Optimization problem.

For the discussion of math. Duh.

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quintopia
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Optimization problem.

Postby quintopia » Fri Apr 24, 2009 9:19 pm UTC

What is the supremum of a*x+(1-a)*(1-x)-sqrt(x*x+(1-x)*(1-x))?

For a=.75, I got about -0.161. I can't really see what this has to do with the parameters.

Actually, it would be good enough to know that the answer is always negative.

Please, hints only, as this is for class.
Last edited by quintopia on Sat Apr 25, 2009 1:57 am UTC, edited 1 time in total.

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Re: Optimization problem.

Postby GreedyAlgorithm » Fri Apr 24, 2009 9:23 pm UTC

Hint #1: b does not appear in the expression you gave. :D
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Re: Optimization problem.

Postby skeptical scientist » Fri Apr 24, 2009 9:48 pm UTC

Hint #2 (actually, this is just a question, but I suspect it may in fact be a hint): do you care about all a and x, or just when they are between 0 and 1?
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quintopia
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Re: Optimization problem.

Postby quintopia » Sat Apr 25, 2009 1:58 am UTC

I've reduced the original problem down to one where i only care about [0,1]

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Re: Optimization problem.

Postby NathanielJ » Sat Apr 25, 2009 3:56 am UTC

If all you care about is whether or not it's negative, then just consider it as a function of a and x and find its maximum in the unit square. Standard multivariable calculus extrema-finding tools should work fine, I think.
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quintopia
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Re: Optimization problem.

Postby quintopia » Sat Apr 25, 2009 4:18 am UTC

Oh, I'm an idiot , I should have simplified the expression more once I converted it to that form. Turns out it's not even a function in a.

Edit: copied it wrong. yes it is.

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quintopia
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Re: Optimization problem.

Postby quintopia » Sat Apr 25, 2009 3:32 pm UTC

I got this really nasty expression for the critical point, and plugging it back in isn't really going to tell me much of anything unless I can simplify it a lot. Anyone have an expression simplifier for very long expressions handy?

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Re: Optimization problem.

Postby NathanielJ » Sat Apr 25, 2009 6:27 pm UTC

quintopia wrote:I got this really nasty expression for the critical point, and plugging it back in isn't really going to tell me much of anything unless I can simplify it a lot. Anyone have an expression simplifier for very long expressions handy?


Mild spoiler:
Spoiler:
Setting the gradient equal to zero, you get that the derivative of the expression with respect to a is really simple and it gives you that x = 1/2. Thus the only three values of x you care about are x = 0, 1/2, 1. Plug those three values of x into your original expression and you get three expressions in a that are *very* easy to maximize.
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Re: Optimization problem.

Postby quintopia » Sat Apr 25, 2009 7:31 pm UTC

I was thinking about the original problem in entirely the wrong way. Of course <x,v> is maximized when x is parallel to the unit vector v, and of course <x,v>-||x|| is non-positive.


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