How to minimize the maximum? Watch

Jundong
Badges: 0
Rep:
?
#1
Report Thread starter 4 years ago
#1
For example, how to decide what 'a' is, in order to minimize the following:

max | 1-a \sigma_i^2|


where \sigma_i is sequence of numbers, or entries of a n-by-1 matrix. Suppose \sigma_n and \sigma_1 are the smallest and the largest elements in that matrix.



I don't even know how to start. Can anyone help me?
0
reply
Smaug123
  • Study Helper
Badges: 13
Rep:
?
#2
Report 4 years ago
#2
(Original post by Jundong)
For example, how to decide what 'a' is, in order to minimize the following:

max | 1-a \sigma_i^2|


where \sigma_i is sequence of numbers, or entries of a n-by-1 matrix. Suppose \sigma_n and \sigma_1 are the smallest and the largest elements in that matrix.



I don't even know how to start. Can anyone help me?
Under what circumstances will |1-a \sigma_i^2| be large? (Never mind "largest" for the moment.)
0
reply
Jundong
Badges: 0
Rep:
?
#3
Report Thread starter 4 years ago
#3
(Original post by Smaug123)
Under what circumstances will |1-a \sigma_i^2| be large? (Never mind "largest" for the moment.)
Sorry to reply you so late.
When I consider this minimax question (never done that before), I am more worried about the minimum.
I think the limit of the whole thing must be less that 1. In order to do that, I think I need to choose a to be something like \frac{1}{\sigma_1 ^2}, where \sigma_1^2 is the largest element of \sigma_i? Then maximum of that thing would be
|1-a \sigma_n^2|
where \sigma_n is the smallest element. So I need to choose something between \frac{1}{\sigma_1^2} and \frac{1}{\sigma_n^2}.

That is what I have got so far.
0
reply
Smaug123
  • Study Helper
Badges: 13
Rep:
?
#4
Report 4 years ago
#4
(Original post by Jundong)
Sorry to reply you so late.
When I consider this minimax question (never done that before), I am more worried about the minimum.
Quite right: I was just trying to get you to play with the problem a bit. Don't worry about that, then.

I think the limit of the whole thing must be less that 1. In order to do that, I think I need to choose a to be something like \frac{1}{\sigma_1 ^2}, where \sigma_1^2 is the largest element of \sigma_i? Then maximum of that thing would be
|1-a \sigma_n^2|
where \sigma_n is the smallest element. So I need to choose something between \frac{1}{\sigma_1^2} and \frac{1}{\sigma_n^2}.

That is what I have got so far.
If you chose a=\frac{1}{\sigma_1^2}, let's suppose \sigma_n = 1 and \sigma_1 = \frac{1}{10}. What, then, is \text{max}(|1-a \sigma_i^2|)?
(I think I've understood your working so far, but I might be wrong. Let me know if my question makes no sense in the light of your current working.)
0
reply
Jundong
Badges: 0
Rep:
?
#5
Report Thread starter 4 years ago
#5
(Original post by Smaug123)
Quite right: I was just trying to get you to play with the problem a bit. Don't worry about that, then.


If you chose a=\frac{1}{\sigma_1^2}, let's suppose \sigma_n = 1 and \sigma_1 = \frac{1}{10}. What, then, is \text{max}(|1-a \sigma_i^2|)?
(I think I've understood your working so far, but I might be wrong. Let me know if my question makes no sense in the light of your current working.)
Yes, your question makes sense to me. So I need to choose something between \frac{1}{\sigma_1^2} and \frac{1}{\sigma_n^2}, right? For example,

\frac{1}{\sigma_1^2+\sigma_n^2}
But so far, we only discussed the logical behind the question. How can I come up with an algebraic answer?
0
reply
Jundong
Badges: 0
Rep:
?
#6
Report Thread starter 4 years ago
#6
And, even I know the correct answer, how can I prove it is the minimax?
0
reply
DFranklin
Badges: 18
Rep:
?
#7
Report 4 years ago
#7
(Original post by Jundong)
Yes, your question makes sense to me. So I need to choose something between \frac{1}{\sigma_1^2} and \frac{1}{\sigma_n^2}, right? For example,

\frac{1}{\sigma_1^2+\sigma_n^2}
But so far, we only discussed the logical behind the question. How can I come up with an algebraic answer?
I don't think you're going to come up with an algebraic answer without talking about the logic of the situation.

(And the logic of the situation should explain to you how to justify your answer).

A heuristic I will offer you: If you have two functions f(x), g(x) and f is decreasing and g is increasing, then if x minimizes max{f(x), g(x)} then f(x) = g(x).
0
reply
Jundong
Badges: 0
Rep:
?
#8
Report Thread starter 4 years ago
#8
(Original post by DFranklin)
I don't think you're going to come up with an algebraic answer without talking about the logic of the situation.

(And the logic of the situation should explain to you how to justify your answer).

A heuristic I will offer you: If you have two functions f(x), g(x) and f is decreasing and g is increasing, then if x minimizes max{f(x), g(x)} then f(x) = g(x).
Thank you!! And Smaug!!

Problem solved!!
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • Cranfield University
    Cranfield Forensic MSc Programme Open Day Postgraduate
    Thu, 25 Apr '19
  • University of the Arts London
    Open day: MA Footwear and MA Fashion Artefact Postgraduate
    Thu, 25 Apr '19
  • Cardiff Metropolitan University
    Undergraduate Open Day - Llandaff Campus Undergraduate
    Sat, 27 Apr '19

Have you registered to vote?

Yes! (273)
38.18%
No - but I will (51)
7.13%
No - I don't want to (51)
7.13%
No - I can't vote (<18, not in UK, etc) (340)
47.55%

Watched Threads

View All
Latest
My Feed