I would like to maximize the function: 1 2 </mfrac> <munderover> &#x221

Lena Bell 2022-07-14 Answered
I would like to maximize the function:
1 2 i = 1 N | x i 1 N |
under the constrains i = 1 N x i = 1 and i ( 1 , . . . , N )
I have done some test for small values of N and I have the feeling that the solution is 1 1 N but I can't figure out how to solve it analytically.
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Answers (1)

Kaylie Mcdonald
Answered 2022-07-15 Author has 19 answers
Look at the point in which the function takes the maximum value. If there are two numbers that lie on the different sides of 1 / N, i.e. x i < 1 / N < x j (otherwise you can make x i a little bit smaller and x j the same little bit larger so that the function value will grow) then either x i = 0 or x j = 1. Lets assume that none of the numbers is equal to one.

Now look at all nonzero numbers. They are either all greater then 1 / N all smaller. The latter is impossible. So all numbers are greater then 1 / N. But then the value of your function is 1 m / N where m is the number of nonzero x i s.
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