Dynamic Programming Related Questions

[sampahmel]

Hi all,

(1.) Can someone tell me the difference between a compact valued and single valued correspondence?

(2.) I have been seeing repeating themes of "continuity on a compact set". Does that imply boundedness and thus possible to attain maximum?

(3.) What's the difference between Supremum (I know it is the l.u.b.) and Maximum?

Thank you for taking time out to read and to answer.

 

[disregardthat]

(3.) What's the difference between Supremum (I know it is the l.u.b.) and Maximum?

Thank you for taking time out to read and to answer.

Some sets such as the open interval (0,1) have no maximum, but it has a least upper bound, namely 1. This is not the maximum value since it is not contained in the set.

 

[sampahmel]

Some sets such as the open interval (0,1) have no maximum, but it has a least upper bound, namely 1. This is not the maximum value since it is not contained in the set.

Does that mean the maximum is (1-G) where G is the smallest real >0

 

[disregardthat]

Does that mean the maximum is (1-G) where G is the smallest real >0

No such G exists.

 

[CellCoree]

I am happy to provide a response to your questions about dynamic programming.

(1.) A compact valued correspondence is a function that maps elements from one set to a subset of another set, where the subset is compact. This means that the output of the function is a closed and bounded set. On the other hand, a single valued correspondence is a function that maps elements from one set to a single element in another set. In other words, it has a unique output for every input.

(2.) Continuity on a compact set does not necessarily imply boundedness. A function can be continuous on a compact set but still be unbounded. However, if a function is continuous on a compact set and bounded, then it is possible to attain a maximum value. This is because a continuous function on a compact set must attain both its maximum and minimum values.

(3.) The supremum of a set is the least upper bound, meaning it is the smallest value that is greater than or equal to all the elements in the set. On the other hand, the maximum of a set is the largest element in the set. So while the supremum may or may not be an actual element in the set, the maximum always is.

I hope this helps clarify your questions. Dynamic programming is a powerful tool in optimization and problem-solving, and understanding these concepts will be beneficial in your studies. Best of luck!