# Gomory function

### From Glossary

A recursively defined class of functions on
to (rational n-vectors to rationals)
using three operations: (1) non-negative, rational linear
combinations, (2) integer rounding, and (3) maximum. Letting
be the integer round-up of , i.e.
the least integer not less than , we say is
a *Gomory function* if it is the identity function, , or
if it has one of the following forms:

- for some and rational;
- ;
- f;

where each is a Chvátal function.

This arises in integer linear programming in several ways. One fundamental result is that the optimal value as a function of is a Gomory function:

Here is an example:

Note: The above definition assumes minimization; if
the ILP is maximization, would be *round-down* (i.e.,
greatest integer not exceeding the value), and condition 3 would be
the point-wise minimum.