Lagrange Dual Function
Lagrangian: Define Lagrangian $(L: \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow \mathbb{R})$ as
$$ L(x, \lambda, \nu)=f_0(x)+\sum_{i=1}^m \lambda_i f_i(x)+\sum_{i=1}^p \nu_i h_i(x), $$
with dom $(L=\mathcal{D} \times \mathbb{R}^m \times \mathbb{R}^p$). Here $(\lambda_i, \nu_i)$ are called Lagrange multipliers. Here $(\lambda)$ and $(\nu)$ are called dual variables or Lagrange multiplier vectors.
Lagrange Dual Function: Define the Lagrange dual function $(g: \mathbb{R}^m \times \mathbb{R}^\rho \rightarrow \mathbb{R})$
$$ g(\lambda, \nu)=\inf {x \in \mathcal{D}} L(x, \lambda, \nu)=\inf {x \in \mathcal{D}}\left(f_0(x)+\sum{i=1}^m \lambda_i f_i(x)+\sum{i=1}^p \nu_i h_i(x)\right) $$