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, where
represents the vector of variables to be determined,
The constraints have to restrict the feasible region to
[1][2] Alternatively, the denominator of the objective function has to be strictly negative in the entire feasible region.
Fractional linear programs have a richer set of objective functions.
Informally, linear programming computes a policy delivering the best outcome, such as maximum profit or lowest cost.
In contrast, a linear-fractional programming is used to achieve the highest ratio of outcome to cost, the ratio representing the highest efficiency.
For example, in the context of LP we maximize the objective function profit = income − cost and might obtain maximum profit of $100 (= $1100 of income − $1000 of cost).
Using LFP we might obtain an efficiency of $10/$50 = 0.2 with a profit of only $10, but only requiring $50 of investment.
Any linear-fractional program can be transformed into a linear program, assuming that the feasible region is non-empty and bounded, using the Charnes-Cooper transformation.
[1] The main idea is to introduce a new non-negative variable
This allows us to require that the denominator of the objective function (
(To understand the transformation, it is instructive to consider the simpler special case with
to the original linear-fractional program can be translated to a solution of the transformed linear program via the equalities Conversely, a solution for
of the transformed linear program can be translated to a solution of the original linear-fractional program via Let the dual variables associated with the constraints
Then the dual of the LFP above is [3][4] which is an LP and which coincides with the dual of the equivalent linear program resulting from the Charnes-Cooper transformation.
The objective function in a linear-fractional problem is both quasiconcave and quasiconvex (hence quasilinear) with a monotone property, pseudoconvexity, which is a stronger property than quasiconvexity.
A linear-fractional objective function is both pseudoconvex and pseudoconcave, hence pseudolinear.
Since an LFP can be transformed to an LP, it can be solved using any LP solution method, such as the simplex algorithm (of George B. Dantzig),[5][6][7][8] the criss-cross algorithm,[9] or interior-point methods.