Optimalstelle

Optimalstelle is a term used in optimization literature to denote a point in the domain of a function where an optimum value is attained. The concept covers both local and global optima as well as unconstrained and constrained problems. In many contexts the word is paired with indicators such as minimum or maximum to specify the type of optimum.

In unconstrained optimization, an optimalstelle x* typically satisfies that the gradient of the objective function f vanishes there: ∇f(x*) = 0. If f is twice differentiable, a positive definite Hessian ∇^2f(x*) implies a strict local minimum, while a negative definite Hessian implies a local maximum. If the function is convex, any local minimum is also a global minimum; if it is concave, any local maximum is a global maximum. Existence of an optimalstelle in unconstrained problems is not guaranteed without further conditions.

In constrained optimization, the feasible set is restricted by constraints, and optimalstellen must satisfy first-order conditions such as the Karush-Kuhn-Tucker (KKT) conditions with Lagrange multipliers. Existence results often rely on continuity of f and compactness of the feasible set; in convex problems, a global optimum exists and, under strict convexity, is unique.

Common methods to find optimalstellen include gradient-based algorithms (gradient descent, Newton’s method) for unconstrained problems and specialized techniques such as interior-point methods or sequential quadratic programming for constrained problems. Examples illustrate the concept: f(x) = x^2 has a global and local optimalstelle at x* = 0 (minimum); f(x) = - (x-2)^2 has a global maximum at x* = 2.