Rodion Kuzmin

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

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

where $g(\mathit{x})>0$ 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 $f,g,h_j,j=1,\dots ,m$ are affine.

Properties

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

$\begin{array}{rl}\underset{\frac{\mathit{y}}{t}\in {\mathbf{S}}_0}{\text{maximize}}& tf(\frac{\mathit{y}}{t})\\ \text{subject to}& tg(\frac{\mathit{y}}{t})\le 1,\\ & t\ge 0.\end{array}$

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

Duality

The Lagrangean dual of the equivalent concave program is

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

References

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