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Can anyone help me to prove the convexity/concavity of following complicated function...? I have tried a lot of methods (definition, 1st derivative etc.), but this function is so complicated, and I finally couldn't prove... however, I plotted with many different parameters, it's always concave to $\rho$...

$$ f\left( \rho \right) = \frac{1}{\lambda }(M\lambda \phi - \rho (\phi - \Phi )\ln (\rho + M\lambda ) + \frac{1}{{{e^{(\rho + M\lambda )t}}\rho + M\lambda }}\cdot( - (\rho + M\lambda )({e^{(\rho + M\lambda )t}}{\rho ^2}t(\phi - \Phi ) ) $$ $$+ M\lambda (\phi + \rho t\phi - \rho t\Phi )) + \rho ({e^{(\rho + M\lambda )t}}\rho + M\rho )(\phi - \Phi )\ln ({e^{(\rho + M\lambda )t}}\rho + M\lambda ))$$

Note that $\rho > 0$ is the variable, and $M>0, \lambda>0, t>0, \phi>0, \Phi>0 $ are constants with any possible positive values...

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Is $f$ the function you wanted to optimize in your previous question? :D Edit: Oh no I didn't realized that $f$ is one-variable here! – Jineon Baek Aug 19 '11 at 9:52
Dear Jineon, you are right, this is a part of the equation in my previous question... and btw, in this function only $\rho$ is variable, others are just any positive values. – Dobby Aug 20 '11 at 4:11
@Dobby: Try separating this function out into a sum of functions $g_i(\rho)$ and see if $g_i$ is convex. – Jacob Oct 24 '11 at 15:11
When $\phi = \Phi$ your function is of the form $A/(B \rho e^{\rho t} + C)$ where $A,B,C,t>0$. This should be convex, not concave. – Robert Israel Mar 22 '12 at 19:21
up vote 2 down vote accepted

I am a newcomer so would prefer to just leave a comment, but alas I see no "comment" button, so I will leave my suggestion here in the answer box.

I have often used a Nelder-Mead "derivative free" algorithm (fminsearch in matlab) to minimize long and convoluted equations like this one. If you can substitute the constraint equation $g(\rho)$ into $f(\rho)$ somehow, then you can input this as the objective function into the algorithm and get the minimum, or at least a local minimum.

You could also try heuristic methods like simulated annealing or great deluge. Heuristic methods would give you a better chance of getting the global minimum if the solution space has multiple local minima. In spite of their scary name, heuristic methods are actually quite simple algorithms.

As for proving the concavity I don't see the problem. You mention in your other post that both $g(\rho)$ and $f(\rho)$ have first and second derivatives, so it should be straightforward, right?

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Dear, thank you so much for your suggestions, I am trying your methods : ) GL! – Dobby Oct 5 '11 at 6:34

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