Polynomial of degree 4, on the right one finds a local optimum, on the left is the global optimum.
In mathematics, a global optimum is a selection from a given domain which yields the highest value, when a specific function is applied. For example, for the function
f(x) = −x2 + 2,
defined on the real numbers, 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.
A typical result of a global optimization task might be something like “after a one hour search, point P has the highest probability of being the globaloptimum of error function F in region R”.
One simple global optimization scheme is the grid search: the merit function M is evaluated at points on a regular grid aligned to the coordinate axes, and the sample with the lowest value of M is taken as an estimate of the global minimum.
Simulated annealing belongs to a class of global optimizers that are called controlled random search methods because the merit-function space is sampled randomly according to a scheme in which the values of several parameters determine the distribution of the random samples.
In mathematics, a globaloptimum 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.
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.
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