Optimization - maximize the sum of distances to the power alpha

[vladimir69]

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

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

$$x_1^2+x_2^2+x_3^2=1$$
$$x_i \geq 0$$ $$\forall i = 1,2,3$$
$$x_i \leq 1$$ $$\forall i =1,2,3$$

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 can't notice anything different at all when i run my program. when i say can't 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.

 

[Nate810]

At alpha = 2, the objective function is minimized when all the points are as far away from each other as possible. This is because at this value of alpha, the distances are squared, so the further apart the points are, the lower the value of the objective function will be.

 

[thepatientmental]

1. The constraints in this problem will affect the distribution of points on the surface of the sphere. The constraint gradients, which represent the rate of change of the constraints with respect to the variables, will determine the direction in which the points can move on the sphere. For example, if the constraint gradient for x_1^2+x_2^2+x_3^2=1 is positive, it means that increasing the value of x_1 will also increase the value of x_2 and x_3 in order to satisfy the constraint. This will result in the points being distributed more towards the positive x_1 direction on the sphere. Similarly, if the constraint gradient is negative, the points will be distributed more towards the negative x_1 direction. The curvature of the constraints will also play a role in determining the shape of the distribution of points on the sphere.

2. When alpha = 2, the objective function has a special property known as scale invariance. This means that if all the points on the sphere are scaled by a constant factor, the value of the objective function will remain the same. This is not the case for other values of alpha, such as 1.5 or 3, where scaling the points will result in a different value for the objective function. This property of scale invariance makes the optimization problem simpler and easier to solve for alpha = 2. Additionally, when alpha = 2, the objective function can be expressed as the negative of the sum of squared distances between the points, which has a well-known minimum at the center of the sphere. This is why you may have been expecting to get -800 as the optimal value for your objective function in your trials. However, this may not always be the case and the optimal value may vary depending on the initial distribution of points and the constraints.