Finding the maximum value of the following equation
$begingroup$
For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation},
where $Mge K$.
We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?
optimization convex-analysis convex-optimization
$endgroup$
add a comment |
$begingroup$
For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation},
where $Mge K$.
We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?
optimization convex-analysis convex-optimization
$endgroup$
$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
1
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53
add a comment |
$begingroup$
For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation},
where $Mge K$.
We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?
optimization convex-analysis convex-optimization
$endgroup$
For $Nin mathbb{N}$, $Min mathbb{N}$, and $Kin mathbb{N}$, $f(K)$ is given by
begin{equation}
f(K) = sumlimits_{i = 1}^{K} {left( {frac{{left( {M - K} right)!left( {M - i} right)!}}{{left( {M - i - K} right)!M!}}} right)^{N - 1} cdot left( begin{array}{c}
K \
i \
end{array} right) cdot left( { - 1} right)^{i - 1} }
end{equation},
where $Mge K$.
We found that $f(K)$ is a convex function by plotting $f(K)$ for varying $K$.
We would like to solve the above equation as a closed-form.
Does anyone help us for solving it or suggesting any ideas?
optimization convex-analysis convex-optimization
optimization convex-analysis convex-optimization
edited Dec 9 '18 at 13:36
Taehoon
asked Dec 5 '18 at 16:42
TaehoonTaehoon
112
112
$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
1
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53
add a comment |
$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
1
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53
$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
1
1
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53
add a comment |
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$begingroup$
I think it would help if we had some context as to where this equation came from. Also, why you are convinced there is a nice closed form.
$endgroup$
– Don Thousand
Dec 5 '18 at 16:51
1
$begingroup$
Also, what maximum value do we have to find?
$endgroup$
– Federico
Dec 5 '18 at 16:53