Karush–Kuhn–Tucker conditions

In <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a>, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are <a href="/facts/First-order_condition/5BmmVBAS">first derivative tests</a> (sometimes called first-order <a href="/facts/Necessary_and_sufficient_conditions/G45aDCY9">necessary conditions</a>) for a solution in <a href="/facts/Nonlinear_programming/cSXGuX2m">nonlinear programming</a> to be <a href="/facts/Optimization_(mathematics)/oRn8Iv5I">optimal</a>, provided that some regularity conditions are satisfied.
Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of <a href="/facts/Lagrange_multipliers/U5mSI5YP">Lagrange multipliers</a>, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a global maximum or minimum over the domain of the choice variables and a global minimum (maximum) over the multipliers. The Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem.
The KKT conditions were originally named after <a href="/facts/Harold_W._Kuhn/6jk6Lnue">Harold W. Kuhn</a> and <a href="/facts/Albert_W._Tucker/nYfsIxem">Albert W. Tucker</a>, who first published the conditions in 1951. Later scholars discovered that the necessary conditions for this problem had been stated by <a href="/facts/William_Karush/bJ3MQzPj">William Karush</a> in his master's thesis in 1939.

Karush–Kuhn–Tucker conditions open-in-new

Karush–Kuhn–Tucker conditions