mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, fractional programming is a generalization of

linear-fractional programming In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a r ...

. The

objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...

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

real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real f ...

s defined on a set

\mathbf_0 \subset \mathbb^n

. Let

\mathbf = \

. The nonlinear program :

\underset \quad \frac,

where

g(\boldsymbol) > 0

\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 In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

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 Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization pro ...

\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 ''f'' is nonnegative may be dropped.

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

* * {{cite journal , title=Fractional programming , last1=Schaible , first1=Siegfried , journal=Zeitschrift für Operations Research , volume=27 , year=1983 , pages=39–54 , doi=10.1007/bf01916898, s2cid=28766871 Optimization algorithms and methods