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Optimization - maximize the sum of distances to the power alpha

  1. Jun 3, 2005 #1
    hi, what i am trying to do is maximize the sum of distances to the power alpha between all the points
    [tex]D_{\alpha} (\mathcal{U}) = \sum_{i=1}^m \sum_{\substack{j=1\\j\neq i}}^m|\mathbf{u}_i - \mathbf{u}_j|^\alpha [/tex]
    on the surface of a sphere of radius 1 where
    [tex]\mathbf{u} \in \mathbb{R}^3[/tex]
    [tex]|\mathbf{u}|[/tex] = the euclidean norm of a vector in [tex]\mathbb{R}^3[/tex]

    i need to find out the following:
    1. what is the effect of the constraint gradients on this problem?

    i got the constraints to be

    [tex] x_1^2+x_2^2+x_3^2=1[/tex]
    [tex]x_i \geq 0[/tex] [tex] \forall i = 1,2,3 [/tex]
    [tex]x_i \leq 1[/tex] [tex] \forall i =1,2,3 [/tex]

    now i think the constraints will affect where the points are distributed around the sphere but i am not sure what effect the constraints gradients will have. is it because the curvature of the constraints also influence where the points will be placed on the sphere?

    2. what is special about alpha = 2 as opposed to alpha = 1.5 or alpha = 3?

    i am thinking something special is supposed to happen at alpha = 2 but cant notice anything different at all when i run my program. when i say cant notice anything different, with each of the three alphas i tried (1.5 , 2 and 3) 20 trials, produced varying values for the objective function. i was under the impression that for alpha = 2 i was supposed to get -800 all the time.
  2. jcsd
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