FirstOrderBedingung
FirstOrderBedingung is a term used in optimization theory to denote the first-order optimality condition. It describes the necessary conditions a solution must satisfy to be a candidate for an optimum in a mathematical optimization problem. The precise form depends on whether the problem is unconstrained or constrained.
In unconstrained optimization, where f: R^n -> R is differentiable, a local minimum occurs only at points
In constrained optimization with equality constraints h_i(x) = 0, the standard approach introduces the Lagrangian L(x, λ) = f(x)
The FirstOrderBedingung is a necessary condition for optimality under mild regularity assumptions, such as differentiability and
Example: minimize f(x,y) = x^2 + y^2 subject to x + y = 1. The Lagrangian is L = x^2 + y^2
In practice, FirstOrderBedingung underpins many optimization algorithms, including gradient-based methods and constrained optimization techniques.
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