Dynamic Programming Related Questions

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Discussion Overview

The discussion revolves around concepts in dynamic programming and mathematical analysis, specifically focusing on correspondences, continuity, and the distinctions between supremum and maximum values. The scope includes theoretical aspects and conceptual clarifications.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the difference between compact valued and single valued correspondences.
  • Another participant questions whether continuity on a compact set implies boundedness and the possibility of attaining a maximum.
  • There is a discussion on the difference between supremum and maximum, with an example of the open interval (0,1) illustrating that it has a supremum (1) but no maximum.
  • A later reply suggests that if the maximum were to be defined, it could be represented as (1-G), where G is the smallest real number greater than 0.
  • Another participant counters that no such G exists, indicating a challenge to the previous claim.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a maximum in certain sets and the implications of continuity on compact sets. The discussion remains unresolved regarding the existence of G and its implications for defining maximum values.

Contextual Notes

Limitations include the lack of clarity on definitions of correspondences and the conditions under which continuity implies boundedness. The discussion also highlights the dependence on the nature of the sets being considered.

sampahmel
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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.
 
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sampahmel said:
(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.
 
Jarle said:
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
 
sampahmel said:
Does that mean the maximum is (1-G) where G is the smallest real >0

No such G exists.
 

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