minimisation

Minimisation is the process of finding the smallest possible value of a quantity or function. In mathematics and applied disciplines, it involves determining an input that yields the minimum of an objective, often subject to constraints.

In mathematics, a minimisation problem seeks x in a domain X that minimizes f(x). A global minimum x* satisfies f(x*) ≤ f(x) for all x in X; a local minimum satisfies the inequality only for points near x*. If f is differentiable, unconstrained minima occur where the gradient is zero and the Hessian is positive semidefinite; constrained problems use Lagrange multipliers or Karush-Kuhn-Tucker conditions.

Common methods include gradient descent, Newton’s method, and conjugate gradient for continuous problems; for convex objectives, any local minimum is also a global minimum. Discrete minimisation appears in combinatorial optimization and uses dynamic programming, branch-and-bound, and integer programming.

Applications span economics (cost minimisation and resource allocation), engineering design, statistics and machine learning (minimising loss or negative log-likelihood), data compression, and model reduction. In software and systems, minimisation can refer to reducing runtime, memory usage, or code size.

Variants include approximations, heuristics, and probabilistic methods such as simulated annealing or stochastic gradient descent, used when problems are large, nonconvex, or noisy.