Lagrangian function

Concept

$$\mathcal{L}(x,\lambda )=f(x)+{\lambda }^{T}g(x)$$

where $\lambda$ is vector of Lagrangian multipliers, $g$ is constrains function, and

$$\nabla \mathcal{L}(x,\lambda )=\left[\begin{array}{c}\nabla f(x)+J_g^T\lambda \\ g(x)\end{array}\right]$$

where $J$ is Jacobian matrix.

$$\nabla H_{\mathcal{L}}(x,\lambda )=\left[\begin{array}{cc}B(x,\lambda )& J_g^T(x)\\ J_g(x)& O\end{array}\right]$$

where $O$ is a matrix with zeros and

$$B(x,\lambda )=H_f(x)+\sum _{i=1}^m{\lambda }_i{H}_{gi}(x)$$

where $H$ is Hessian matrix.