# Compatibility theory

### From Glossary

The idea that a solution's *character* does not
change for a particular perturbation. In linear programming the character
could be an optimal basis, and the theory is
concerned with whether a particular basis remains optimal when the
data is changed in a prescribed direction. A *Fundamental Theorem
of Basis Compatibility* is the following:

is an admissible direction for perturbing if, and only if, it is compatible with some equilibrium basis.

The *range of compatiblity* of a basis, , for a direction, ,
is the greatest step for which remains optimal:

The *basis spectrum* is the greatest range: