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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.


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

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

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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