I am reading Boyd's book on convex optimization.
Could you assisst me in understanding what this expression means:
$$\text{minimize} \ \ \text{max}_{i=1,...,k}|a_i^Tx-b_i|$$
This is what I think what it means: We iterate through all i's and find such i that maximizes the absolute value of $a_i^Tx-b_i$ we then want to find such $x$ that minimizes that absolute value for that given $i$. Is this correct?
You can think of the problem geometrically.
For example, in a 2 dimensional space, each $|a_i^Tx-b_i|$ is a line. So $\max_i|a_i^Tx-b_i|$ is the line that lies above all the other ones, and finally $$ \min\left\{ \max_i|a_i^Tx-b_i| \right\} $$ is the set of lines that always lie above the other ones, but at the lowest height: it is the convex hull described by the lines.