KarushKuhnTuckervoorwaarden

Karush–Kuhn–Tucker conditions, commonly called the KKT conditions, are a set of necessary conditions for a solution to be optimal in constrained optimization problems. They extend the method of Lagrange multipliers to include inequality constraints and are named after William Karush, who introduced them in 1939, and Harold W. Kuhn and Albert W. Tucker, who popularized them in 1951.

Consider the problem of minimizing a function f(x) subject to equality constraints h_i(x) = 0 for i = 1,...,m and inequality constraints g_j(x) ≤ 0 for j = 1,...,p. Let L(x, λ, μ) = f(x) + λ^T h(x) + μ^T g(x) be the Lagrangian, where λ and μ are the multipliers for the equality and inequality constraints, respectively.

The KKT conditions are:

- Primal feasibility: h_i(x) = 0 for all i, and g_j(x) ≤ 0 for all j.

- Dual feasibility: μ_j ≥ 0 for all j.

- Complementary slackness: μ_j g_j(x) = 0 for all j.

- Stationarity: ∇f(x) + ∑_i λ_i ∇h_i(x) + ∑_j μ_j ∇g_j(x) = 0.

Under a constraint qualification (such as Slater’s condition), these conditions are necessary for a local optimum. If the problem is convex (f and g convex, h affine) and Slater’s condition holds, the KKT conditions are also sufficient. The framework is widely used in optimization algorithms and in applications across economics, engineering, and machine learning.