I hope you can help me.
I would like to know which are the conditions to transform a maximization problem into a minimization one.
I have the following problem
$ Q: \max_{x,y} ~ f(y) / g(x) ~~ s.t. ~ (x,y) \in \Omega, $
where $ f(y) > 0 $ is linear, $ f(x) > 0 $ is convex, $ \Omega $ is a convex set.
I am tempted to recast $ Q $ as a minimization problem as shown below
$ P: \min_{x,y} ~ g(x)/f(y) ~~ s.t. ~ (x,y) \in \Omega, $
where the problem is convex since $ g(x)/f(y) $ is a quadratic-over-linear function.
I have made some simulations for a few different functions and I note that they are equivalenet. But I cannot say for sure if this equivalence will always hold. By pure math manipulation I have not been able to go from $Q$ to $P$. Do you know the conditions for this to always be true?