What Is The Golden Search Value?

The Golden Search value is derived from the golden ratio, ϕphi 𝜙 (phi), which is approximately 1.618034. The algorithm itself, properly known as the golden section search, uses this ratio to efficiently find the minimum or maximum of a unimodal function within a specified interval. The specific search values, or probe points, used in each step of the golden section search are placed at a distance related to ϕphi

𝜙

from the ends of the current search interval.

Detailed analysis of the golden search value

The "golden search value" refers to the constant proportion used in the golden section search method. This iterative optimization algorithm progressively narrows the search interval for an extremum (a minimum or maximum).

The golden ratio (ϕphi

𝜙

)

The search is founded on the golden ratio, ϕphi

𝜙

, an irrational number with a deep history in mathematics and aesthetics.

• Formula: The exact value of the golden ratio is expressed as ϕ=1+52phi equals the fraction with numerator 1 plus the square root of 5 end-root and denominator 2 end-fraction

  𝜙=1+5√2

  .

• Approximate value: The decimal approximation of ϕphi

  𝜙

  is 1.61803398875....

• The reciprocal: The reciprocal of the golden ratio, 1/ϕ1 / phi

  1/𝜙

  , is approximately 0.618034. This is the value most commonly used for positioning the internal points in the search algorithm.

The golden section search algorithm

The algorithm operates on a "unimodal" function—one that has a single local extremum within a given interval. Here is a step-by-step breakdown of the process for finding a minimum:

1. Define the search space: The algorithm begins with a defined search interval, [a,b]open bracket a comma b close bracket

   [𝑎,𝑏]

   , where the minimum is known to exist.

2. Calculate initial interior points: The algorithm places two initial interior points, x1x sub 1

   𝑥1

   and x2x sub 2

   𝑥2

   , symmetrically inside the interval. The placement is determined by the golden ratio's reciprocal:

   • x1=a+(1−1/ϕ)(b−a)x sub 1 equals a plus open paren 1 minus 1 / phi close paren open paren b minus a close paren

     𝑥1=𝑎+(1−1/𝜙)(𝑏−𝑎)

   • x2=a+(1/ϕ)(b−a)x sub 2 equals a plus open paren 1 / phi close paren open paren b minus a close paren

     𝑥2=𝑎+(1/𝜙)(𝑏−𝑎)

   • Note that 1−1/ϕ=1/ϕ21 minus 1 / phi equals 1 / phi squared

     1−1/𝜙=1/𝜙2

     , which is approximately 0.382.

3. Evaluate the function: The function is evaluated at these two points, yielding f(x1)f of open paren x sub 1 close paren

   𝑓(𝑥1)

   and f(x2)f of open paren x sub 2 close paren

   𝑓(𝑥2)

   .

4. Narrow the interval: The algorithm uses the function evaluations to shrink the interval.

   • If f(x1)<f(x2)f of open paren x sub 1 close paren is less than f of open paren x sub 2 close paren

     𝑓(𝑥1)<𝑓(𝑥2)

     , the new interval becomes [a,x2]open bracket a comma x sub 2 close bracket

     [𝑎,𝑥2]

     . This is because the minimum must lie within this smaller range.

   • If f(x1)>f(x2)f of open paren x sub 1 close paren is greater than f of open paren x sub 2 close paren

     𝑓(𝑥1)>𝑓(𝑥2)

     , the new interval becomes [x1,b]open bracket x sub 1 comma b close bracket

     [𝑥1,𝑏]

     for the same reason.

5. Repeat iteratively: The process is repeated with the new, smaller interval. The key advantage of using the golden ratio is that one of the two interior points for the new iteration is already in place from the previous step. This saves a function evaluation with each iteration.

Efficiency and comparison with other methods

The golden section search is a robust method for optimization, but it has trade-offs compared to other techniques.