Lagrangian Duality

Why it matters

Lagrangian theory shapes convex optimization. Understanding shapes theory.

The architecture

Lagrangian L(x, lambda) = f(x) + sum lambda_i g_i(x).

KKT: stationarity + feasibility + complementary slackness.

How it works end to end

KKT necessary + sufficient (convex).

Dual problem = max over lambda of dual function.

Slater's condition ensures strong duality.