minimisationsproblem

Minimisationsproblem, often translated as a minimization problem, is the task of finding a decision vector x that minimizes an objective function f(x) over a feasible set X. The problem is usually written as minimize f(x) subject to x in X, or with explicit constraints g_i(x) ≤ 0 and h_j(x) = 0. If there are no constraints besides domain bounds, it is an unconstrained minimisation; with constraints, it is a constrained minimisation.

A solution is a point where no nearby feasible point yields a smaller value. A global minimum gives the lowest possible objective over all feasible points; a local minimum is the smallest value in a neighborhood. Existence of a solution depends on problem data; smoothness, coercivity, and compactness of the feasible set help guarantee existence.

Methods range from analytical to numerical. For smooth f, gradient-based methods such as gradient descent or Newton’s method are common; Lagrange multipliers handle constraints. Convex minimisation, where f is convex and the feasible set is convex, has the property that any local minimum is global. Linear programming solves problems with linear f and linear constraints; integral or combinatorial problems use specialized algorithms such as branch-and-bound or cutting planes. General nonconvex problems may be NP-hard.

Applications span economics, engineering, operations research and machine learning, including portfolio optimisation, production planning, resource allocation, and loss minimisation in predictive modelling.

The term minimisationsproblem is used in the Nordic languages; in English, minimization problem is common.