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Topic: Tricky Quadratic Optimization Problem
Replies: 14   Last Post: Mar 10, 2007 2:53 AM

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Paul Abbott

Posts: 1,437
Registered: 12/7/04
Re: Tricky Quadratic Optimization Problem
Posted: Mar 2, 2007 3:31 AM
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In article <1172726025.019663.109420@j27g2000cwj.googlegroups.com>,
"junoexpress" <MTBrenneman@gmail.com> wrote:

> I am trying to optimize a function of a single variable which has a
> very simple form.
>
> F(x) = A*[cos(x) - r*cos(c)]^2 + B*[sin(x)-r*sin(c)]^2
>
> where A,B,r,c are all known constants with A,B, and r all positive.
> (I know from experience with this problem that numerically A and B
> have about the same value, r is close to 1, and x should be close to
> c).


For a close to b, the form

((a + b) (1 + r^2) + (a - b) (Cos[2x] + r^2 Cos[2c]) -
4 a r Cos[c] Cos[x] - 4 b r Sin[c] Sin[x])/2

may be useful in that (a - b) is small. If a == b, then this reduces to

b (1 + r^2 - 2 r Cos[c - x])

and if c == x, then one obtains

b (r - 1)^2

Perhaps series expansion about a == b, x == c, r == 1, would be useful?
The leading terms (dependent on the order of expansion) of this are

(r - 1)^2 (b + (a - b) Cos[c]^2) + (a - b) (r - 1)(x - c) Sin[2 c] + ...

Cheers,
Paul

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Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
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