Linear-fractional programming: Difference between revisions

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==Duality==

Let the [[duality (optimization)|dual variables]] associated with the constraints <math>A\mathbf{y} - \mathbf{b} t \leq \mathbf{0}</math> and <math>\mathbf{d}^T \mathbf{y} + \beta t - 1 = 0</math> be denoted by <math>\mathbf{u}</math> and <math>\lambda</math>, respectively. Then the dual of the LFP above is <ref>{{cite journal|last1=Schaible |first1=Siegfried |title=Parameter-free Convex Equivalent and Dual Programs| url=http://www.springerlink.com/content/kv02870752256413/ |journal=Zeitschrift für Operations Research |volume=18 |year=1974 |issue=5 |pages=187–186|ref=harv|doi=10.1007/BF02026600|mr=351464}}</ref><ref>{{cite journal|title=Fractional programming&nbsp;I: Duality |last1=Schaible |first1=Siegfried | journal=Management Science |volume=22 |issue=8 |pages=858–867 |year=1976|jstor=2630017|mr=421679|ref=harv |doi=10.1287/mnsc.22.8.858}}</ref>

:<math>

\begin{align}