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I have a function F(x,y) and a constraint G(x,y), I am trying to maximize F but the normal Lagrange Multiplier method can't be solved analytically; it's too complicated, even for mathematica. Are there other strategies for doing this? Also I have an idea about the answer. Is it possible to prove this is a maximum through other means?

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Why don't you try numerical optimization software, e.g. fmincon in MATLAB? –  emrea Sep 25 '12 at 21:22
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If numerical optimization is not an option for you, then you certainly needs to say somewhat more about your problem! As is is stated, it is to broad. –  kjetil b halvorsen Sep 25 '12 at 22:38
    
There is nothing that can be said in this generality. What kind of functions are $F$ and $G$? –  Qiaochu Yuan Sep 26 '12 at 22:56
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