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

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

Duality

The Lagrangian dual of the equivalent concave program is

{\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}}

Notes

↑ Schaible, Siegfried (1974). "Parameter-free Convex Equivalent and Dual Programs". Zeitschrift für Operations Research. 18 (5): 187–196. doi:10.1007/BF02026600. MR 0351464. S2CID 28885670.

References

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.

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[1] Schaible, Siegfried (1974). "Parameter-free Convex Equivalent and Dual Programs". Zeitschrift für Operations Research. 18 (5): 187–196. doi:10.1007/BF02026600. MR 0351464. S2CID 28885670.

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