Minimum

Minimum is a concept used in mathematics and related fields to denote the smallest value within a set or function.

In a partially ordered set, the minimum of a nonempty subset S is an element m in S such that m ≤ s for every s in S. If such an element exists, it is called the minimum of S. A set may have no minimum even if it has a lower bound; for example, the open interval (0, 1) has an infimum of 0 but no minimum. In the real numbers, a nonempty set that is bounded below has an infimum (greatest lower bound), and if the infimum lies within the set, it is the minimum.

When dealing with real-valued functions, the global or absolute minimum is a point where the function attains its smallest value over its domain. A local minimum is a point where the function is no larger than nearby points. If a function is differentiable, a local minimum interior to the domain typically has a zero derivative, and a positive second derivative (in one variable) or a positive semidefinite Hessian (in multiple variables) confirms a minimum.

Optimization problems often seek to minimize a function under certain constraints. Convexity plays a key role: a convex function on a convex domain has a unique global minimum. Numerical methods such as gradient descent or interior-point algorithms are commonly used to locate minima in practice.

The term also appears in statistics and data analysis, where the minimum is the smallest observed value in a sample, and in computer science as the operation of selecting the smallest element. The minimum contrasts with the maximum and with the concept of an infimum or lower bound in more advanced contexts.