first, we have a function: with variables of , (m=1,2,...,M)
we know that the , can be considered as constants.
Now, based on we have a function of that to the maximize the minimum of , by optimally allocating the as
,
where ,
Actually, we can alternatively express the above function of z as
where the now becomes a variable which is the sum of the (m=1,2,...,M).
then, we want to prove is continuous to x and the function f(') returns the maiximized minimum value of with an input variable of .
for the z=f(x), I have prove it is a strictly monotonically increasing function to x, but I failed to prove the continuity of it.
Thanks
Please forgive my ignorance but I am still having trouble understanding. Here is the gist of what I've gotten as of now:
We have a fixed set of constants (the s) and a function by a function (I understand the function itself). Clearly the order of the tuple is very relevant since for example .
Now here is where I get lost. So you would agree that the above implies that we could maximize our function based on the order of our xs. But, this maximization depends on the fixed s and so really it seems that where where are the values which maximize the function for the values of alpha and beta specified.
So, how can we fix the alpha and betas? Is this function even well defined?
sorry about my ambiguous definition of .
actually, in the function of , the are predetermined in my model regardness of the or we can say are independent to the .
please have a look of my proof for the monotonicity of the funtion, then may be you can know about the problem.
In my problem , I need to focus on the continuity between and , instead of that between and and there is no closed form solution for , so this make the problem difficult.
I'm sorry to say problems like this (those which take time to understand...especially because there HAS to be some kind of background to this problem) usually go unanswered on forums. People just don't have the time and aren't willing to put the effort into them. I wish you the best of luck with this problem and if you have any smaller bits of the problem that you would like help on (or another problem for that matter) please don't hesitate to re-post. Most of your question will be answered.