I want to know what is the exact definition of Lagrange-type function.
For example i have the problem $$\min f_0(x)\;\;\text{ s.t } h_i(x)=0 ,\; g_j(x)\leq0$$ where $i=1,2\ldots,m$ and $j=1,2,\ldots,r$, and $f_0,h_i,g_j:\mathbb{R}^n\to\mathbb{R}$
and we can consider this Lagrange function $$L(x,\theta,\mu)=f_0(x)+\sum_{i=1}^m\theta_ih_i(x)+\sum_{j=1}^r\mu_jg_j(x)$$
but I have seen that also we can also consider the augmented Lagrangian.
I have thought about this definition as: A Lagrange-type function is a function that satisfies certain properties. But what are those properties? Or a function is a Lagrange-type function if it has a certain structure ,(for example a cuadratic form has the structure $x^TAx$, where $A$ is a symmetric matrix), but what is this structure?
A Lagrange-type function is the Lagrangian of a certain constrained optimization problem.
Consider for example the following problem: \begin{equation}\label{eq:1} \min f(x) \quad \mbox{s.t. } g(x) = 0, h(x) \le 0.\tag{1} \end{equation} The function $L(x,\lambda,\mu) = f(x) + \lambda g(x) + \mu h(x)$ (with $\mu \ge 0$) is a Lagrange-type function because it is the Lagrangian of problem \eqref{eq:1}.
Is $L(x,\lambda,\mu,\rho) = f(x) + \lambda g(x) + \mu h(x) + \rho(g(x))^2$ (known as the augmented Lagrangian of \eqref{eq:1}) a Lagrangian-type function? Yes it is, because it is the Lagrangian of the following problem (that is equivalent to \eqref{eq:1}): \begin{equation}\label{eq:2} \min f(x) \quad \mbox{s.t. } g(x) = 0, (g(x))^2 = 0, h(x) \le 0.\tag{2} \end{equation}
Same for \begin{align} L(x,\lambda,\mu,\rho,\alpha,\beta) = &f(x) + \lambda g(x) + \mu h(x)\\ & +\rho(g(x))^2 + \alpha\sin(g(x)) + \beta(\cos(g(x)) - 1) \end{align} or \begin{align} L(x,\lambda,\mu,\rho,\alpha,\beta,\gamma,\delta) = &\exp(f(x)) + \lambda g(x) + \mu h(x) \\ &+\rho(g(x))^2 + \alpha\sin(g(x)) + \beta(\cos(g(x)) - 1) \\ &-\gamma (h(x))^2 + \delta (h(x))^3, \end{align} where $\mu,\gamma, \delta\ge 0$. (Try to find the problems for which these are Lagrangians.)