Consider a linear program (in inequality form) where one of the parameters is not known with certainty:
minimize $ \quad c^T x $
subject to $ \quad a_i^T x \leq b_i, \quad a_i\in \xi_i=\{\bar{a}_i+P_iu|\;||u||_2 \leq 1\}, \quad i=1,\dots,m . $
I.e., the $ a_i $ are known to lie in ellipsoids.
It is possible to rewrite the linear constraint $ a_i^T x \leq b_i $ as
$ \sup \{ a_i^Tx|a_i\in \xi_i \}\leq b_i $
The LHS of the inequality can further be expressed as
$ \sup \{ a_i^Tx|a_i\in \xi_i \}=\bar{a}_i^Tx+\sup\{ u^TP_i^Tx|\;||u||_2\leq 1 \} $
$ \qquad \qquad \qquad \quad \; \;=\bar{a}_i^Tx+||P_i^Tx||_2. $
However, I do not see why this last equality holds. Can someone help me?
In one direction, the Cauchy-Schwarz inequality will do the trick. In the other direction, you can always pick $u$ in the same direction as $P_{i}^{T}x$, so Cauchy-Schwarz is tight.