Hooke-Jeeves method

An updated version of the Hooke-Jeeves method, which is based on onedimensional searches and the Rosenbrock device for tackling narrow valleys, is presented in Sections 3.2.1 and 3.2.2, respectively. 

The Opmov method will be introduced coupled with the Simplex method in Section 3.2.4. Chapter 5 will describe its new version that exploits object-oriented programming and parallel computing, making it appealing for handling very challenging problems of small dimensions. 

1) The method starts the search from xq. An auxiliary vector y is initialized by xq. 

2) n one-dimensional searches across Cartesian axes are performed, obtaining the series of points xi, X2. x . 

5) A one-dimensional search is carried out across the direction p starting from Xn. 

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The computation result yield acetaldehyde concentration as function of time. The value of kinetics parameters, ki, ka, k3 were adjusted to minimize the sum of square of error between the predicted and measured concentration using Hooke Jeeve method [3]. 

There are many different types of search routines used to locate optimum operating conditions. One approach is to make a large number of runs at different combinations of temperature, reaction time, hydrogen partial pressure, and catalyst amount, and then run a multivariable computer search routine (like the Hooke-Jeeves method or Powell method). A second approach is to formulate a mathematical model from the experimental results and then use an analytical search method to locate the optimum. The formulation of a mathematical model is not an easy task, and in many cases, this is the most critical step. Sometimes it is impossible to formulate a mathematical model for the system, as in the case of the system studied here, and an experimental search must be performed. 

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive. 

Below, we describe four algorithms that are able to handle small and medium dimension problems even in the presence of relatively narrow valleys without using any gradient or Hessian the Rosenbrock method (1960), the Hooke-Jeeves method (1961), the Simplex method (Spendley et al, 1962 Nelder and Mead, 1965), and the Optnov method (Buzzi-Ferraris, 1967). Note that their current structure is slightly different from the original one. 

Without going into too much detail, the original Rosenbrock and Hooke-Jeeves methods do not perform one-dimensional searches along the axes, but they use opportune steps Oj that are modified according to their success (step increase) or failure (step decrease and sign changing). 

The Hooke-Jeeves method (and its variants) presents the following pros and cons. 

The Hooke-Jeeves method (as is the case for all methods) must be always coupled with other methods. 

The optimum seeking methods which have been found to be particularly useful are the modified Fibonacci search (search by golden section) for one-dimensional searches and the Hooke-Jeeves search for multi-dimensional searches. Beveridge and Schechter (8) give a complete description of these searches. 

The results presented in Fig. 5 were used to evaluate the theoretical models derived in section 2. The equations were fitted to the experimental data with a linear optimization program, that uses the least square method and the Hook-Jeeves algorithm [22]. The sum of the least square deviations were taken as the criteria for the applicability of the models. Only the algorithms that led to the best fittings are presented in the text. 

The pattern searches implement the method of Hooke and Jeeves using a line search and the method of Rosenbrock [6]. The Hooke and Jeeves algorithm that was used in this work can be thought of in two separate phases an exploratory search and a pattern search. The exploratory search successively perturbs each parameter and tests the resulting performance. The pattern search steps along the (iTfc+i-a ) direction, which is the direction between the last two points selected by the exploratory search. When both positive and negative perturbations of the parameters do not result in enhanced performance, the perturbation size is decreased. When the perturbation size is less than an arbitrary termination factor, e, the algorithm stops. 

Each of these algorithms was used to adaptively update a two-weight FIR filter and stabilize a Rijke-tube combustor through acoustic actuation. Both the gradient descent and pattern search methods proved quite effective and produced 40 to 50 dB of attenuation of the instability peak. For example, Fig. 18.6 shows the power spectral density of the uncontrolled tube and the system controlled with the Hooke and Jeeves algorithm. The steady-state results of all the... 

To help overcome this problem, the method of Hooke and Jeeves (1961) involves both exploratory moves and pattern moves (acceleration moves) with discrete steps along the search directions. The discrete steps eliminate the need for a line search. In the exploratory move phase, starting at a point X , a modified cyclic coordinate search is performed in which, if possible, the objective function is reduced once along each of the coordinate directions using a prescribed discrete step length. This leads to a new point x +j and establishes a direction of improvement. A pattern move is then performed in the direction x +j - x, leading to an intermediate point y. Now, starting from y, another exploratory move yields the point x +2- If < /(Xa +i) then an improvement has been found... 

The so-called direct search method, described by Hooke and Jeeves, is used for the minimum of the function F. The above mentioned authors Anthony and Himmelblau also studied different possibilities of selecting equilibrium criteria and thus also of choosing the function f i = 1,2,. ..,iV or F. One of the existing possibilities is to choose the function F equal to the overall enthalpy of the system. However, the state of equilibrium may also be characterized by R linearly independent stoichiometric equations, satisfying the relations (see (3.29))... 

Minimum-time joint trajectory is a constrained non-linear optimization problem with a single objective function. The optimization procedure used in this work is the non-linear optimization search method with goal programming based on the Modified Hooke and Jeeves Direct Search Method [13]. 

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