argminy

Argminy is the notation for the set of arguments that minimize a function y: X → R over a domain X. More precisely, argminx y(x) denotes the set { x in X | y(x) = min_{z in X} y(z) }. In many texts the notation is written as argmin_x y(x), with the minimum value denoted by min_{x in X} y(x). The term is used in optimization to identify the points at which the objective function attains its smallest value.

Existence and uniqueness depend on properties of the function and the domain. If X is compact and y is continuous, the minimum exists and the argmin is nonempty. If y is strictly convex on a convex X, the argmin is unique. When the function has flat regions, is non-convex, or the domain is unbounded or constrained in complex ways, the argmin can consist of multiple points or be empty in degenerate cases.

Computation and methods. If y is differentiable, stationary points satisfy the condition ∇y(x) = 0 and, under suitable second-order conditions, yield local minima. In convex problems, any local minimum is globally optimal, and efficient algorithms exist (gradient methods, Newton methods, interior-point methods). For non-differentiable or constrained problems, subgradients or KKT (Karush-Kuhn-Tucker) conditions are employed. In non-convex settings, numerical optimization may converge to local minima, so global optimality often requires problem reformulation, convexification, or global optimization techniques.

Examples. For y(x) = (x−2)², argminx y(x) = {2}. For y(x) = sin²x over the real line, argminx y(x) = {kπ | k ∈ Z}. These illustrate how argminy can be a single point or an infinite set, depending on the function and domain.

See also: argmax, convex optimization, gradient descent, KKT conditions.