minimizers

Minimizers are points in a domain at which a function attains its smallest value. For a function f: X → R, a point x* ∈ X is a global minimizer if f(x*) ≤ f(x) for every x ∈ X. A local minimizer satisfies f(x*) ≤ f(x) for all x in some neighborhood of x*. If the inequality is strict for all x ≠ x*, x* is a strict minimizer. A function may have multiple minimizers, especially when the problem is non-convex. In convex optimization, every local minimizer is global, and if the function is strictly convex there is a unique minimizer.

In constrained problems, minimizers must satisfy not only the objective but also the constraints. Optimality conditions such as the Karush-Kuhn-Tucker (KKT) conditions for differentiable problems with inequality constraints and Lagrange multiplier methods for equality constraints provide practical characterizations of minimizers.

Existence of minimizers is a foundational concern. A classic result (Weierstrass) states that if the domain X is compact and f is continuous, a global minimizer exists. More general existence results rely on properties like coercivity (f tends to infinity as ||x|| → ∞) and lower semicontinuity, which ensure the attainment of minimum values in broader settings.

Minimizers arise in many fields, including mathematics, economics, engineering, and machine learning. In optimization algorithms, one seeks points where the gradient vanishes (stationary points) or where subgradients satisfy optimality conditions, with convexity providing stronger guarantees such as uniqueness of the minimizer.