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.

Let ${\displaystyle f,g,h_{j},j=1,\ldots ,m}${\displaystyle f,g,h_{j},j=1,\ldots ,m} be real-valued functions defined on a set ${\displaystyle \mathbf {S} _{0}\subset \mathbb {R} ^{n}}${\displaystyle \mathbf {S} _{0}\subset \mathbb {R} ^{n}}. Let ${\displaystyle \mathbf {S} =\{{\boldsymbol {x}}\in \mathbf {S} _{0}:h_{j}({\boldsymbol {x}})\leq 0,j=1,\ldots ,m\}}${\displaystyle \mathbf {S} =\{{\boldsymbol {x}}\in \mathbf {S} _{0}:h_{j}({\boldsymbol {x}})\leq 0,j=1,\ldots ,m\}}. The nonlinear program

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

where ${\displaystyle g({\boldsymbol {x}})>0}${\displaystyle g({\boldsymbol {x}})>0} on ${\displaystyle \mathbf {S} }${\displaystyle \mathbf {S} }, is called a fractional program.

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 ${\displaystyle f,g,h_{j},j=1,\ldots ,m}${\displaystyle f,g,h_{j},j=1,\ldots ,m} are affine.

The function ${\displaystyle q({\boldsymbol {x}})=f({\boldsymbol {x}})/g({\boldsymbol {x}})}${\displaystyle 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.

By the transformation ${\displaystyle {\boldsymbol {y}}={\frac {\boldsymbol {x}}{g({\boldsymbol {x}})}};t={\frac {1}{g({\boldsymbol {x}})}}}${\displaystyle {\boldsymbol {y}}={\frac {\boldsymbol {x}}{g({\boldsymbol {x}})}};t={\frac {1}{g({\boldsymbol {x}})}}}, any concave fractional program can be transformed to the equivalent parameter-free concave program ^[1]

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

If g is affine, the first constraint is changed to ${\displaystyle tg({\frac {\boldsymbol {y}}{t}})=1}${\displaystyle tg({\frac {\boldsymbol {y}}{t}})=1} and the assumption that g is positive may be dropped. Also, it simplifies to ${\displaystyle g({\boldsymbol {y}})=1}${\displaystyle g({\boldsymbol {y}})=1}.

The Lagrangian dual of the equivalent concave program is

${\displaystyle {\begin{aligned}{\underset {\boldsymbol {u}}{\text{minimize}}}\quad &{\underset {{\boldsymbol {x}}\in \mathbf {S} _{0}}{\operatorname {sup} }}{\frac {f({\boldsymbol {x}})-{\boldsymbol {u}}^{T}{\boldsymbol {h}}({\boldsymbol {x}})}{g({\boldsymbol {x}})}}\\{\text{subject to}}\quad &u_{i}\geq 0,\quad i=1,\dots ,m.\end{aligned}}}${\displaystyle {\begin{aligned}{\underset {\boldsymbol {u}}{\text{minimize}}}\quad &{\underset {{\boldsymbol {x}}\in \mathbf {S} _{0}}{\operatorname {sup} }}{\frac {f({\boldsymbol {x}})-{\boldsymbol {u}}^{T}{\boldsymbol {h}}({\boldsymbol {x}})}{g({\boldsymbol {x}})}}\\{\text{subject to}}\quad &u_{i}\geq 0,\quad i=1,\dots ,m.\end{aligned}}}

• Avriel, Mordecai; Diewert, Walter E.; Schaible, Siegfried; Zang, Israel (1988). Generalized Concavity. Plenum Press.
• Schaible, Siegfried (1983). "Fractional programming". Zeitschrift für Operations Research. 27: 39–54. doi:10.1007/bf01916898. S2CID 28766871.

• Optimization algorithms and methods