Re: Re: Constrained Optimization

• To: mathgroup at smc.vnet.net
• Subject: [mg57778] Re: [mg57762] Re: Constrained Optimization
• From: Andrzej Kozlowski <andrzej at akikoz.net>
• Date: Wed, 8 Jun 2005 03:21:26 -0400 (EDT)
• References: <d7mj30\$bqm\$1@smc.vnet.net> <d7pb7q\$t80\$1@smc.vnet.net> <d7rk7r\$blu\$1@smc.vnet.net> <d812pv\$cog\$1@smc.vnet.net> <200506070959.FAA28782@smc.vnet.net>
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```On 7 Jun 2005, at 18:59, Caspar von Seckendorff wrote:

> Thanks also for pointing out how to use ForAll[...] to get the upper
> bound (Maxim) and the maximizing x (Andrzej Kozlowski). Being new to
> Mathematica, I have not worked with this function before, but it seems
> that you  can do a lot with it...
>

All that is based on "quantifier elimination over the Reals"-a
beautiful mathematical theory due to the great Polish logician Alfred
Tarski. However, his original approach was not computationally
feasible. Mathematica,  I believe, uses Cylindrical Algbraic
Decomposition, an algorithm  due to Collins, which has polynomial
complexity if the number of variables is kept constant but is double
exponential in the number of variables. This has been implemented for
Mathematica by Adam Strzebonski of WRI and the University of Cracow
(quite appropriately ;-))
There is a much faster algorithm due to Basu and Roy, but I don't
think it has been implemented in any symbolic algebra program.

Andrzej Kozlowski

```

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