Minimization

Minimization is a mathematical process of finding the smallest value of a function and, often, the point at which that value is attained. In optimization, a minimization problem seeks x in a domain X to minimize an objective function f: X → R. The minimum value is min f(x), and a corresponding point x* is a minimizer.

Minimization problems are often categorized as unconstrained, where x ranges over a space such as R^n, and constrained, where x must satisfy constraints. The minimizer may be global (the smallest value over the entire domain) or local (smallest value within a neighborhood). Not every function has a minimizer on every domain; existence depends on properties such as continuity, compactness, and coercivity.

For differentiable functions, necessary conditions for an interior minimizer include that the gradient of f at x* is zero. Second-order conditions involve the Hessian being positive semidefinite. For constrained problems, the method of Lagrange multipliers provides conditions that combine the gradient of f with gradients of the constraints.

Convexity plays a central role: if f is convex on a convex domain, any local minimum is global; if f is strictly convex, the minimizer is unique. In practice, many minimization tasks are solved numerically by gradient-based methods (gradient descent, Newton's method) or by specialized algorithms for linear or convex problems (linear programming, interior-point methods).

Common contexts include science and engineering, economics, and machine learning, where one seeks to minimize loss, cost, or risk. The concept is dual to maximization and to the broader field of optimization.