Golden section

The golden section search is the optimization analog of a binary search. It is used for functions of a single variable, F a). It is faster than a random search, but the difference in computing time will be trivial unless the objective function is extremely hard to evaluate. 

Walser, H. (2001) The Golden Section. Mathematical Association of American, Washington, DC. 

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 nature of the relationships and constraints in most design problems is such that the use of analytical methods is not feasible. In these circumstances search methods, that require only that the objective function can be computed from arbitrary values of the independent variables, are used. For single variable problems, where the objective function is unimodal, the simplest approach is to calculate the value of the objective function at uniformly spaced values of the variable until a maximum (or minimum) value is obtained. Though this method is not the most efficient, it will not require excessive computing time for simple problems. Several more efficient search techniques have been developed, such as the method of the golden section see Boas (1963b) and Edgar and Himmelblau (2001). 

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 Golden Ratio, Golden Point and Golden Sections... 

The Golden Sections of the Ground State Bohr Radius for Hydrogen... 

Ionic Radii as the Golden Sections of Inter-Atomic Distances... 

This showed that, in general, for at r atom (A) the inter-atomic distance d(AA) can be divided into two Golden sections, which give the anioiuc and cationic radii,... 

The teeth of a comb are not really a very important item in life. Nobody ever cared to consider a comb from such an angle.. .. It could also be compared to a golden section, when you have a different [proportionate] relation of length to lines. That s very farfetched, but I am always attracted by the farfetched.. . . The comb becomes [a metaphor of] the generation of space, space generated by [the number and width of] the teeth.. . . There is a possibility, as I said, of generating space from a flat surface. You can do it with any surface. ... 

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

The efficiency of the golden section stems from the special value of the ratio i. We require the ratio of the larger of the two segments to the total length of the interval be the same as the ratio of the smaller to the larger segment,... 

Fig. 2.7. Notations used in golden-section search derivation...

The golden section search guarantees that each new function evaluation will reduce the uncertainty interval to a length of >. times the previous interval. This is comparable to, but not as good as interval halving in the bisection method of solving a nonlinear equation. You can easily calculate that to attain an error tolerance EP we need... 

Example 2.2.1 Optimal drug dosing by golden section search... 

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

Although the golden section search works quite well, it is obviously not the best available for a given number of function evaluations. For example, with only two evaluations allowed it is better to choose the internal points close to the midpoint of the initial interval, as we already discussed. The idea can... 

To avoid these problems Brent (ref. 11) suggested a combination of the parabolic fit and the golden section bracketing technique. The main idea is to apply equation (2.20) only if (i) the next estimate falls within the most recent bracketing interval (ii) the movement from the last estimate is less than half the step taken in the iteration before the last. Otherwise a golden section step is taken. The following module based on (ref. 12) tries to avoid function evaluation near a previously evaluated point. 

Parabolic interpolation is more effective than golden section search for this problem, because the function is of parabolic character in the vicinity of the minimum. To show a counterexample we slightly change the approximate objective function (2.19) and define by... 

The only parameter that has been fixed in the above three sequential stages is the HRAT. We can subsequently update the H RAT by performing a one-dimensional search using the golden section search method, which is shown as the outside loop in Figure 8.20. 

Remark 1 Steps (i) and (ii) are applied to the overall HEN without decomposing it into subnetworks. It is assumed, however, that we have a fixed HRAT for which we calculated the minimum utility cost. The HRAT can be optimized by using the golden section search in the same way that we described it in Figure 8.20. 

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