Homogeneous Fractional Programming

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

Chapter 33 in Conjugate Duality in Economic Analysis, 2026, pp 243-247 from Springer

Abstract: Abstract We analyze the maximization of the ratio of a concave positively homogeneous function to a convex positively homogeneous function: $$\sup _{\boldsymbol {x}\neq \mathbf {0}}\frac {g\left ( \boldsymbol {x}\right ) }{f\left ( \boldsymbol {x}\right ) }=\sup _{\boldsymbol {x}\neq \mathbf {0}}\frac {-\delta _{C}^{\ast }\left ( -\boldsymbol {x}\right ) }{\delta _{D}^{\ast }\left ( \boldsymbol {x}\right ) },$$ in which C and D are nonempty, closed, convex sets, neither equal to the entire space. We define a “standard homogeneous fractional program,” the situation in many economic applications. Because in general the ratio is not a concave function of x, reduce it to a Karush-Kuhn-Tucker maximization: $$\sup _{\boldsymbol {x}\neq \mathbf {0}}\left [ \frac {g\left ( \boldsymbol {x}\right ) }{f\left ( \boldsymbol {x}\right ) }\right ] =\sup _{\boldsymbol {x}}\left [ g\left ( \boldsymbol {x}\right ) -\delta _{f\left ( \boldsymbol {x}\right ) \leq z}\right ] \!,$$ in which the perturbation base value z=1. Under general conditions, the zero-maximum condition $$0=\max _{\boldsymbol {x}\neq \mathbf {0}}\left [ g\left ( \boldsymbol {x}\right ) -z^{\ast }f\left ( \boldsymbol {x}\right ) \right ] {\text { for some }}z^{\ast }>0.$$ is attained by a choice x if and only if x solves the homogeneous fractional program and z* is the optimum value. Alternatively, to solve primal and dual problems can be insightful.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_33

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