Karush–Kuhn–Tucker conditions

The Karush–Kuhn–Tucker conditions are first-order conditions for constrained optimization, combining stationarity, feasibility, dual feasibility, and complementary slackness at candidate optima.

Lagrangian function: A function combining an objective with constraints weighted by multipliers. Its gradient yields the stationarity equation in the KKT conditions.
Constrained optimization: Optimization of an objective over a set restricted by equality or inequality conditions. KKT conditions address optima when the feasible set has explicit constraints.
Sequential quadratic programming: A nonlinear optimization method that repeatedly solves quadratic approximations with linearized constraints. Its subproblems approximate the KKT system of the original problem.
Fritz John conditions: Necessary first-order conditions for constrained optimization that allow a zero objective multiplier. They remain applicable when standard constraint qualifications for KKT fail.
William Karush: An American mathematician whose 1939 master's thesis derived conditions for constrained extrema. His thesis contains an early form of the conditions later bearing his name.
Complementary slackness: A condition requiring paired nonnegative quantities to have product zero. It links each inequality constraint's slack to its multiplier.
Gradient: A vector of partial derivatives pointing in the direction of greatest local increase. Objective and constraint gradients enter the stationarity relation.
Interior-point method: An optimization algorithm that approaches constrained solutions through the interior of the feasible region. Primal-dual variants drive residuals of KKT conditions toward zero.
Weierstrass theorem: A theorem guaranteeing extrema for continuous functions on compact sets. It ensures existence, while KKT conditions characterize local candidates through derivatives.
Harold W. Kuhn: An American mathematician who co-developed the named conditions for nonlinear programming. His joint 1951 paper helped establish their role in optimization.

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