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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.
Definition
Let
f
,
g
,
h
j
,
j
=
1
,
⋯
m
{\displaystyle f,g,h_{j},j=1,\cdots m}
real-valued functions defined on a set
S
0
⊂
R
n
{\displaystyle \mathbf {S} _{0}\subset \mathbb {R} ^{n}}
S
=
{
x
∈
S
0
:
h
j
(
x
)
≤
0
,
j
=
1
,
⋯
,
m
}
{\displaystyle \mathbf {S} =\{{\boldsymbol {x}}\in \mathbf {S} _{0}:h_{j}({\boldsymbol {x}})\leq 0,j=1,\cdots ,m\}}
nonlinear program
maximize
x
∈
S
f
(
x
)
g
(
x
)
,
{\displaystyle {\underset {{\boldsymbol {x}}\in \mathbf {S} }{\text{maximize}}}\quad {\frac {f({\boldsymbol {x}})}{g({\boldsymbol {x}})}},}
where
g
(
x
)
>
0
{\displaystyle g({\boldsymbol {x}})>0}
S
{\displaystyle \mathbf {S} }
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 . The linear fractional program is a special case of a concave fractional program where all functions
f
,
g
,
h
j
,
j
=
1
,
⋯
m
{\displaystyle f,g,h_{j},j=1,\cdots m}
References
Avriel, Mordecai; Diewert, Walter E.; Schaible; Zang, Israel (1988). Generalized Concavity . Plenum Press. 
