Optimalwerte

Optimalwerte (plural of Optimalwert) is a term used in optimization and related fields to denote the values of a decision variable or vector that optimize an objective function under a specified set of constraints. In English, this is typically referred to as the optimal values or the optimal value of the objective function, together with the corresponding optimal solution.

Definitions: An optimal value is the best achievable value of the objective function given the feasible region. A global optimum is the best possible across all feasible points; a local optimum is the best within a neighborhood.

Common methods: Analytical calculus finds critical points by setting derivatives to zero; second-order conditions decide minima vs maxima. For constrained problems, methods include Lagrange multipliers and KKT conditions. In convex optimization, any local optimum is global. Numerical methods such as gradient descent, Newton's method, simplex or interior-point methods search for optimal values when closed-form solutions are unavailable.

Examples: If f(x) = -(x-2)^2 + 5, the optimal value is 5 at x = 2. In linear programming, maximize c^T x subject to Ax ≤ b; the optimal value is the maximum attainable objective value, attained at an optimal vertex of the feasible polyhedron.

Applications and notes: Optimalwerte are central in economics, engineering, logistics, machine learning. Existence and boundedness determine whether optimal values exist. Sensitivity analysis studies how optimal values change with parameters; dual problems provide alternative characterizations.