Supremum over ellipsoid set

Question

In Boyd's Convex Optimization Textbook, page 157, it is stated:

$ \mathrm{sup}\{a_i^T x\; |\; a_i\in\mathcal{E}_i \} = \bar a_i^T x + \mathrm{sup}\{u^T P_i^T x\; |\; \lVert u \rVert_2 \leq 1 \} = \bar a_i^T x + \lVert P_i^T x \rVert_2 $

But I do not see how

$\mathrm{sup}\{u^T P_i^T x\; |\; \lVert u \rVert_2 \leq 1 \} = \lVert P_i^T x \rVert_2$ ?

Also, is it supremum over $a_i$ ?

Answer 1

It looks like a mistype on the authors' part. What they are really trying to say is this:

$\forall i=1,2,\ldots,$ and $\forall x \in \mathbb R^n$, we have

\begin{equation} \begin{split}\sup\{a_i^Tx \text{ s.t } a_i \in \mathcal E_i\} &= \sup\{x^T(\bar{a}_i + P_iu) \text{ s.t } \|u\| \le 1\}\\ &= x^T\bar{a}_i + \sup\{x^TP_iu \text{ s.t } \|u\| \le 1\} \\ &=: x^T\bar{a}_i + \|P_ix\|_* \text { (definition of dual norm of the vector $v := P_iu$)}\\ &= x^T\bar{a}_i + \|P_ix\| \text{ (the $\ell_2$-norm is self-dual)}. \end{split} \end{equation}