Function minimization and maximization are the process of finding the smallest and largest value of a given function. Let's talk briefly about that value.

If the value is within a given range, then it is called the local extrema; if it is within the entire domain of a function then it is called the global extrema. Let's say we have a function f, and it's defined against a domain X. The maximum, or global, point at x* is f(x*) is greater than or equal to f(x) for all x in the domain X. Conversely, the function's global minimum point at x* is f(x*) is less than or equal to f(x) for all x in the domain X.

In a simpler fashion, the maximum point is also called the maximum value, and the minimum point is called the minimum value, of the function. The global maximum or minimum is either the highest or lowest function value in the entire domain space (search space), and the local maximum or minimum is the highest or lowest value in a defined neighborhood within that search space (it is not allowed to reside on the boundary at all), as follows:

In this simple illustration, D is the global minimum and G is the global maximums. A, C, and E are local maximums (it is important to note that a function can have more than one global or local maximum or minimum). B and F are considered local minimum. X, Y, and Z exist around the minimum value F, since the value of Y is less than both X and Z:

Let's take a real example. Let's say we are using the function sin(x). The maximum value for this function is +1, and the minimum value would be -1. Therefore, we have the global minimum and maximum. Sin(x) can take on any value between negative and positive infinity, but over all of these values, the maximum can only be +1 and, the minimum can only be -1.

If we then restrict the search space (global domain) to between 0 and 90 (sometimes people call this the interval), sin(x) will now have a minimum of 0, and its value will be 0. However, the global or maximum value will now be 90 and the value is 1, because we restricted our search space to between 0 and 90. All values of sin(x) will lie between 0 and 1, within the interval of 0 to 90.