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 $f,g,h_{j},j=1,\cdots m$ 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 ${\boldsymbol {y}}={\frac {\boldsymbol {x}}{f({\boldsymbol {x}})}};t={\frac {1}{g({\boldsymbol {x}})}}$, any concave fractional program can be transformed to the equivalent parameter-free concave program

${\begin{aligned}{\underset {{\frac {\boldsymbol {y}}{t}}\in \mathbf {S} _{0}}{\text{maximize}}}\quad &tf({\boldsymbol {x}})\\{\text{subject to}}\quad &tg({\boldsymbol {x}})\leq 1,\\&t\geq 0.\end{aligned}}$

If g is affine, the first constraint is changed to $tg({\boldsymbol {x}})=1$.

References

• Avriel, Mordecai; Diewert, Walter E.; Schaible; Zang, Israel (1988). Generalized Concavity. Plenum Press. {{cite book}}: Unknown parameter |first 3= ignored (|first3= suggested) (help)