Under which conditions maximizing a ratio of functions is equivalent to minimizing its reciprocal?
Assume two non-linear functions, , respectively, both positive and monotone non-decreasing, is concave, is convex.
I am trying to maximize their ratio, , 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?