objectivefunktion

An objectivefunktion, or objective function, is a central concept in optimization. It is a scalar-valued function that encodes the goal of the optimization problem. The function assigns a real number to each admissible decision vector x, and the aim is to find x that minimizes or maximizes this value. The location of x is governed by constraints that define feasible solutions, such as equalities and inequalities.

In mathematical form, the problem is often written as: minimize f(x) subject to x in X, where f: R^n -> R is the objective function and X represents the feasible set, defined by g_i(x) <= 0 and h_j(x) = 0. Some problems are maximization problems, in which case one may maximize f(x) or equivalently minimize -f(x). The decision variables x can be continuous, discrete, or a mix (mixed-integer programming). The problem's structure, for example whether f is linear, convex, differentiable, or stochastic, influences the choice of solution methods.

Common types include linear objective functions in linear programming, convex objectives in convex optimization, and nonconvex objectives in nonlinear programming. In data-driven contexts, the objective function is often a loss or cost to be minimized, such as the sum of squared errors, cross-entropy, or negative log-likelihood. In economics or operations research, objective functions may reflect profit, cost, energy, or risk.

Properties such as convexity, smoothness, and constraint qualifications affect existence and uniqueness of solutions and the tractability of algorithms like gradient descent, simplex, or interior-point methods.