Minimum of linear combination of logarithms
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I have to minimize the following function $sum a_i log(1-b_ix)$ on $0leq xleq bar{x}$ with $a_i in R$ and $0leq b_i<1$. In my case all the $a_i$ have the same positivity except one, but i don t know if it s rilevant. Doing the derivate comes out that is usefull know the zeros of $sum_i prod_{jneq i} c_i (1-xb_j)$ so if this is a polinomial with known root it would be very usefull.
real-analysis
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
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show 2 more comments
$begingroup$
I have to minimize the following function $sum a_i log(1-b_ix)$ on $0leq xleq bar{x}$ with $a_i in R$ and $0leq b_i<1$. In my case all the $a_i$ have the same positivity except one, but i don t know if it s rilevant. Doing the derivate comes out that is usefull know the zeros of $sum_i prod_{jneq i} c_i (1-xb_j)$ so if this is a polinomial with known root it would be very usefull.
real-analysis
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
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Are the $b_i$ positive?
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– saulspatz
Dec 4 '18 at 15:26
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yes, all $b_i$ positive and less than 1
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– Matteo Cacciola
Dec 4 '18 at 17:20
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And are you saying that exactly one of the $a_i$ is negative?
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– saulspatz
Dec 4 '18 at 17:23
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yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
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I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42
|
show 2 more comments
$begingroup$
I have to minimize the following function $sum a_i log(1-b_ix)$ on $0leq xleq bar{x}$ with $a_i in R$ and $0leq b_i<1$. In my case all the $a_i$ have the same positivity except one, but i don t know if it s rilevant. Doing the derivate comes out that is usefull know the zeros of $sum_i prod_{jneq i} c_i (1-xb_j)$ so if this is a polinomial with known root it would be very usefull.
real-analysis
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
$endgroup$
I have to minimize the following function $sum a_i log(1-b_ix)$ on $0leq xleq bar{x}$ with $a_i in R$ and $0leq b_i<1$. In my case all the $a_i$ have the same positivity except one, but i don t know if it s rilevant. Doing the derivate comes out that is usefull know the zeros of $sum_i prod_{jneq i} c_i (1-xb_j)$ so if this is a polinomial with known root it would be very usefull.
real-analysis
real-analysis
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
edited Dec 4 '18 at 17:23
Matteo Cacciola
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
asked Dec 4 '18 at 14:16
Matteo CacciolaMatteo Cacciola
11
11
$begingroup$
Are the $b_i$ positive?
$endgroup$
– saulspatz
Dec 4 '18 at 15:26
$begingroup$
yes, all $b_i$ positive and less than 1
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:20
$begingroup$
And are you saying that exactly one of the $a_i$ is negative?
$endgroup$
– saulspatz
Dec 4 '18 at 17:23
$begingroup$
yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
$begingroup$
I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42
|
show 2 more comments
$begingroup$
Are the $b_i$ positive?
$endgroup$
– saulspatz
Dec 4 '18 at 15:26
$begingroup$
yes, all $b_i$ positive and less than 1
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:20
$begingroup$
And are you saying that exactly one of the $a_i$ is negative?
$endgroup$
– saulspatz
Dec 4 '18 at 17:23
$begingroup$
yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
$begingroup$
I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42
$begingroup$
Are the $b_i$ positive?
$endgroup$
– saulspatz
Dec 4 '18 at 15:26
$begingroup$
Are the $b_i$ positive?
$endgroup$
– saulspatz
Dec 4 '18 at 15:26
$begingroup$
yes, all $b_i$ positive and less than 1
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:20
$begingroup$
yes, all $b_i$ positive and less than 1
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:20
$begingroup$
And are you saying that exactly one of the $a_i$ is negative?
$endgroup$
– saulspatz
Dec 4 '18 at 17:23
$begingroup$
And are you saying that exactly one of the $a_i$ is negative?
$endgroup$
– saulspatz
Dec 4 '18 at 17:23
$begingroup$
yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
$begingroup$
yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
$begingroup$
I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42
$begingroup$
I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42
|
show 2 more comments
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$begingroup$
Are the $b_i$ positive?
$endgroup$
– saulspatz
Dec 4 '18 at 15:26
$begingroup$
yes, all $b_i$ positive and less than 1
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:20
$begingroup$
And are you saying that exactly one of the $a_i$ is negative?
$endgroup$
– saulspatz
Dec 4 '18 at 17:23
$begingroup$
yes, all $a_i$ are positive except one
$endgroup$
– Matteo Cacciola
Dec 4 '18 at 17:32
$begingroup$
I don't think you'll be able to get a general solution. Can you list the specific $a_i$,b_i?$
$endgroup$
– saulspatz
Dec 4 '18 at 17:42