objektivfunksjon

Objektivfunksjon is a central concept in optimization. It refers to a function that assigns a real-valued scalar to each possible decision vector x, representing the cost, loss, or utility that should be optimized. The objective function encapsulates the goal of the problem, such as minimizing production cost or maximizing profit.

In a typical optimization problem, the objective function is minimized or maximized subject to a set of constraints. A standard form is: minimize f(x) (or maximize f(x)) subject to x belonging to a feasible set X, defined by equality and inequality constraints g_i(x) ≤ 0 and h_j(x) = 0. The feasible set may encode resource limitations, safety requirements, or regulatory rules. The variables x are the decision variables, which may be continuous, discrete, or a mix.

Properties of the objektivfunksjon influence how the problem can be solved. Differentiability enables gradient-based methods, while convexity guarantees that any local optimum is global. Other important properties include smoothness, Lipschitz continuity, and sparsity. The choice of formulation—linear, quadratic, nonlinear, or convex—determines the applicable algorithms and the computational effort required.

In practice, objective functions are used across fields such as operations research, machine learning, and economics. A single objective yields a conventional optimization problem, while multiple objectives may be handled through scalarization or Pareto efficiency concepts. Common examples include minimizing squared error in regression, minimizing total cost in logistics, and maximizing utility in economic models.