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

$\underset{\boldsymbol{x} \in \mathbf{S}}{\text{maximize}} \quad \frac{f(\boldsymbol{x})}{g(\boldsymbol{x})},$

where $g(\boldsymbol{x}) > 0$ on $\mathbf{S}$, 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 $f, g, h_j, j=1, \ldots, m$ are affine.

Properties

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

$\begin{align} \underset{\frac{\boldsymbol{y}}{t} \in \mathbf{S}_0}{\text{maximize}} \quad & t f(\frac{\boldsymbol{y}}{t}) \\ \text{subject to} \quad & t g(\frac{\boldsymbol{y}}{t}) \leq 1, \\ & t \geq 0. \end{align}$

If g is affine, the first constraint is changed to $t g(\frac{\boldsymbol{y}}{t}) = 1$ and the assumption that f is nonnegative may be dropped.

Duality

The Lagrangean dual of the equivalent concave program is

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

Notes

1. Schaible, Siegfried (1974). "Parameter-free Convex Equivalent and Dual Programs". Zeitschrift für Operations Research 18 (5): 187–196. doi:10.1007/BF02026600. MR 351464.

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.