argmina

Argmina is a term sometimes used in mathematics to denote the set of arguments that minimize a function. It is formed as a plural reference to argmin and is less common than the singular expression “argmin” but appears in some texts and discussions of optimization problems.

Definition and notation: For a real-valued function f defined on a domain X, the argmin of f is the set Argmin f = { x in X | f(x) = min_{y in X} f(y) }. When the minimum is unique, Argmin f contains a single element, often written as x*,; otherwise it can be an interval or a more general set, in particular not necessarily connected.

Properties and examples: If f is convex on a convex set X and has a unique minimizer, Argmin f = {x*}. Example: f(x) = (x-2)^2 on R has Argmin f = {2}. Example of multiple minimizers: f(x) = 0 for x in [0,1], f(x) = (x-2)^2 elsewhere; then Argmin f = [0,1]. Applications include statistical estimation, machine learning, and operations research. In differentiable cases, interior minima satisfy ∇f(x) = 0 and, for constrained problems, Karush-Kuhn-Tucker conditions apply.

Relation to broader optimization: Argmina is related to the concept of argmax, as both describe the input values that optimize a given objective function.