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,
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,
The first summation id a finite geometric series and so its sum can be written as:
The second summation can also be written in terms of the sum of a finite geometric series since:
Once you have the objective free of summations I would consider looking for the maximum for a continuous variable in the first instance.