# How to minimize the maximum?Watch

#1
For example, how to decide what 'a' is, in order to minimize the following:

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

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

where is sequence of numbers, or entries of a n-by-1 matrix. Suppose and 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 be large? (Never mind "largest" for the moment.)
0
#3
(Original post by Smaug123)
Under what circumstances will 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 to be something like , where is the largest element of ? Then maximum of that thing would be

where is the smallest element. So I need to choose something between and .

That is what I have got so far.
0
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 to be something like , where is the largest element of ? Then maximum of that thing would be

where is the smallest element. So I need to choose something between and .

That is what I have got so far.
If you chose , let's suppose and . What, then, is ?
(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.)
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#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 , let's suppose and . What, then, is ?
(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 and , right? For example,

But so far, we only discussed the logical behind the question. How can I come up with an algebraic answer?
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#6
And, even I know the correct answer, how can I prove it is the minimax?
0
4 years ago
#7
(Original post by Jundong)
Yes, your question makes sense to me. So I need to choose something between and , right? For example,

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).
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#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!!
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