constraints, in the following form arg min x f(x), s.t. gi(x) = 0, hj(x) ≤ 0 The necessary and sufficient conditions (also known as the KKT conditions) for ∂L(x∗, λ∗, β∗) ∂x = 0, ∂L(x∗, λ∗, β∗) ∂β = 0, gi(x∗) = 0, hj(x∗) ≤ 0, J j=1 β∗ j hj(x∗) = 0, λi, βj ≥ 0 The last condition is sometimes written in the equivalent form: β∗ j hj(x∗) = 0. 14 / 40