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This is a question that originated with World of Warcraft. I have a solution, but I don't know where to look up other problems of the same kind for a better explanation.

There is a plane with p axis and q axis.

There is a function f(x,y) = 2px+qy^2 such that f(x,y) is only defined on the curve x^3+y^3=1.

Claim: There are regions of the plane for which f has exactly one critical point and exactly three critical points.

The proposed solution was: F(x,y,k)=2px+qy^2+k(x^3+y^3-1)

This was not explained, but the last term multiplies k by zero. I think this is some kind of Lagrange multiplier method, but it's not like any use of that method that I've seen before.

The solution continued: Fx=2p+3kx^2=0 Fy=2qy++3ky^2=0 I assume that Fx means "derivative of F with respect to x" rather than a simple product.

This part of the solution looked like a Karusch-Kuhn-Tucker method, but I have only seen simple examples of that.

The most applicable book I have found for this kind of problem is Optimization Concepts and Applications in Engineering by Belegundu and Chandrupatla.

But perhaps this is not a KKT method specifically - perhaps it's just advanced calculus.

I have one book that touches on this kind of problem but doesn't really explain much. I want to find a book (or better yet, a free online wiki) that explains this kind of problem thoroughly.

Thanks

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closed as too localized by Qiaochu Yuan, Will Jagy, S. Carnahan Aug 8 '11 at 3:47

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It looks like this is not research-level. You should ask these kinds of questions at math.stackexchange.com in the future. –  Qiaochu Yuan Aug 7 '11 at 1:51
    
math.stackexchange.com seems to be a more appropriate home for your question. –  S. Carnahan Aug 8 '11 at 3:48
    
Whoops, sorry, I'll put it on math.stackexchange.com next time. –  zhai2nan2 Aug 8 '11 at 8:55

1 Answer 1

up vote 1 down vote accepted

It is using KKT. For details, I'd recommend the Convex Optimization book by Boyd and Vandenberghe.

As a side note, this is not research level mathematics, and I would like to vote to close, but since I am still a new user, I don't know how to do it and I just leave this comment as an answer (can't comment yet either).

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In order to vote to close you need 3000 reputation points. Also, I think it is entirely appropriate to leave this as an answer as you cannot comment yet. –  Tony Huynh Aug 7 '11 at 1:04
    
Thank you! I will look for that Boyd and Vandenberghe book! –  zhai2nan2 Aug 8 '11 at 8:53

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