 Linear-fractional programming

Linear-fractional programming minimizes or maximizes a ratio of affine functions over a linear polyhedron, with the denominator required to be positive:For a threshold $t$, the sublevel condition is equivalent to the linear inequality $(c-td{)}^{T}x\le t\beta -\alpha$. Its sublevel sets are therefore convex, although the ratio need not be a convex function. The positivity restriction makes this multiplication preserve the inequality direction.^[ref]

The Charnes-Cooper transformation rescales by the denominator and introduces one extra variable to produce a linear program. Positive transformed scale gives an actual original feasible point; zero scale can encode a recession limit and must be interpreted separately when discussing attainment.

Linear-fractional programming (LFP) is a type of mathematical optimization problem where the objective function is a ratio of linear functions.