In our lectures notes, we had:
Definition: The function $$L: X \times Y^{*} \rightarrow \mathbb{R}_{\infty},$$ $$-L(x, p^{*}) = \sup_{p\in Y}\left\{ \langle p \vert x \rangle - \Phi(x, p)\right\}$$ is called the Lagrangian of problem $(\mathcal{P})$ relative to the given perturbations.
And as context what the perturbation $\Phi$ looks like:
Let $J: X\rightarrow \mathbb{R}_{\infty}$ be of the form $J(x) = F(x) + G(Ax)$ with convex, lower semicontinuous, proper maps $F:X\rightarrow\mathbb{R}_{\infty}$ and $G:Y\rightarrow\mathbb{R}_{\infty}$ and linear bounded operator $A: X\rightarrow Y$. We introduce the perturbation $\Phi: X\times Y \rightarrow \mathbb R_{\infty}$, $\Phi(x, p) = F(x) + G(Ax - p)$.
Concerning the primal problem $(\mathcal P)$:
Let $J: X \rightarrow \mathbb R_{\infty}$ be of the form $J(x) = F(x) + G(Ax)$ with convex, lower semicontinuous, proper maps $F: X\rightarrow \mathbb R_{\infty}$ and $G: Y \rightarrow \mathbb R_{\infty}$ and linear bounded operator $A: X\rightarrow Y$. We introduce the perturbation $\Phi: X\times Y \rightarrow \mathbb R_{\infty}$, $\Phi(x, p) = F(x) + G(Ax - p)$, where $\Phi$ is proper, convex and lower semicontinuous. [...]
Defintion: The primal problem is defined as $$\inf_{x\in X}\Phi(x, 0) = \inf_{x\in X}\Big( F(x) + G(Ax)\Big).$$
My question is: The above def. from our lecture cannot be correct, because on the LHS, an element $p^{*}$ of the dual space $Y^{*}$ appears, but not on the RHS. Does anybody know the correct definition? I could not find it on Wikipedia.
