Optimization Problem with a decreasing constraint

ARo

Jun 2018
1
0
Spain
I need to find this maximum,

$$
\max_{a\leq \frac{b}{1-db}}\frac{1}{1+da}(-a\log(a)-(1-a)\log(1-a))
$$
where $b\rightarrow 0$, $d=\frac{n}{b}$ and $n\in\mathbb{R}$ is a constant.
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Here is what I've done,

first I tried to simplify the problem, so I studied the case where $b$ is still going to $0$ but $d$ is a constant. And the maximum is (obviously) archived just taking $a$ to be the higher bound.

With the complete problem (where $d=\frac{n}{b}$ and $b$ is going to $0$), I started for doing some plots of specific cases, and it seems that the evaluation of the higher bound is again the maximum... (At least in the cases I saw) but of course this is just to have an idea, and it is not a proof...

For the proof I tried to optimize the expression evaluated in some $xb$ (because the value of $a$ I think will depend on $b$) but I didn't get any interesting.

Any help will be appreciated!
 

mathman

Forum Staff
May 2007
6,928
772
The condition $a\le \frac{b}{1-db}, \ d=\frac{n}{b}$ is the same as $a\le \ \frac{b}{1-n}$ which becomes $a\le 0$ in the limit as $b \to 0$. You now have a problem in that $a \lt 0$ gives a complex number, so you need to use $a=0$ and expression = 0.