Minima

Minima is the plural form of minimum, referring to points in the domain of a function where the function value is smallest with respect to nearby values (local minima) or overall on the domain (global minima). The corresponding smallest values are called minimum values.

A local minimum of a function f occurs at a point x0 if there exists a neighborhood around x0 such that f(x) ≥ f(x0) for all x in that neighborhood. If the inequality is strict for all nearby x ≠ x0, the local minimum is called a strict local minimum. A global (or absolute) minimum is a point x0 where f(x0) ≤ f(x) for every x in the domain; the associated value f(x0) is the global minimum value.

In one-variable calculus, a common test is that if f is differentiable and f'(x0) = 0, then a local minimum may occur. If additionally f''(x0) > 0, x0 is a local minimum; if f''(x0) < 0, x0 is a local maximum; if f''(x0) = 0, the test is inconclusive. In several variables, a similar criterion uses the gradient and the Hessian: a critical point where the gradient is zero can be a local minimum if the Hessian is positive definite.

Existence of minima depends on the domain and the function. A function may be bounded below but fail to attain a minimum value, so the infimum exists without a minimizer. For continuous functions on compact (closed and bounded) domains, minima are guaranteed to exist. Common examples include f(x) = (x − 2)^2, which has a global minimum at x = 2, and f(x, y) = (x − 1)^2 + (y − 2)^2, with a global minimum at (1, 2).