Is it possible to transform a convex optimisation with the max-affine cost into either LP, QP, or conic programming?

Question

Question

I wonder if it's possible to transform a convex optimisation problem with the max-affine cost into either LP (linear programming), QP (quadratic programming), or (CP) conic programming. That is, the given problem can be expressed as

$$ \min_{x} \quad \max{c_{i}^{\intercal}x}, $$ where $c_{i}$'s are prescribed constant vectors ($i=1, 2, \ldots, N$).

My approach

I guess that it can be transformed into an LP as $$ \min_{t, x} t \\ \text{subject to} \\ c_{i}^{\intercal} x\leq t \quad (i=1, 2, \ldots, N) $$ by introducing a slack variable $t$ but I'm not sure it's correct.

Answer 1

In Section 4.3.1 [1], the transformation is introduced as piecewise-linear minimisation, and it can be transformed into a linear programming problem as I wrote.

[1] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK ; New York: Cambridge University Press, 2004.