minimiser

A minimiser (British English) or minimizer (American English) is a term used in mathematics and related fields to denote an element of a domain at which a given function attains its minimum value. Formally, for a real-valued function f defined on a set X, a point x* in X is a minimiser if f(x*) ≤ f(x) for all x in X. If the inequality is strict for all x ≠ x*, x* is a strict minimiser.

Minimisers can be classified by how far their minimum value extends across the domain. A global minimiser achieves the smallest value of f over the entire domain, while a local minimiser attains the smallest value within a neighborhood around the point. In convex optimization, every local minimiser is also a global minimiser, which simplifies analysis.

Existence and characterization depend on the problem. If X is compact and f is continuous, a minimum exists (Weierstrass theorem). For differentiable f on a smooth domain, interior minimisers satisfy the first-order condition ∇f(x*) = 0; second-order conditions involve the Hessian being positive semidefinite (positive definite for a strict local minimiser). In constrained problems, necessary and sufficient conditions are expressed through Karush-Kuhn-Tucker (KKT) conditions.

Common methods to find minimisers include gradient descent and Newton-type algorithms for unconstrained problems, and specialized algorithms for convex optimization, such as interior-point or proximal methods. In the calculus of variations, a minimiser is a function that minimizes a functional, often found via the Euler–Lagrange equation.

Minimisers are central in optimization, statistics, economics, and engineering, where determining the configuration that minimizes a loss, cost, or energy is a fundamental objective.