# Compatibility theory

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:

$LaTeX: h$ is an admissible direction for perturbing $LaTeX: (b, c)$ if, and only if, it is compatible with some equilibrium basis.

The range of compatiblity of a basis, $LaTeX: B$, for a direction, $LaTeX: h$, is the greatest step for which $LaTeX: B$ remains optimal:

$LaTeX: \sup \{t : B \;\mbox{ is optimal for the LP defined by }\; r + th\}.$

The basis spectrum is the greatest range:

$LaTeX: \sup\{ \mbox{range}(B; h) : B \;\mbox{ is optimal for the LP defined by }\; r\}.$