Linear-fractional programming

In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1.

Formally, a linear-fractional program is defined as the problem of maximizing (or minimizing) a ratio of affine functions over a polyhedron,

${\displaystyle {\begin{aligned}{\text{maximize}}\quad &{\frac {\mathbf {c} ^{T}\mathbf {x} +\alpha }{\mathbf {d} ^{T}\mathbf {x} +\beta }}\\{\text{subject to}}\quad &A\mathbf {x} \leq \mathbf {b} ,\end{aligned}}}$

where ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ represents the vector of variables to be determined, ${\displaystyle \mathbf {c} ,\mathbf {d} \in \mathbb {R} ^{n}}$ and ${\displaystyle \mathbf {b} \in \mathbb {R} ^{m}}$ are vectors of (known) coefficients, ${\displaystyle A\in \mathbb {R} ^{m\times n}}$ is a (known) matrix of coefficients and ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ are constants. The constraints have to restrict the feasible region to ${\displaystyle \{\mathbf {x} |\mathbf {d} ^{T}\mathbf {x} +\beta >0\}}$, i.e. the region on which the denominator is positive. Alternatively, the denominator of the objective function has to be strictly negative in the entire feasible region.