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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!

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2 Answers 2

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.

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Doesn't the $\rho t$ objective make your formulation nonlinear? Still, the KKT conditions should be linear. –  Mike Spivey Dec 7 '11 at 23:26
Sorry, it's a quadratic program. Thanks. I edited my answer to correct this. The objective is nonconvex indeed and finding a global minimizer is NP-hard. –  Dominique Dec 8 '11 at 2:41

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.

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