Math Help - Discrete Time Optimization

1. Discrete Time Optimization

Hello All,

I basically just want to know if it is possible to find a closed form solution to the following situation...

I want to find the maximum of two summations added together. The catch is, I want to maximize with respect to the limits of the summation- which is in the function as well (I am summing over time, hence I want to find the time which maximizes the expression). AND I want the upper limit of one summation and the lower limit on the next summation to be the value time value that I found. So in short, I am summing over t, I want to optimize with respect to t, and I want to sum to t* on the first and from t* to infinity on the next.

Any help, even if its just a suggestion of where to look, would be warmly appreciated,

Sincerely,

Nick

2. Originally Posted by salohcin
Hello All,

I basically just want to know if it is possible to find a closed form solution to the following situation...

I want to find the maximum of two summations added together. The catch is, I want to maximize with respect to the limits of the summation- which is in the function as well (I am summing over time, hence I want to find the time which maximizes the expression). AND I want the upper limit of one summation and the lower limit on the next summation to be the value time value that I found. So in short, I am summing over t, I want to optimize with respect to t, and I want to sum to t* on the first and from t* to infinity on the next.

Any help, even if its just a suggestion of where to look, would be warmly appreciated,

Sincerely,

Nick
As I understand this you are looking for $t^*$ which maximises:

$h(t^*)=\left[\sum_{t=1}^{t^*}f(t)\right] + \left[\sum_{t=t^*+1}^{\infty}g(t) \right]$

though I don't know how to solve this in general, possibly more detail might help.

CB

3. Hi Giordano,

Thanks very much for your response. That is precisely what I am trying to optimize. Let f(t)=(x-t)r^t (where r is between 0 and 1). The second summation could actually be expressed as r^(t*+1). Do you have any idea (no matter how vague it might be) how one would solve this problem? I have no idea where to even begin. Thanks again,

Nick

4. Originally Posted by salohcin
Hi Giordano,

Thanks very much for your response. That is precisely what I am trying to optimize. Let f(t)=(x-t)r^t (where r is between 0 and 1). The second summation could actually be expressed as r^(t*+1). Do you have any idea (no matter how vague it might be) how one would solve this problem? I have no idea where to even begin. Thanks again,

Nick
So you have:

$
h(t^*)=\left[\sum_{t=1}^{t^*} xr^t \right] - \left[\sum_{t=1}^{t^*}tr^t \right]+r^{t^*+1}
$

The first summation id a finite geometric series and so its sum can be written as:

$\sum_{t=1}^{t^*} xr^t =xr \frac{1-r^{t^*}}{1-r}$

The second summation can also be written in terms of the sum of a finite geometric series since:

$\sum_{t=1}^{t^*}tr^t=r \sum_{t=1}^{t^*}tr^{t-1}=r \frac{d}{dr}\left[ \sum_{t=1}^{t^*}r^t\right]$

Once you have the objective free of summations I would consider looking for the maximum for $t^*$ a continuous variable in the first instance.

CB