Fibonacci method

The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

The method used to collocate the points based on (2.19) is called the Fibonacci method, since Fibonacci s series plays a part in the procedure. 

It is possible to evaluate a priori the efficiency of the Fibonacci method for example, given Ll=18, delta=0.0001, and N = 10, the final interval is... 

Unfortunately, some of these advantages may also turn out be disadvantages at times too. The Fibonacci method minimizes the maximum final interval of uncertainty only if the required function evaluations are accomplished. 

If the Fibonacci method sequence selected for an assigned number of points is stopped before completion, the method may not perform as efficiently than other algorithms. 

Many functions have interesting, highly exploitable features (e.g., continuity and derivability). Specifically, many can be well approximated by means of quadratic functions as their minimum is approached. Conversely, the Fibonacci method does not discriminate between functions and takes all of them in the same way the worst one. 

We might also think about coupling the Fibonacci method with other algorithms that can exploit the function s features Fibonacci is a slow but sure method and... 

It is not possible to combine the Fibonacci method with more efficient alternatives its intrinsic nature means it must start from an interval with no points inside it, unless properly positioned in line with the Fibonacci series. 

It is also possible to modify the Fibonacci method to obtain a series that minimizes the maximum final interval of uncertainty when a point is already positioned in the starting interval. In this case, the final interval is not univocally determined by the number of points, but different widths can be obtained depending on the position of the first point and the function to be minimized. 

BzzMath library does not combine the Fibonacci method with other more efficient alternatives. 

The Fibonacci method is of paramount theoretical, practical, and educational importance but is rarely used in its original form in a general-purpose program. 

Another method based on function comparison is the golden section method. The golden section method was proposed before the Fibonacci method. The golden section also exploits the position of the point still inside the new interval of uncertainty, in a sequential search. 

II. 12-6.88 = 4.24 far from t. Whatever the best point, the final interval of uncertainty after four points is 4.24, vhereas the Fibonacci method yields a final range equal to 4 (or less with a value of d < 1). Nevertheless, if the search is stopped at the third point, the interval of uncertainty is 6.88 for the golden section and 7 for the Fibonacci method, which is only optimized for four points and a value of d = 1. 

The golden section method shares the following pros with the Fibonacci method. 

Unfortunately, the golden section method suffers from the same problem as the Fibonacci method. The selection of the series starts from an empty interval, unless an existing point is positioned within it in line with golden section philosophy. 

The Fibonacci method can be adapted, although only in a partially satisfactory way, when a point is already located in the initial interval. 

Various optimum search methods exist for the minimization of objective functions, which can be used for the estimation of kinetic constants [3], for example, the Fibonacci method, the golden section method, the Newton-Raphson method, the Levenberg-Marquardt method, and the simplex method. Recently, even genetic algorithms have been... 

Since many patients in phase I clinical trials are treated at doses of chemotherapeutic agents that are below the biologically active, they have a reduced the chance to get therapeutic benefit. Therefore we decided to adopt an accelerated dose escalation design followed by a modified Fibonacci method to reduce the number of such patients . 

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