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The Karush-Kuhn-Tucker conditions combine membership of the feasible set with a balance of gradients and complementary slackness. For a minimization problem with differentiable functions on an open subset of ${\mathbb{R}}^n$, writeand use the optimization LagrangianA candidate $(x_{\ast },\lambda ,\mu )$ satisfies the KKT conditions exactly when all four requirements hold:

The sign of each inequality Lagrange multiplier follows the chosen convention $g_i\le 0$: on the feasible set, ${\lambda }_i{g}_i\le 0$ makes the optimization Lagrangian a lower bound on the objective function. An equality contributes zero on the feasible set for either sign of its Lagrange multiplier. Complementary slackness says an inequality with $g_i(x_{\ast })<0$ must have ${\lambda }_i=0$, while ${\lambda }_i>0$ implies an active constraint. An active constraint can still have zero Lagrange multiplier; the quadratic minimization with a moving upper bound below gives an explicit example.

The name records William Karush's 1939 master's thesis and the 1951 work of Harold W. Kuhn and Albert W. Tucker.^[ref]

William Karush obtained necessary conditions for inequality-constrained optimization in his1939 master's thesis. The result was independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker, producing the name Karush-Kuhn-Tucker conditions.^[ref] Their distinguishing idea is that an inequality constraint can be active or inactive: a nonnegative Lagrange multiplier pairs with its constraint through complementary slackness. A constraint qualification is needed for the usual necessary-condition theorem; a stationary Lagrangian equation alone is insufficient.

Biography: www.math.u-szeged.hu/~hajnal/courses/MSc_Optimalizalas/2014/Cottle%20-%20William%20Karush%20and%20the%20KKT%20theorem.pdf