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I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done...

$ Max \frac {1}{||x||} $

is converted into

$ Min \frac{1}{2} x^Tx $

How was this step achieved..? Many thanks in advance !

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1 Answer 1

up vote 2 down vote accepted

I know nothing about VSM, but probably this was a constrained optimization problem where it is known that the solution is bounded away from $0$ (unconstrained it seems to make little sense). Recall that  

$ ( ||x||)^2 = <x , x> = x^{T} x$.

Now, if $x$ is bounded away from 0, then maximizing $\frac{1}{|| x ||}$ is the same as minimizing $|| x ||$ (subject to the constraint I assume is missing; and maximizing the square root of a value leads to the same solution as maximizing the value). The factor $\frac{1}{2}$ is just for convenience to make the derivative prettier. After you calculate a solution with the factor, you back the solution w/o factor out from it.

That's my guess; but as said, I know nothing about this field. 

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1  
You are correct about both, 1] there are constraints that essentially rule x being 0 and 2] the 1/2 is just to make the derivative pretty. –  TenaliRaman Jan 6 '13 at 3:23
    
@gnometorule : thanks for clarifying :) –  Jugesh Sundram Jan 8 '13 at 4:03
    
@TenaliRaman : thanks for verifying :) –  Jugesh Sundram Jan 8 '13 at 4:04

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