Golden section search

Diagram of a golden section search

The golden section search is a technique for finding the extremum (minimum or maximum) of a unimodal function by successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio. The algorithm is closely related to a Fibonacci search (also described below) and to a binary search. Fibonacci search and Golden section search were introduced by Kiefer (1953). (see also Avriel and Wilde (1966)).

Basic idea

The diagram above illustrates a single step in the technique for finding a minimum. The functional values of f(x) are on the vertical axis, and the horizontal axis is the x parameter. The value of f(x) has already been evaluated at the three points: x₁, x₂, and x₃. Since f₂ is smaller than either f₁ or f₃, it is clear that a minimum lies inside the interval from x₁ to x₃ (since f is unimodal).

The next step in the minimization process is to "probe" the function by evaluating it at a new value of x, namely x₄. It is most efficient to choose x₄ somewhere inside the largest interval, i.e. between x₂ and x₃. From the diagram, it is clear that if the function yields f_4a then a minimum lies between x₁ and x₄ and the new triplet of points will be x₁, x₂, and x₄. However if the function yields the value f_4b then a minimum lies between x₂ and x₃, and the new triplet of points will be x₂, x₄, and x₃. Thus, in either case, we can construct a new narrower search interval that is guaranteed to contain the function's minimum.

Probe point selection

From the diagram above, it is seen that the new search interval will be either between x₁ and x₄ with a length of a+c , or between x₂ and x₃ with a length of b . The golden section search requires that these intervals be equal. If they are not, a run of "bad luck" could lead to the wider interval being used many times, thus slowing down the rate of convergence. To ensure that b = a+c, the algorithm should choose x₄ = x₁ + (x₃ − x₂).

However there still remains the question of where x₂ should be placed in relation to x₁ and x₃. 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 x₁,x₂,x₄ or x₂,x₄,x₃. By maintaining the same proportion of spacing throughout the algorithm, we avoid a situation in which x₂ is very close to x₁ or x₃, and guarantee that the interval width shrinks by the same constant proportion in each step.

Mathematically, to ensure that the spacing after evaluating f(x₄) is proportional to the spacing prior to that evaluation, if f(x₄) is f_4a and our new triplet of points is x₁, x₂, and x₄ then we want:

$\frac{c}{a}=\frac{a}{b}.$

However, if f(x₄) is f_4b and our new triplet of points is x₂, x₄, and x₃ then we want:

$\frac{c}{(b-c)}=\frac{a}{b}.$

Eliminating c from these two simultaneous equations yields:

$\left(\frac{b}{a}\right)^2=\frac{b}{a}+1$

or

$\frac{b}{a}=\varphi$

where φ  is the golden ratio:

$\varphi= \frac{1+\sqrt{5}}{2}= 1.618033988\ldots$

The appearance of the golden ratio in the proportional spacing of the evaluation points is how this search algorithm gets its name.

Termination condition

In addition to a routine for reducing the size of the bracketing of the solution, a complete algorithm must have a termination condition. The one provided in the book Numerical Recipes in C is based on testing the gaps among x₁, x₂, x₃ and x₄, terminating when within the relative accuracy bounds:

$|x_3 - x_1| < \tau (|x_2| + |x_4|)\,$

where τ is a tolerance parameter of the algorithm and | x | is the absolute value of x. The check is based on the bracket size relative to its central value, because that relative error in x is approximately proportional to the squared absolute error in f(x) in typical cases. For that same reason, the Numerical Recipes text recommends that $\tau = \sqrt{\epsilon}$ where $\epsilon$ is the required absolute precision of f(x).

Recursive algorithm

double phi = (1 + Math.sqrt(5)) / 2;
double resphi = 2 - phi;
 
// a and c are the current bounds; the minimum is between them.
// b is a center point
// f(x) is some mathematical function elsewhere defined
// a corresponds to x1; b corresponds to x2; c corresponds to x3
// x corresponds to x4
 
public double goldenSectionSearch(double a, double b, double c, double tau) {
    double x;
    if (c - b > b - a)
      x = b + resphi * (c - b);
    else
      x = b - resphi * (b - a);
    if (Math.abs(c - a) < tau * (Math.abs(b) + Math.abs(x))) 
      return (c + a) / 2; 
    if (f(x) < f(b)) {
      if (c - b > b - a) return goldenSectionSearch(b, x, c, tau);
      else return goldenSectionSearch(a, x, b, tau);
    }
    else {
      if (c - b > b - a) return goldenSectionSearch(a, b, x, tau);
      else return goldenSectionSearch(x, b, c, tau);
    }
  }

To realise the advantage of golden section search, the function f(x) would be implemented with caching, so that in all invocations of goldenSectionSearch(..) above, except the first, f(x₂) would have already been evaluated previously — the result of the calculation will be re-used, bypassing the (perhaps expensive) explicit evaluation of the function. Together with a slightly smaller number of recursions, this 50% saving in the number of calls to f(x) is the main algorithmic advantage over Ternary search.

Fibonacci search

A very similar algorithm can also be used to find the extremum (minimum or maximum) of a sequence of values that has a single local minimum or local maximum. 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. For this reason, the sequence variant of golden section search is often called Fibonacci search.

See also

• Fibonacci search technique
• Brent's method

References

• Kiefer, J. (1953), "Sequential minimax search for a maximum", Proceedings of the American Mathematical Society 4 (3): 502–506, doi:10.2307/2032161, JSTOR 2032161, MR0055639 
• Avriel, Mordecai; Wilde, Douglass J. (1966), "Optimality proof for the symmetric Fibonacci search technique", Fibonacci Quarterly 4: 265–269, MR0208812 

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 10.2. Golden Section Search in One Dimension", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, http://apps.nrbook.com/empanel/index.html#pg=492