In mathematical optimization, fractional programming is a generalization of linear-fractional programming.
The objective function in a fractional program is a ratio of two functions that are in general nonlinear.
The ratio to be optimized often describes some kind of efficiency of a system.
be real-valued functions defined on a set
The nonlinear program where
, is called a fractional program.
A fractional program in which f is nonnegative and concave, g is positive and convex, and S is a convex set is called a concave fractional program.
If g is affine, f does not have to be restricted in sign.
The linear fractional program is a special case of a concave fractional program where all functions
The function
is semistrictly quasiconcave on S. If f and g are differentiable, then q is pseudoconcave.
In a linear fractional program, the objective function is pseudolinear.
By the transformation
, any concave fractional program can be transformed to the equivalent parameter-free concave program[1] If g is affine, the first constraint is changed to
and the assumption that g is positive may be dropped.
The Lagrangian dual of the equivalent concave program is