argmin

Argmin is a mathematical concept used to denote the argument(s) that minimize a given function. For a function f defined on a domain X, the argmin is the set of points x in X such that f(x) is as small as possible: argmin_{x in X} f(x) = { x in X | f(x) ≤ f(y) for all y in X }. The minimum value of f is min f (or inf f if the minimum is not attained); argmin specifies the locations where that minimum is achieved.

Existence of an argmin depends on the domain and the function. If f is continuous on a compact domain X, an argmin exists by the extreme value theorem. In general, the infimum of f may not be attained, in which case the argmin is empty. On finite domains, the argmin is simply the set of minimizers among the finitely many points.

Examples illustrate the idea. The function f(x) = (x − 2)^2 on the real line has argmin {2}, with minimum value 0. A function with multiple minimizers is f(x) = (x^2 − 1)^2 on the real line, whose argmin is {−1, 1} with minimum value 0.

In optimization and machine learning, the argmin represents the parameters or inputs that minimize a loss or objective function. Numerically, approximate argmin is sought using methods such as grid search, gradient-based optimization, or proximal and subgradient methods. If f is convex, every local minimizer is global, and the argmin is a single point or a convex set; for nonconvex or nondifferentiable functions, the argmin may be non-unique or only approximately attained, with algorithms converging to a stationary point rather than a global minimum.

Notes: x* or x^* is often used to denote a chosen minimizer; the argmin may refer to a set when multiple minimizers exist, and the domain should be specified. See also min, minimization, convex optimization.