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Creative Problem Solving
We want to optimize the exponents labeled with k's
to maximize the probability given by the formula.
To visualize this, imagine a set of components
as stacks of redundancies. Each component has
an number of backup copies k, and a probability
of any one of them failing q. See the picture at right.
The k's are the heights (not labeled)
Lagrange
Multipliers
Lagrange multipliers are the usual way
to optimize subject to constraints, so let's
see what that would look like.
Ugh. That's ugly. Notice the form of the partial
derivatives in the matrix. I abused the log notation by "factoring it out". Maybe there's some symmetry property that could simplify this?
Symmetry
First, let's make the notation more
readable by introducing q.
Now suppose our system was near the optimal value of k1,k2.... Let's consider just two of those factors and see what we can learn.
Conjecture:
The k's are chosen so that all the
(1-q^k) factors are the same value!
Let's look at an example.
Example
Let's try a simple example and see
if that works. Here k1+k2=10
Well, it's plausible. Let's see if we can prove it.
Finding
Extremals
Suppose we try to find the optimal value of x for:
Differentiate wrt x and set to zero,
but that gives us another mess. Hmmm.
Use the fact that maximizing the function maximizes the log.
Taylor
Series
We can simplify that when q's are close to zero by using a Taylor series at x=1, where ln(1-x) is about x
Now we're ready to differentiate and set to zero.
We did a lot of stuff there. Does an example work?
So far so good. The example checks out.
So our solution has the following form, subject to the constraint that the k's add up to K.
Wait! We saw this form in the big matrix. These are derivatives!
The right symmetry property is: partial derivatives wrt k's are all equal at the solution!
Here we go:
"tune" the values to get near integers. The formula clearly works in this example.
Can we derive this symmetry property from the Lagrange equation? Try an example.
This method takes a lot less work than Lagrange in this special case.
There are some minor mistakes in the handwritten stuff, and the assumption that q's are close to one is important. If that isn't true, the formulas don't work. So we have an approximate solution at this point.
Creative Problem Solving
How do we get from (?) to (!) ?
Solution
Problem