Under which conditions maximizing a ratio of functions is equivalent to minimizing its reciprocal?Assume two non-linear functions, f ( x ), g ( x ) respectively, both positive and monotone non-decreasing, f ( x ) is concave, g ( x ) is convex.I am trying to maximize their ratio, f ( x ) g ( x ) , subject to some inequality constraints. I do not have these functions in closed form but I noticed experimentally that minimizing their reciprocal ratio gives me the same solution as maximizing their ratio. I would like to understand better why this happen. Are there any known conditions for this result?

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Under which conditions maximizing a ratio of functions is equivalent to minimizing its reciprocal?Assume two non-linear functions,   f  (  x  ),   g  (  x  ) respectively, both positive and monotone non-decreasing,   f  (  x  ) is concave,   g  (  x  ) is convex.I am trying to maximize their ratio,             f      (      x      )              g      (      x      )      , subject to some inequality constraints. I do not have these functions in closed form but I noticed experimentally that minimizing their reciprocal ratio gives me the same solution as maximizing their ratio. I would like to understand better why this happen. Are there any known conditions for this result?

Annalise Baldwin · Accepted Answer

One sufficient condition: the constraints don&#039;t bind and the first ratio is strictly concave and second strictly convex (you can twice derivate these ratios to characterize these). Moreover, the   f and   g are twice differentiable. Then both the unique maximum of             f      (      x      )              g      (      x      )       and unique minimum of             g      (      x      )              f      (      x      )       is given by the first order condition      f    x    (  x  )  g  (  x  )  =      g    x    (  x  )  f  (  x  )  .

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