants in kinetic and thermodynamic equations that are highly nonlinear. An example of this type of application is given in reference 9. 

Figure 14-3 Rosenbrock s method for determining the optimum reactor conditions.

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)... 

If no success is found with AUTO, then try Rosenbrock (STIFF) and adjust by the same procedure. Oscillations can sometimes be seen by zooming in on a graph often these are a sign of integration problems. Sometimes some variables look OK but others oscillate, so look at all of them if problems arise. Un-... 

R2. Rosenbrock, H. H., and Storey, C., Computational Techniques for Chemical Engineers. Macmillan (Pergamon), New York, 1966. 

Interaction among control loops in a multivariable system has been the subject of much research over the last 20 years. Various types of decouplers were explored to separate the loops. Rosenbrock presented the inverse Nyquist array (INA) to quantify the amount of interaction. Bristol, Shinskey, and McAvoy developed the relative gain array (RGA) as an index of loop interaction... 

Rosenbrock (Computer-Aided Control System Design, Academic Press, 1974) was one of the early woikers in the area of multivariable control. He proposed the use of INA plots to indicate the amount of interaction among the loops. 

Rosenbrock H, Hagemeyer CE, Singec I, Knoth R, Volk B. 1999. Testosterone metabolism in rat brain is differentially enhanced by phenytoin-inducible cytochrome P450 isoforms. J Neuroendocrinol 11 597-604. 

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. 

The Rosenbrock search routine in the computer program employed by Balke and Hamielec was found to converge to the correct optimum values of Cl and C2 within approximately 200 iterations. 

Two adjustable parameters of fhe equafions can be found by an optimization technique using Marquardt s or Rosenbrock s maximum likelihood method of minimizafion... 

M72 Solution of stiff differential equations semi-implicit Runge-Kutta method with backsteps Rosenbrock-Gottwa1d-Wanner 7200 7416... 

EX241 2.4.1 Rosenbrock problem by Nelder-Mead method M34... 

EX242 2.4.2 Rosenbrock problem by Davidon-Fletcher-Powell method M36... 

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. ... 

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

We need to solve s sets of nonlinear equations, but Rosenbrock devised a much simpler procedure. Linearization of the m-th set of equations in (5.30) around the point... 

B.A. Gottwald and G. Wanner, A reliable Rosenbrock-integrator for stiff differential equations, Computing, 26 (1981) 335-357. 

Shinskey has discussed in detail both interaction and decoupling in general(33) and in respect to the distillation process<34). ROSENBROCK 35 has presented a more theoretical treatment of the problem. 

---