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
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.
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?
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.