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Primal feasibility: $g_{i} (x_{*}) \leq 0$ and $h_{j} (x_{*}) = 0$ , so $x_{*}$ is a feasible point.
Dual sign feasibility: $λ_{i} \geq 0$ . The equality Lagrange multipliers $μ_{j}$ are unrestricted real numbers.
Stationarity: the gradient of the optimization Lagrangian in the decision variables vanishes,
$\nabla_{x} L (x_{*}, λ, μ) = \nabla f (x_{*}) + \sum_{i} λ_{i} \nabla g_{i} (x_{*}) + \sum_{j} μ_{j} \nabla h_{j} (x_{*}) = 0.$
(3)
Complementary slackness: $λ_{i} g_{i} (x_{*}) = 0$ for every inequality optimization constraint.^[ref]

The sign of each inequality Lagrange multiplier follows the chosen convention

g_{i} \leq 0

λ_{i} g_{i} \leq 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_{*}) < 0

must have

λ_{i} = 0

, while

λ_{i} > 0