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Yeah, I looked into this yesterday night. I noticed that, for maxima with all energy investments being positive, the derivatives of the individual mex od functions are all equal.
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1 |
Yeah, I looked into this yesterday night. I noticed that, for maxima with all energy investments being positive, the derivatives of the individual mex od functions are all equal.
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| 2 |
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| 3 |
That is, say you have 2 mexes with base income m_a and m_b and you invest e_a and e_b into them for OD, then at the maximum OD return you will have
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That is, say you have 2 mexes with base income m_a and m_b and you invest e_a and e_b into them for OD, then at the maximum OD return you will have
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derivative(m_a * (sqrt(1 + e_a/4) - 1)) = derivative(m_b * (sqrt(1 + e_b/4) - 1)),
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derivative(m_a * (sqrt(1 + e_a/4) - 1)) = derivative(m_b * (sqrt(1 + e_b/4) - 1)),
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| 6 |
or with the actual derivatives:
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or with the actual derivatives:
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m_a / (4 sqrt(e_a + 4)) = m_b / (4 sqrt(e_b + 4))
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m_a / (4 sqrt(e_a + 4)) = m_b / (4 sqrt(e_b + 4))
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| 9 |
(Still held true for 4 different mexes, so I would assume that it applies to any number).
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(Still held true for 4 different mexes, so I would assume that it applies to any number).
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| 11 |
Now, confusingly enough, if any of the energy investments at the global maximum were negative, the above didn't hold. I wasn't awake enough to figure it out any deeper and also don't have enough time on my hands for this at the moment.
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Now, confusingly enough, if any of the energy investments at the global maximum were negative, the above didn't hold. I wasn't awake enough to figure it out any deeper and also don't have enough time on my hands for this at the moment.
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| 13 |
I currently can't come up with an algorithm that does not possibly lead to having to redo the calculation each time the optimum would involve a negative energy investment (which you could then safely set to 0 and start over). Since this is likely more "difficult" (as in, computationally expensive) than we want, a different approach would be needed. I'm not sure if iteratively constructing or approaching the optimum would be a good idea.
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I currently can't come up with an algorithm that does not possibly lead to having to redo the calculation each time the optimum would involve a negative energy investment (which you could then safely set to 0 and start over). Since this is likely more "difficult" (as in, computationally expensive) than we want, a different approach would be needed. I'm not sure if iteratively constructing or approaching the optimum would be a good idea.
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Studying the locus curve of the optimal investments per mex (depending on the amount of e available for OD) in the case with just two mexes (= two dimensional search space [two e inputs] with one dimensional constraint [sum of investments is fixed], it can even be visualized) might be enlightnening.
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Studying the locus curve of the optimal investments per mex (depending on the amount of e available for OD) in the case with just two mexes (= two dimensional search space [two e inputs] with one dimensional constraint [sum of investments is fixed], it can even be visualized) might be enlightnening.
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Have a look at (for example) [url=http://www.wolframalpha.com/input/?i=maximize+3+%28sqrt%281+%2B+a%2F4%29+-+1%29+%2B+1+%28sqrt%281+%2B+b%2F4%29+-+1%29+with+a%2Bb+%3D+x]this[/url]. As expected, the maximum total OD involves a square root of the total energy invested, the optimal energy investments per mex are linear functions (this sort of surprised me).
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Have a look at (for example) [url=http://www.wolframalpha.com/input/?i=maximize+3+%28sqrt%281+%2B+a%2F4%29+-+1%29+%2B+1+%28sqrt%281+%2B+b%2F4%29+-+1%29+with+a%2Bb+%3D+x]this[/url]. As expected, the maximum total OD involves a square root of the total energy invested, the optimal energy investments per mex are linear functions (this sort of surprised me).
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| 19 |
Finding how those functions depend on the involved mex incomes would be quite helpful, but an analytical derivation would be even better.
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Finding how those functions depend on the involved mex incomes would be quite helpful, but an analytical derivation would be even better.
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| 20 |
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| 21 |
W|A
breaks
down
when
trying
to
add
more
mexes
(
taking
the
problem
to
the
next
dimension,
literally)
,
which
sort
of
limits
my
investigative
power
here.
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21 |
W|A
breaks
down
when
trying
to
add
more
mexes
(
taking
the
problem
to
the
next
dimension,
literally)
,
which
sort
of
limits
my
investigative
power
here.
I'd
mainly
be
interested
if
the
optimal
energy
investments
still
depend
linearly
on
the
total
energy.
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What's the difference between a +10 mex and 5 +2 mexes?