Global optimum
Encyclopedia
In mathematics
, a global optimum is a selection from a given domain which yields either the highest value or lowest value (depending on the objective), when a specific function
is applied. For example, for the function
defined on the real number
s, the global optimum occurs at x = 0, when f(x) = 2. For all other values of x, f(x) is smaller.
For purposes of optimization
, a function must be defined over the whole domain, and must have a range which is a totally ordered set, in order that the evaluations of distinct domain elements are comparable.
By contrast, a local optimum is a selection for which neighboring selections yield values that are not greater. The concept of a local optimum
implies that the domain is a metric space
or topological space
, in order that the notion of "neighborhood" should be meaningful.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a global optimum is a selection from a given domain which yields either the highest value or lowest value (depending on the objective), when a specific function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
is applied. For example, for the function
- f(x) = −x2 + 2,
defined on the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the global optimum occurs at x = 0, when f(x) = 2. For all other values of x, f(x) is smaller.
For purposes of optimization
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
, a function must be defined over the whole domain, and must have a range which is a totally ordered set, in order that the evaluations of distinct domain elements are comparable.
By contrast, a local optimum is a selection for which neighboring selections yield values that are not greater. The concept of a local optimum
Local optimum
Local optimum is a term in applied mathematics and computer science.A local optimum of a combinatorial optimization problem is a solution that is optimal within a neighboring set of solutions...
implies that the domain is a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
or topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, in order that the notion of "neighborhood" should be meaningful.