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 [math]\displaystyle{ f, g, h_j, j=1, \ldots, m }[/math] be real-valued functions defined on a set [math]\displaystyle{ \mathbf{S}_0 \subset \mathbb{R}^n }[/math]. Let [math]\displaystyle{ \mathbf{S} = \{\boldsymbol{x} \in \mathbf{S}_0: h_j(\boldsymbol{x}) \leq 0, j=1, \ldots, m\} }[/math]. The nonlinear program

[math]\displaystyle{ \underset{\boldsymbol{x} \in \mathbf{S}}{\text{maximize}} \quad \frac{f(\boldsymbol{x})}{g(\boldsymbol{x})}, }[/math]

where [math]\displaystyle{ g(\boldsymbol{x}) \gt 0 }[/math] on [math]\displaystyle{ \mathbf{S} }[/math], 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 [math]\displaystyle{ f, g, h_j, j=1, \ldots, m }[/math] are affine.

Properties

The function [math]\displaystyle{ q(\boldsymbol{x}) = f(\boldsymbol{x}) / g(\boldsymbol{x}) }[/math] 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 [math]\displaystyle{ \boldsymbol{y} = \frac{\boldsymbol{x}}{g(\boldsymbol{x})}; t = \frac{1}{g(\boldsymbol{x})} }[/math], any concave fractional program can be transformed to the equivalent parameter-free concave program^[1]

[math]\displaystyle{ \begin{align} \underset{\frac{\boldsymbol{y}}{t} \in \mathbf{S}_0}{\text{maximize}} \quad & t f\left(\frac{\boldsymbol{y}}{t}\right) \\ \text{subject to} \quad & t g\left(\frac{\boldsymbol{y}}{t}\right) \leq 1, \\ & t \geq 0. \end{align} }[/math]

If g is affine, the first constraint is changed to [math]\displaystyle{ t g(\frac{\boldsymbol{y}}{t}) = 1 }[/math] and the assumption that g is positive may be dropped. Also, it simplifies to [math]\displaystyle{ g(\boldsymbol{y}) = 1 }[/math].

Duality

The Lagrangian dual of the equivalent concave program is

[math]\displaystyle{ \begin{align} \underset{\boldsymbol{u}}{\text{minimize}} \quad & \underset{\boldsymbol{x} \in \mathbf{S}_0}{\operatorname{sup}} \frac{f(\boldsymbol{x}) - \boldsymbol{u}^T \boldsymbol{h}(\boldsymbol{x})}{g(\boldsymbol{x})} \\ \text{subject to} \quad & u_i \geq 0, \quad i = 1,\dots,m. \end{align} }[/math]

Notes

References

• Avriel, Mordecai; Diewert, Walter E.; Schaible, Siegfried; Zang, Israel (1988). Generalized Concavity. Plenum Press. 
• Schaible, Siegfried (1983). "Fractional programming". Zeitschrift für Operations Research 27: 39–54. doi:10.1007/bf01916898. 

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