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Maximization of a |F|^2

  1. Aug 31, 2009 #1
    Will anyone please help me to solve the problem:

    F(x_1,x_2,...,x_n) is a complex valued function and each x_i are real (may be positive too) numbers.

    I have to find the maximum of |F| (or |F|^2) w.r.t. x_i.

    What are the set of constraints? I don't think it will be exactly as
    [tex]\frac{\partial |F|}{\partial x_i}=0[/tex]

    Please provide some helpful reference.

    Thanks and Regards.
     
  2. jcsd
  3. Sep 1, 2009 #2
    Some friends told me that it was correct and the set of constraints are
    [tex]\frac{\partial |F|}{\partial x_i}=0,\quad \forall i=1(1)n.[/tex]

    The reason they provides is that we can always consider [tex]f=|F|[/tex] as a real valued function from
    [tex]\mathbfl{R}^n\to \mathbfl{R}[/tex]

    Please clarify me.
     
  4. Sep 2, 2009 #3
    You're partially correct. Finding the critical points will not distinguish between max, min, and saddle points. The technical way to do it is to find the Hessian and show that it's negative definite. Although given the level of the original post, that may not mean much.
     
  5. Sep 2, 2009 #4
    Thanks for the reply.
    I'm only interested in the condition for critical point (Let me assume that it is given that |F| has a maximum)-and I don't need to check the characteristic of the critical point (saddle point/maxima/minima).
    Is it correct what I said [ GRAD(|F|)=0 is the condition for critical points ] in this case?

    Please, clarify me.
     
  6. Sep 2, 2009 #5
    If your function is continuously differentiable, then yes.
     
  7. Sep 2, 2009 #6
    @zhentil,
    OOps...its very difficult to check the differentiability etc..(a generalized multidimensional form of Cauchy-Riemann equations are to be satisfied etc..). For my case, the function has no singularity in its domain of definition.

    @thornahawk (GP)

    My problem is :
    [tex]\max_{|x_i|\le k_i}|F(x_1,x_2,...,x_n)|[/tex] where F is a given complex function (means [tex] F:\mathbf{R}^n\to\mathbf{C}[/tex]).

    Now, my question is:


    Is the above problem is equivalent to (i.e., they are the same upto a square)
    [tex]\max_{|x_i|\le k_i}[U^2+V^2][/tex] where [tex]F=U+iV,~U,V:\mathbf{R}^n\to\mathbf{R}[/tex]?

    If this is correct, then can I assume [tex]U,V\ge0[/tex] in the condition for critical points
    [tex]U\frac{\partial U}{\partial x_i}+V\frac{\partial V}{\partial x_i}=0,~i=1(1)n[/tex]

    The explicite form of F shows it has no singularity for [tex]|x_i|\le k_i[/tex]

    Thanks in advance.
     
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