Fractional programming


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.

Definition

Let be real-valued functions defined on a set. Let. The nonlinear program
where on, is called a fractional program.

Concave fractional programs

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.

Transformation to a concave program

By the transformation, any concave fractional program can be transformed to the equivalent parameter-free concave program
If g is affine, the first constraint is changed to and the assumption that f is nonnegative may be dropped.

Duality

The Lagrangian dual of the equivalent concave program is