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 $\mathbf_0 \subset \mathbb^n$. Let $\mathbf = \$. The nonlinear program

:$\underset \quad \frac,$

where $g(\boldsymbol) > 0$ on $\mathbf$, 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 $q(\boldsymbol) = f(\boldsymbol) / g(\boldsymbol)$ 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 $\boldsymbol = \frac; t = \frac$, any concave fractional program can be transformed to the equivalent parameter-free concave program

:$\begin \underset \quad & t f\left(\frac\right) \\ \text \quad & t g\left(\frac\right) \leq 1, \\ & t \geq 0. \end$

If ''g'' is affine, the first constraint is changed to $t g(\frac) = 1$ and the assumption that ''g'' is positive may be dropped. Also, it simplifies to $g(\boldsymbol) = 1$.

Duality

The Lagrangian dual of the equivalent concave program is
:$\begin \underset \quad & \underset \frac \\ \text \quad & u_i \geq 0, \quad i = 1,\dots,m. \end$

Notes

References

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{{Major subfields of optimization

Optimization algorithms and methods