What does the term "canonical problem" mean generally?

Question

What does it mean to say a "canonical problem" in mathematical modelling?

Answer 1

"Canonical" means according to the canon. The canon is a standard form of hymns in orthodox churches. In mathematics it is an aphorism for "as we normally do it", a standard formulation or standard approach.

E.g. we could represent the equivalence classes of $\mathbb{Z}/3\mathbb{Z}$ as $\{[0],[1],[2]\}$ or as $\{[-1],[0],[1]\}$. The first one could be called a canonical representation, as it is the one which is normally used.

Answer 2

If you're referring to canonical form, then in linear programming that is a problem written as :

$$\max\{z=c^Tx : Ax \leq b ,\ x \geq 0 \}$$

$$\text{with $A \in \mathbb{R}^{m \times n} ,\ c \in \mathbb{R}^{ n} ,\ b \in \mathbb{R}^{m }$ and $x$ a vector of variables in $\mathbb{R}^{n} $}$$