# Optimizing with a logistic function

I have a system in which I want to maximize the value of some function $$f(x_T, y_T)$$.

The time evolution of the system is described by some functions: \begin{align} \frac{dx}{dt}&=\alpha \frac{x}{x+y}+\ldots \\ \frac{dy}{dt}&=\beta \frac{y}{x+y}+\ldots \\ \end{align} If I discretize the system it linearizes nicely as a multi-period problem, except for the ratios $$x/(x+y)$$. (Examples of such a system might occur in fisheries models or SIR models.)

The ratios are nice because increasing $$x$$ at a given time leads to greater values of $$x$$ at future times, regardless of the value of $$y$$ (assuming $$y$$ is otherwise fixed); the same holds true for $$y$$. That is, although the function ratio saturates to 1, it doesn't impose any upper limits on what values $$x$$ and $$y$$ should take.

Is there some clever way to linearize this function or approximate it via convex constraints? Alternately, is there some formulation which would give a similar behaviour to that described in the previous paragraph?

• Are x and y integer? Or could they be made integer? If so, then you could introduce an auxiliary variable $z = x/(x+y)$, and then you can linearize the term $xz$ and $yz$ exactly using a big-M formulation (or indicator constraints). It's not going to be pretty, but it would work. – Richard Sep 24 '20 at 10:57
• @Richard (another one!): I'm afraid x and y are continuous, but they are constrained to be >=0. – Richard Sep 24 '20 at 14:53
• Is there any additional function or variable (in the model) that appears in the evolution constraints and/or the objective $f(x_T, y_T)$? E.g. you are creating some desired effect over the system (harvesting rate, vaccination rate) along time versus the system just working on its own. – dhasson Sep 25 '20 at 0:03