golden section search

The golden section search requires that these intervals be equal.

The golden ratio is key to the golden section search.

For this reason, the sequence variant of golden section search is often called Fibonacci search.

Fibonacci search and Golden section search were discovered by Kiefer (1953).

Golden section search (similar to ternary search, useful if evaluating f takes most of the time per iteration)

Golden section search conceptually resembles PS in its narrowing of the search-range, only for single-dimensional search-spaces.

Of the various methods of dividing the interval, golden section search is particularly simple and effective, as the interval proportions are preserved regardless of how the search proceeds:

The golden section search chooses the spacing between these points in such a way that these points have the same proportion of spacing as the subsequent triple or .

Alternating the parabolic iterations with a more robust method (golden section search is a popular choice) to choose candidates can greatly increase the probability of convergence without hampering the convergence rate.

The golden section search is a technique for finding the extremum (minimum or maximum) of a strictly unimodal function by successively narrowing the range of values inside which the extremum is known to exist.

In order to approximate the probe positions of golden section search while probing only integer sequence indices, the variant of the algorithm for this case typically maintains a bracketing of the solution in which the length of the bracketed interval is a Fibonacci number.

To realise the advantage of golden section search, the function would be implemented with caching, so that in all invocations of above, except the first, would have already been evaluated previously - the result of the calculation will be re-used, bypassing the (perhaps expensive) explicit evaluation of the function.

However, he also made significant contributions to other areas of statistics and optimization, including the introduction of golden section search (his master's thesis work) the Dvoretzky-Kiefer-Wolfowitz inequality and the Bahadur-Ghosh-Kiefer representation (with R. R. Bahadur and J. K. Ghosh)