Minimipiste

Minimipiste is a term used in mathematics and optimization to denote a point where a function attains its minimum value on a given domain. It can be local, where the function value is not exceeded in a neighborhood, or global, where the value is the smallest over the entire domain. A minimipiste may be strict (the inequality is strict for nearby points) or non-strict.

In differentiable problems, a common necessary condition for a local minimipiste is that the gradient vanishes: the gradient ∇f(x*) equals zero. For twice differentiable functions, the Hessian matrix at x* being positive semidefinite suggests a local minimum, while a positive definite Hessian guarantees a strict local minimum. In one dimension, a local minimum occurs at x* where f'(x*) = 0 and f''(x*) > 0. If the function is convex, any local minimum is also a global minimum.

Examples include f(x) = (x − 4)^2 + 2, which has a global minimipiste at x = 4 with value 2, and f(x, y) = x^2 + y^2, which has a unique global minimipiste at (0, 0). Non-differentiable points can require subgradients; in convex optimization, subgradient methods can locate minimipiste even without differentiability.

Numerical methods to find minimipiste include gradient descent, Newton’s method, and interior-point methods for constrained problems. In discrete or combinatorial settings, other approaches such as dynamic programming or metaheuristics may be used. Minimipiste concepts underpin many applications in physics, economics, engineering, and machine learning, where the goal is to minimize energy, cost, or loss functions.