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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>
• Sender: owner-wri-mathgroup at wolfram.com

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

• References:
  • Re: Constrained Optimization
    • From: Caspar von Seckendorff <seckendorff@alphatec.de>