Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed.
For convenience, the relevant part is shown as:
I am confused in two things:
The conjugacy relation between the primal function $p$ and the dual function $q$.
How to derive (1.47)
Note: $Q(\mu) = \underset{u\in R^r}{u^T\mu - P(u)} = P^*(\mu)$
My work:
- The derivation is in this link. It seems we cannot say $p(u)=q^*(u)$
- To (1.47), my derivation is:
a. The left-hand side of (1.47) can be written as (Fenchel duality framework)
\begin{equation}
\begin{aligned}
&{\text{min}}
& & p(u)+P(Iu)\\
& \text{s.t.} & & u \in R^r \\
\end{aligned}
\end{equation}
b. It is equivalent to
\begin{equation}
\begin{aligned}
&{\text{min}}
& & p(u_1)+P(u_2)\\
& \text{s.t.} & & u_1, u_2 \in R^r \\
& & & u_2=Iu_1 \\
\end{aligned}
\end{equation}
c. Form the dual function:
\begin{align*}
&\ \ \ \ \underset{u_1,u_2}{\text{inf}}\{p(u_1)+P(u_2)+\mu^T(u_2-u_1)\}\\ &= \underset{u_1}{\text{inf}}\{p(u_1)-\mu^T u_1\} + \underset{u_2}{\text{inf}}\{P(u_2)+\mu^T u_2\}\\ &= -\underset{u_1}{\text{sup}}\{\mu^T u_1-p(u_1)\} - \underset{u_2}{\text{inf}}\{(-\mu)^T u_2 - P(u_2)\} \\ &= -p^*(\mu) - P^*(-\mu)\end{align*}
d. Form the dual problem:
\begin{align*}
& \underset{\mu}{\text{sup}} \{-p^*(\mu) - P^*(-\mu)\} \end{align*}
I do not know the following steps to obtain the right-hand side of (1.47).
It's actually easy. Let's do it from first principles. First observe that \begin{eqnarray} \underset{\mu \ge 0}{\text{sup }}\mu^T(g(x)-u) = \begin{cases}0, &\mbox{ if }g(x) \le u,\\+\infty, &\mbox{ otherwise.}\end{cases} \end{eqnarray}
Now, \begin{eqnarray} \begin{split} \text{LHS of 1.47} &= \underset{u \in \mathbb{R}^r}{\text{inf }}p(u) + P(u) = \underset{u \in \mathbb{R}^r}{\text{inf }}\underset{x \in X, g(x) \le u}{\text{inf }}f(x) + P(u) \\ &= \underset{u \in \mathbb{R}^r}{\text{inf }}\underset{x \in X}{\text{inf }}f(x) + \underset{\mu \ge 0}{\text{sup }}\mu^T(g(x)-u) + P(u)\\ &=\underset{\mu \ge 0}{\text{sup }}\underbrace{\underset{x \in X}{\text{inf }}f(x) + \mu^Tg(x)}_{q(\mu)} + \underbrace{\underset{u \in \mathbb{R}^r}{\text{inf }}P(u) - \mu^Tu}_{-Q(\mu)}\\ &=\underset{\mu \ge 0}{\text{sup }}q(\mu) - Q(\mu) = \text{RHS of 1.47} \end{split} \end{eqnarray}
This should hopefully answer your question.