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From: Walter Roberson on 28 May 2010 18:09
Bruno Luong wrote:
> Luis Felipe <luispipe16(a)gmail.com> wrote in message
> <8445cd34-c732-4010-aa1c-5a20d6b4b5cf(a)f14g2000vbn.googlegroups.com>...

>> Have you ever seen this optimization problem ?
>> n
>> max prod (p_i)^a_i
>> i=1
>>
>> n
>> subject to sum p_i = 1
>> i=1
>>
>> where a_i is a positive constant for i=1,...,n

> Not sure, but this is trivial to solve
> Maximizing prod(p.*a) is equivalent to maximizing f(p) := sum(a*log(p))
> The gradient of f is a./p.

I'm not certain, Bruno, whether your solution is intended to work when the a_i
are potentially different from each other?
From: Bruno Luong on 28 May 2010 18:25
>
> I'm not certain, Bruno, whether your solution is intended to work when the a_i
> are potentially different from each other?

I believe as long as a a >= 0, it works, otherwise the solution does not exist. But I let those detail for OP to play with.

Bruno
From: Matt J on 28 May 2010 18:29
Walter Roberson <roberson(a)hushmail.com> wrote in message <htpf09$ec7$1(a)canopus.cc.umanitoba.ca>...

> I'm not certain, Bruno, whether your solution is intended to work when the a_i
> are potentially different from each other?

Seems fine to me, provided that we also have the constraint p>=0, which the OP never verified. What doesn't look right to you?
From: Walter Roberson on 28 May 2010 19:23
Matt J wrote:
> Walter Roberson <roberson(a)hushmail.com> wrote in message
> <htpf09$ec7$1(a)canopus.cc.umanitoba.ca>...
>
>> I'm not certain, Bruno, whether your solution is intended to work when
>> the a_i are potentially different from each other?
>
> Seems fine to me, provided that we also have the constraint p>=0, which
> the OP never verified. What doesn't look right to you?

The statement that the gradient was a./p -- I'm not sure that is right when
the a(:) are different. Gradient is slope, which would be length(a)-1 values
in between the other values, and the slope between positions K and K+1 would
seem to more naturally depend upon a(K) and a(K+1)
From: Roger Stafford on 28 May 2010 19:55
Walter Roberson <roberson(a)hushmail.com> wrote in message <htpjbp$ko0$1(a)canopus.cc.umanitoba.ca>...
> The statement that the gradient was a./p -- I'm not sure that is right when
> the a(:) are different. Gradient is slope, which would be length(a)-1 values
> in between the other values, and the slope between positions K and K+1 would
> seem to more naturally depend upon a(K) and a(K+1)

Bruno's solution is correct, Walter. It is a standard problem in Lagrange undetermined multiplier. At the maximum you have to satisfy two simultaneous differential conditions:

a1/p1*dp1 + a2/p2*dp2 + .... + an/pn*dpn = 0
dp1 + dp2 + dp3 + .... + dpn = 0

which forces a1/p1, a2/p2, etc to all be equal. The solution then falls out easily.

Roger Stafford
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