KKT (Karush–Kuhn–Tucker) Conditions

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First derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

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Given the Lagrangian of the general problem, the KKT conditions are given by

$$\begin{array}{rlr}{\nabla }_x\mathcal{L}(x^{\ast },{\lambda }^{\ast })& =0& \text{Stationary}\\ c_i(x^{\ast })& =0,i\in \mathcal{E}& \text{Primal Feasibility}\\ c_i(x^{\ast })& \ge 0,i\in \mathcal{I}& \text{Primal Feasibility}\\ {\lambda }_i^{\ast }& \ge 0,i\in \mathcal{I}& \text{Dual Feasibility}\\ {\lambda }_i^{\ast }{c}_i(x^{\ast })& =0,i\in \mathcal{E}\cup \mathcal{I}& \text{Complementary Condition}\end{array}$$

^375397 We usually denote inequality multipliers with $s$ and equality multipliers with $\lambda$.

The complementary condition is also called complementary slackness.