Golden section method

EX221 2.2.1 Optimum dosing by golden section method M25... 

REM EX. 2.2.1 OPTIMUM DOSINE BY GOLDEN SECTION METHOD 104 REM MERGE M25... 

An example of a line search technique that does not use derivatives is the golden section method, which seeks to find the minimum of a function /(x) on an interval [a, d. The interval [a, d is called... 

Let aj < bj < Cj < dj, where [a, is the interval of uncertainty at iteration k and assume that [<3i, di] = [a, d. Since functional evaluations are the most expensive step in the process, the golden section method reduces the amount of overall work by intelligently choosing symmetric points bj and Cj so that they can be reused on successive iterations, as illustrated in Figure 8. This is achieved by using the relationships b = Xa + (1 - X)d and = (1 - A)% + Xdj where A = 0.618. Observe that b and are simply expressed as convex combinations of % and idj A summary of the golden section algorithm follows ... 

Figure 8 Successive Iterations of the Golden Section Method.

Unconstrained optimization (nonlinear programming), 2546-2553 classictil methods, 2546-2547 conjugate gradient methods, 2552-2553 golden section method, 2547-2549 line search techniques for, 2547 multidimensional search techniques for, 2549-2552... 

When the NLP problem consists of only one decision variable (or can be reduced to one), with lower and upper bounds, the optimal solution can be found readily with a spreadsheet, or by one of several structured and efficient search methods, including region elimination, derivative based, and point estimation, as described in detail by Reklaitis et al. (1983). Of the search methods, the golden-section method (involving region elimination) is reasonably efficient, reliable, easily implemented, and widely used. Therefore, it is described and illustrated by example here. 

The optimal values of and were found using the alternating variable descent method. The golden section method was used to the find one-dimensional minimum along the axis. The procedure was considered to be completed at 0(z) <5= 10-4. 

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. 

The golden section method requires that the ratio between two successive... 

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

The golden section method can be stopped after any number of iterations without ( j becoming less efficient. ... 

It might seem like a good idea to couple the golden section method with other algorithms that can exploit the function s features. 

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 golden section method is of paramount theoretical, practical, and educational importance but is rarely used in a general-purpose program. 

Write a program that uses the golden section method to minimize the function... 

The search within the range of uncertainty adopts cubic interpolation as its basic method. If such a method is inefficient, the parabolic method is adopted. If this method is also not as high performance as the golden section method, the new point is inserted into the middle of the largest subintervals. 

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... 

The step length should be chosen such that a sufficient improvement is assured and the length is not too short. For this purpose, a good step length must satisly the so-called Wolfe conditions. There are several methods to determine the step length among them the interpolation method, the golden section method, and the Fibonacci s method are the most commonly used. 

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