Let be real-valued functions defined on a set . Let . The nonlinear program
where on , 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 are affine.
Properties
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.
Duality
The Lagrangian dual of the equivalent concave program is