minimisant

Minimisant is a term used primarily in French-language mathematics to denote a point, vector, or function value that minimizes a given objective function. In English-language texts, the corresponding terms are minimizer or arg min, while the adjective form is minimizing. The noun minimisant is most often encountered in discussions of optimization problems written in French.

Definition and usage

Given an objective function f defined on a domain D, a point x* in D is a minimisant (minimizer) if f(x*) ≤ f(x) for all x in D. If the inequality is strict for all x ≠ x*, x* is a unique minimisant. In unconstrained problems, D is typically the ambient space; in constrained problems, D is the feasible set.

Global and local minimizers

A global minimisant minimizes f over the entire domain D. A local minimisant minimizes f within a neighborhood around x*, and there may exist other, globally optimal or nonoptimal, points elsewhere. The existence of minimizers depends on the properties of f and D, such as compactness and lower semicontinuity.

Mathematical conditions

Existence results often rely on the Weierstrass or direct methods: a lower semicontinuous function on a compact set attains its minimum. If f is differentiable, first-order conditions state that a minimisant x* satisfies ∇f(x*) = 0 (for unconstrained problems), with second-order conditions involving the Hessian to guarantee a local minimum. In constrained problems, Lagrange multipliers characterize stationary minimizers under constraints.

Examples

For f(x) = (x − 2)^2 on the real line, x* = 2 is the minimisant with f(x*) = 0. In higher dimensions, minimizing a quadratic form or a convex function yields a global minimisant, often unique if the function is strictly convex.

See also

Minimization problem; minimizer; argmin; global vs local minimum; calculus of variations. Minimisant is more commonly used in French contexts; in English, minimizer or argmin is preferred.