KarushKuhnTuckeri

KarushKuhnTuckeri, often abbreviated as KKT, refers to a fundamental set of conditions used in optimization theory to find optimal solutions for constrained minimization problems. These conditions are named after the mathematicians William Karush, Abraham Kuhn, and Albert Tucker. The KKT conditions are a generalization of the method of Lagrange multipliers to problems with inequality constraints.

For a minimization problem, the KKT conditions state that if a point is a local minimum and satisfies certain regularity conditions (like constraint qualifications), then there must exist a set of Lagrange multipliers such that the gradient of the Lagrangian function is zero at that point. The Lagrangian function is formed by adding a weighted sum of the constraint functions to the objective function. The weights are the Lagrange multipliers, which are associated with the constraints.

The KKT conditions include primal feasibility, dual feasibility, and the complementary slackness condition. Primal feasibility ensures that the proposed solution satisfies all the original constraints. Dual feasibility requires that the Lagrange multipliers associated with inequality constraints be non-negative. Complementary slackness links the primal and dual variables, stating that for each inequality constraint, either the constraint is active (holds with equality) or its corresponding Lagrange multiplier is zero, or both. These conditions are necessary for optimality under certain assumptions and, in some cases, are also sufficient.