Golden-Section Search

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. 

To know that a minimum exists, we must hnd three points aj i a ax such that F(ai t) is less than either F(a i ) or F a. Suppose this has been done. Now choose another point amin anew amax and evaluate F anew)- If F a e ) F(ai t), then becomes the new interior point. Otherwise a sw will 

To know that a minimum exists, we must find three points amin aint amax such that F(aint) is less than either F(amin) or F(amax). Suppose this has been done. Now choose another point amin anew amax and evaluate F(anew). If F(anew) F(aint), then anew becomes the new interior point. Otherwise anew will become one of the new endpoints. Whichever the outcome, the result is a set of three points with an interior minimum and with a smaller distance between the endpoints than before. This procedure continues until the distance between amin and amax has been narrowed to an acceptable extent. The way of choosing anew is not of critical importance, but the range narrows fastest if a ew is chosen to be at 0.38197 of the distance between the interior point and the more distant of the endpoints amin and amax. 

The golden-section search method determines the optimal solution to a bounded objective function that is one-dimensional and unimodal. However, the function need not be continu- 

M + 1 objective functions. Thus in 10 steps, the optimal solution is located in an interval that is less than 1% of the distance between the lower and upper bounds. In 20 steps, that interval is reduced to less than 0.01% of that distance. 

EXAMPf-t Design of Heat Exchanger to Minimize Annual Costs 

Average specific heat of light gas oil = 0.50 Btu/lb-°F Average specific heat of crude oil = 0.45 Btu/lb-°F 

Use a floating-head, shell-and-tube heat exchanger for areas greater than 200 ft  

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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... 

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... 

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. 

Indirect methods solve the necessary conditions for an optimum (looking at the shape of the function) directly via iteration. Region elimination techniques such as Fibonacci and Golden Section searches use function evaluations only to delete a portion of the independent variable range at each iteration. Interpolation techniques use polynomial fitting (quadratic or cubic oftentimes) to predict the location of the optimum. 

A similar approach is used in the Golden Section Search Technique which uses as its basis a symmetrical placement of search points located at an arbitrary distance from each side of the search area.t This method can eliminate 99.9... 

Glass pipe and fittings, cost of, 505-506 Golden-section search technique, 406-407... 

The golden-section search, sometimes called the golden-mean search, is as simple to implement as the regular search, hut is more computationally efficient if < 0.29. In the golden-section search, only one new point is added at each cycle. 

The optimization problem is one-dimensional with a nonlinear objective function, which may be discontinuous, depending on the heat-exchanger area. The single decision variable is bounded. Therefore, the golden-section search is suitable for determining the optimal solution. The calculations can be carried out conveniently in the following manner for each selection of rLGo,out ... 

Table 18.2 Golden-Section Search Results for Example 18.3...

Be able to use the golden-section search to solve a constrained NLP problem in one decision variable. 

Golden-section search. Use 10 steps of the golden-section search method to find the optimal dimensions for the cylindrical reactor vessel in Example 16.11. In that example, the dimensions of the vessel are given as the inside diameter, D = 6.5 ft, and tangent-to-tangent length, Z, = 40 ft. These dimensions are not... 

The comjjutational effort involved in gradient evaluation is often in the order of function evaluation. In such cases, the golden section search (without any gradient calculation) may be more efficient. 

Comparison methods such as Fibonacci s method and golden section search exploit function unimodality within a specific interval of uncertainty with the aim... 

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