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 June 11th, 2018, 12:07 AM #1 Newbie   Joined: Jun 2018 From: Spain Posts: 1 Thanks: 0 Optimization Problem with a decreasing constraint 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. __________________________________________________ _________________________ 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! June 11th, 2018, 12:58 PM #2 Global Moderator   Joined: May 2007 Posts: 6,806 Thanks: 716 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. Tags calcs, constraint, decreasing, limits, maximization, optimization, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post anuribs Linear Algebra 1 November 19th, 2015 08:42 AM sk3blue Calculus 2 February 15th, 2015 11:51 PM helloprajna Economics 0 February 17th, 2013 11:04 PM Justin Lo Applied Math 0 December 8th, 2009 02:40 AM shack Linear Algebra 3 December 17th, 2007 01:11 PM

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