The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them. If the only extremum on the interval is on a boundary of the interval, it will converge to that boundary point. The method operates by successively narrowing the range of values on the specified interval, which makes it relatively slow, but very robust. The technique derives its name from the fact that the algorithm maintains the function values for four points whose three interval widths are in the ratio φ:1:φ, where φ is the golden ratio. These ratios are maintained for each iteration and are maximally efficient. Excepting boundary points, when searching for a minimum, the central point is always less than or equal to the outer points, assuring that a minimum is contained between the outer points. The converse is true when searching for a maximum. The algorithm is the limit of Fibonacci search (also described below) for many function evaluations. Fibonacci search and golden-section search were discovered by Kiefer (1953) (see also Avriel and Wilde (1966)).

The discussion here is posed in terms of searching for a minimum (searching for a maximum is similar) of a unimodal function. Unlike finding a zero, where two function evaluations with opposite sign are sufficient to bracket a root, when searching for a minimum, three values are necessary. The golden-section search is an efficient way to progressively reduce the interval locating the minimum. The key is to observe that regardless of how many points have been evaluated, the minimum lies within the interval defined by the two points adjacent to the point with the least value so far evaluated.

The diagram above illustrates a single step in the technique for finding a minimum. The functional values of ${\displaystyle f(x)}$ are on the vertical axis, and the horizontal axis is the x parameter. The value of ${\displaystyle f(x)}$ has already been evaluated at the three points: ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, and ${\displaystyle x_{3}}$. Since ${\displaystyle f_{2}}$ is smaller than either ${\displaystyle f_{1}}$ or ${\displaystyle f_{3}}$, it is clear that a minimum lies inside the interval from ${\displaystyle x_{1}}$ to ${\displaystyle x_{3}}$.

The next step in the minimization process is to "probe" the function by evaluating it at a new value of x, namely ${\displaystyle x_{4}}$. It is most efficient to choose ${\displaystyle x_{4}}$ somewhere inside the largest interval, i.e. between ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$. From the diagram, it is clear that if the function yields ${\displaystyle f_{4a}>f(x_{2})}$, then a minimum lies between ${\displaystyle x_{1}}$ and ${\displaystyle x_{4}}$, and the new triplet of points will be ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, and ${\displaystyle x_{4}}$. However, if the function yields the value ${\displaystyle f_{4b}<f(x_{2})}$, then a minimum lies between ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$, and the new triplet of points will be ${\displaystyle x_{2}}$, ${\displaystyle x_{4}}$, and ${\displaystyle x_{3}}$. Thus, in either case, we can construct a new narrower search interval that is guaranteed to contain the function's minimum.

From the diagram above, it is seen that the new search interval will be either between ${\displaystyle x_{1}}$ and ${\displaystyle x_{4}}$ with a length of a + c, or between ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$ 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 ${\displaystyle x_{4}=x_{1}+(x_{3}-x_{2})}$.

However, there still remains the question of where ${\displaystyle x_{2}}$ should be placed in relation to ${\displaystyle x_{1}}$ and ${\displaystyle x_{3}}$. 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 ${\displaystyle x_{1},x_{2},x_{4}}$ or ${\displaystyle x_{2},x_{4},x_{3}}$. By maintaining the same proportion of spacing throughout the algorithm, we avoid a situation in which ${\displaystyle x_{2}}$ is very close to ${\displaystyle x_{1}}$ or ${\displaystyle x_{3}}$ and guarantee that the interval width shrinks by the same constant proportion in each step.

Mathematically, to ensure that the spacing after evaluating ${\displaystyle f(x_{4})}$ is proportional to the spacing prior to that evaluation, if ${\displaystyle f(x_{4})}$ is ${\displaystyle f_{4a}}$ and our new triplet of points is ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, and ${\displaystyle x_{4}}$, then we want

However, if ${\displaystyle f(x_{4})}$ is ${\displaystyle f_{4b}}$ and our new triplet of points is ${\displaystyle x_{2}}$, ${\displaystyle x_{4}}$, and ${\displaystyle x_{3}}$, then we want