discontinuous optimization I'm solving the following problem:
$$ \max_\rho \;\; \rho \; \min\left[\left(
  \frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$
where all constants and variables are defined over $\mathbb{R}^+$.
I feel like there must be a standard way to deal with these kinds of weird optimization functions? I don't want to solve this numerically, I need to do comparative statics at some point.
Even if you don't solve this one, any pointers would be super helpful. Thanks!
 A: Here's an equivalent formulation of your problem:
$$
\begin{aligned}
 \max_{\rho, t} & \rho t \\
 \text{s.t.} & t \leq \left( \frac{bn}{an-bm} \right) (a-m-\rho), \\
             & t \leq \frac{b}{a} (a-p+\rho), \\
             & \rho \geq 0.
\end{aligned}
$$
This is a quadratic program. You can use the KKT conditions to find a solution.
A: By bringing the $\rho$ inside the minimum, this is equivalent to finding the maximum value of the minimum of two parabolas. So the optimal point has to be either a local maximum of a parabola or one of the intersections of the parabolas. This reduces the problem to checking four values of $\rho$ and choosing the one that gives the maximum. That is, if
$$
\rho^* = \arg \max_\rho \left[ \min( c_1 \rho - c_2 \rho^2, d_1 \rho - d_2 \rho^2) \right ]
$$
Then assuming $\max(c_2,d_2) > 0$ (so at least one of the parabolas is concave, which is true in your case), we have
$$
 \rho^* \in \left\{ 0, \frac{c_1}{2 c_2}, \frac{d_1}{2 d_2}, \frac{c_1-d_1}{c_2-d_2} \right\}
$$
It should be straightforward algebraic manipulation to express the coefficients in terms of your constants, and then plug in the possible values of $\rho^*$ and compare them. Unfortunately it seems messy however.
