ArgmaxxA

ArgmaxxA is a compact notation sometimes used in optimization and applied mathematics to denote the argmax of a function with respect to a variable x, with A acting as a parameter. Symbolically, for a function f: X × Θ → R, argmaxxA f(x; A) denotes the set of maximizers of f over x in X for a fixed A ∈ Θ. When a unique maximizer exists, this set is a singleton, often written x*(A).

Formally, define M(A) = argmaxx∈X f(x; A) = { x ∈ X : f(x; A) ≥ f(y; A) for all y ∈ X }. The existence and size of M(A) depend on the domain X and the properties of f. If X is compact and f is continuous in x for each A, M(A) is nonempty; uniqueness follows from strict concavity in x or similar conditions.

Examples help illustrate the concept. If f(x; θ) = - (x − θ)^2 with X = R, the maximizers are x = θ, so M(θ) = {θ}. A multi-maximum example is f(x; α) = - (x^2 − 1)^2 + αx; depending on α, M(α) may contain one or multiple values.

Applications span optimization, economics, and machine learning. ArgmaxxA commonly determines optimal actions, decisions, or predicted labels by selecting the argument that maximizes an objective given parameters A. In practice, the argmax is not differentiable, leading to approximations such as softmax or log-sum-exp relaxations for use in optimization and learning.