minimizer

Minimizer is a point x* in the domain of a function f that minimizes f. Formally, x* is a minimizer if f(x*) ≤ f(x) for all x in the domain. If the inequality holds for every x, x* is a global minimizer; if it only holds for x in a neighborhood of x*, x* is a local minimizer. The set of minimizers is called the argmin of f and is written as argmin f = { x in D | f(x) = inf_{x in D} f(x) }.

Existence: On a nonempty compact domain D and with f continuous, a global minimizer exists (Weierstrass). If f is coercive (f(x) → ∞ as ||x|| → ∞) on a non-compact domain, a global minimizer may exist as well. Local minimizers can occur even when no global minimizer exists.

Characterizations: If f is differentiable on an open set and x* is a local minimizer in the interior, Fermat's rule gives grad f(x*) = 0. Second-order conditions use the Hessian: positive semidefinite for local minimizers, positive definite for strict local minimizers. If f is convex on a convex domain, every local minimizer is global, and if f is strictly convex, the minimizer is unique.

Examples: The quadratic f(x) = (x-3)^2 has a global minimizer at x* = 3 with minimum value 0. In statistics and machine learning, many estimators are defined as minimizers of a loss or risk function.

Computational notes: In algorithms, convergence properties depend on convexity, smoothness, and constraints. Common methods include gradient-based approaches, Newton-type methods, and proximal algorithms for non-smooth problems.