The KKT conditions are necessary conditions for optimality in nonlinear programming with inequality and equality constraints. For problem: min f(x) subject to g(x) ≤ 0, h(x) = 0, at optimal point x*: (1) Stationarity: ∇f(x*) + Σλᵢ∇gᵢ(x*) + Σμⱼ∇hⱼ(x*) = 0; (2) Primal feasibility: g(x*) ≤ 0, h(x*) = 0; (3) Dual feasibility: λᵢ ≥ 0; (4) Complementary slackness: λᵢgᵢ(x*) = 0. Under constraint qualification (LICQ, MFCQ), KKT conditions are necessary. For convex problems, they are also sufficient. The multipliers λᵢ, μⱼ represent shadow prices. KKT conditions generalize Lagrange multipliers to inequality constraints and are fundamental to nonlinear optimization theory and algorithms.