Fractional Programming

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 34 in Conjugate Duality in Economic Analysis, 2026, pp 249-254 from Springer

Abstract: Abstract We generalize the analysis of a homogeneous fractional program to the case in which the denominator and numerator are not positively homogeneous: $$\sup _{\boldsymbol {x}}\frac {g\left ( \boldsymbol {x}\right ) }{f\left ( \boldsymbol {x}\right ) },$$ for a concave function g and a convex function f. In general the ratio g/f is not a concave function, but for a “standard fractional program,” one can set up an equivalent concave maximization. We focus on the concave and nondecreasing value function of the following Karush-Kuhn-Tucker problem: $$V\left ( z\right ) :=\sup _{\boldsymbol {x}}\left [ g\left ( \boldsymbol {x}\right ) -\delta _{f\left ( \boldsymbol {x}\right ) \leq z}\right ] \!.$$ We show $$\sup _{\boldsymbol {x}}\frac {g\left ( \boldsymbol {x}\right ) }{f\left ( \boldsymbol {x}\right ) }=\sup _{z>0}\frac {V\left ( z\right ) }{z}.$$ The optimum value is the unique z* such that $$V_{\ast }\left ( z^{\ast }\right ) =0$$ . Alternatively, even if the numerator and the denominator in the fractional program are not positively homogeneous, nevertheless one can reduce the analysis to the positively homogeneous case, via epi multiplication. The zero-supremum condition $$0=\sup _{\boldsymbol {x}}\left [ g\left ( \boldsymbol {x}\right ) -z^{\ast }f\left ( \boldsymbol {x}\right ) \right ] {\text { for some }}z^{\ast }>0.$$ holds if and only if z* is the optimum value of the fractional program.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_34

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DOI: 10.1007/978-3-032-21396-9_34

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