Linear-fractional programming: Difference between revisions

Revision as of 14:43, 9 February 2021

Revision as of 14:46, 9 February 2021

Line 17:

Under the assumption that the feasible region is non-empty and bounded, the Charnes-Cooper transformation<ref name="CC"/>

:<math>\mathbf{y} = \frac{1}{\mathbf{d}^T\top \mathbf{x} + \beta} \cdot \mathbf{x}\;;\;\; t = \frac{1}{\mathbf{d}^T\top \mathbf{x} + \beta}</math>

translates the linear-fractional program above to the equivalent linear program:

Line 23:

:<math>

\begin{align}

\text{maximize} \quad & \mathbf{c}^T\top \mathbf{y} + \alpha t \\

\text{subject to} \quad & A\mathbf{y} \leq \mathbf{b} t \\

& \mathbf{d}^T\top \mathbf{y} + \beta t = 1 \\

& t \geq 0.

\end{align}

</math>

Then the solution for <math>\mathbf{{math|'''y'''}}</math> and <math>{{mvar|t </math>}} yields the solution of the original problem as

:<math>\mathbf{x}=\frac{1}{t}\mathbf{y}.</math>