Local vs Global Maximum
The greatest value of a set or a function is known as maximum. Consider the set {ai | i ∈ N}. The element ak where ak ≥ ai for all i is known as the maximum element of the set. If the set is ordered it becomes the last element of the set.
For example, take the set A={1,6,9,2,4,8,3}. Considering all the elements, 9 is greater than every other element in the set. Therefore, it is the maximum element of the set. By ordering the set, we get A={1,2,3,4,6,8,9}. In the ordered set, 9 (the maximum element) is the last element.
Local Maximum
The greatest value in a subset or a range of a function is known as the local maximum. It is the largest value for the given subset or the range, but there can be other elements larger than that outside the noted range or the subset. There can be many local maxima in the range of the function or the universal set.
Consider the set of integers 1 to 10, S={1,2,3,4,5,6,7,8,9,10}. A is a subset of the S. Maximum of A (9) is not the maximum for the whole set, which is 10. Hence 9 is a local maximum.
Global Maximum
The largest overall value of a function or a set is known as the global maximum. Is set S, 10 is the global maximum. This element is larger than any value of the set. If it’s a function it`s larger than any other value of the function over the whole domain of the set (greatest element in the codomain). Global maximum of a function or a set is unique (for that particular case).
In the case of a function, at the maximum value the gradient of the function is zero. The gradient just before the maximum is positive and just after that is negative. This is used as a test to find local maxima in functions (First derivative test).
What is the difference between Global Maximum and Local Maximum?
• Maximum is the greatest element in a set or a range of a function.
• Global maximum is the greatest value among the overall elements of a set or values of a function.
• Local maximum is the greatest element in a subset or a given range of a function.
• Global maximum is unique while the local maximum is not. There may be more than one local maximum. If there is only one local maximum, then it is the global maximum.