Hadamard inequality and lagrange multipliers
- From: "Carl R." <solrac140@xxxxxxxxxxx>
- Date: Sun, 20 Apr 2008 13:08:58 -0700 (PDT)
Let alpha, beta >1 , such that 1/alpha + 1/beta = 1, a_i be real
numbers with a_i >=0, i=1...n.
I want to find the minimum of the function:
u(x_1, ,... , x_n) = [ sum ( a_i ^ alpha) ] ^ (1/alpha) * [ sum
( x_i ^ beta) ] ^ ( 1/ beta) subject
to the constraint sum (a_i * x_i ) = A where A is a fixed real number.
So I used lagrange multipliers and ended up with x_k = A / ( a_k +
sum( a_i / a_k) ^(1/(beta-1)) where the sum runs from all those i not
equal to k).
But then I put this value in u(x_1, ..., x_n) and end up with a nasty
expression. It seems
that the minimum is attained at A because the next problem asks to
prove the Hadamard inequality using the previous problem:
sum( a_i b_i ) <= [sum ( a_i ^alpha) ] ^(1/alpha) * [ sum ( b_i
^beta) ] ^(1/beta) which can be
easily obtained by putting x_i = b_i in the above problem.
Can you please help me to find the value of each x_k ? I cannot
simplify it any further..how can you obtain A? it seems that
everything must cancel out but I am not able to cancel out terms.
Thanks in advance
.
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