linguistic:voice_change
Matrix representation

The necessary conditions can be written with Jacobian matrices of the constraint functions. Let g (x): R n → R m {\displaystyle \mathbf {g} (x):\,\!\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}} be defined as g (x) = (g 1 (x), …, g m (x)) ⊤ {\displaystyle \mathbf {g} (x)=\left(g_{1}(x),\ldots,g_{m}(x)\right)^{\top }} and let h (x): R n → R ℓ {\displaystyle \mathbf {h} (x):\,\!\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{\ell }} be defined as h (x) = (h 1 (x), …, h ℓ (x)) ⊤ {\displaystyle \mathbf {h} (x)=\left(h_{1}(x),\ldots,h_{\ell }(x)\right)^{\top }}. Let μ = (μ 1, …, μ m) ⊤ {\displaystyle {\boldsymbol {\mu }}=\left(\mu _{1},\ldots,\mu _{m}\right)^{\top }} and λ = (λ 1, …, λ ℓ) ⊤ {\displaystyle {\boldsymbol {\lambda }}=\left(\lambda _{1},\ldots,\lambda _{\ell }\right)^{\top }}. Then the necessary conditions can be written as: