Nonlinear Optimization Problem

[Cheah Meng Yew]

1) qn here represent a Function for Optimization
2) := means definition
3)p means problem
so basically it means
a problem is defined by a function q and a single variable x that starts from index n = 1 until n=infinity .
I do not know if my interpretation of it is correct and also do not get what the max means ?

 

[Romsek]

You have a countable sequence \(\displaystyle x_n \ni 0 \leq x_n,~\text{ and }\sum \limits_{n=1}^\infty x_n \leq r,~r > 0\)
In English you have an infinite indexed sequence of non-negative real numbers whose sum is less than or equal to some real number \(\displaystyle r\)

Now you also have an infinite indexed sequence of what you call functions for optimization. I don't really know what that means but whatever.
I'll just assume they are functions of a real variable.

[MATH]p^* = \max\{x_n\} \sum \limits_{n=1}^\infty q_n(x_n)[/MATH]
Here an optimization function is assigned to each sequence element. You evaluate that for each element and sum the results.
If your optimization functions are properly designed this sum will be bounded for any given sequence meeting the requirements above.
If the results are bounded then there is a maximum result corresponding to one or more sequences.

\(\displaystyle p^*\) is this maximum result

 

[Cheah Meng Yew]

So the max here is not a mathematical operator ,and it represents the maximum optimization of a given problem ? Where the optimization function is assigned to each possible case of condition for variable x ( give that it is real number) and the maximum optimization is the sum of all the possible optimization of feasible variable ?

 

[Romsek]

Not quite. Pick a sequence \(\displaystyle x_n\) that meets the two criteria given.

There's an infinite number of them. You have your set of \(\displaystyle q_n\) somehow so you can evaluate \(\displaystyle \sum_n q_n(x_n)\)

This is going to vary for different sequence examples \(\displaystyle x_n\)

Assuming these sums are bounded, there there is going to be some maximum sum, across all the possible examples of \(\displaystyle x_n\)

\(\displaystyle p^*\) is that maximum sum. It's possible it could be generated by multiple sequence examples.

 

[Romsek]

Assuming these sums are bounded, there there is going to be some maximum sum, across all the possible examples of \(\displaystyle x_n\)
\(\displaystyle p^*\) is that maximum sum. It's possible it could be generated by multiple sequence examples.

Actually this isn't quite right. The sums could be bounded but there might not exist a least upper bound and the bound might not be attained by any of the sequence examples. Nevertheless part of the idea is to design your \(\displaystyle q_n\) so things behave nicely.

 

[Cheah Meng Yew]

Actually this isn't quite right. The sums could be bounded but there might not exist a least upper bound and the bound might not be attained by any of the sequence examples. Nevertheless part of the idea is to design your \(\displaystyle q_n\) so things behave nicely.

Ya I get it already , thanks for your patience .