KKT conditions

The KKT (Karush-Kuhn-Tucker) conditions generalize "gradient equals zero" to constrained problems. At a constrained optimum x^* (under a constraint qualification), there exist multipliers \lambda^*\ge0,\ \nu^* with: stationarity \nabla f_0(x^*)+\sum_i\lambda_i^*\nabla f_i(x^*)+\sum_j\nu_j^*\nabla h_j(x^*)=\mathbf 0; primal feasibility (x^* satisfies every constraint); dual feasibility (\lambda_i^*\ge0); and complementary slackness (\lambda_i^* f_i(x^*)=0 for every i). See kkt-conditions in the theorem registry for the full statement and proof.