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This is related to a question I had asked earlier, with the distinction that earlier I did not have a non-linear objective functional to minimize. The problem is reproduced below with added information. I used MATLAB to solve the problem posed in that question.

Why I need to minimize the objective functional given below: As it turns out, the constraints I had put on the vectors were not enough to guarantee that the vectors be "as different" from each other as possible. Some of the vectors were completely identical. Therefore, I though it might help to maximize the angles between the vectors, kind of driving them towards orthogonality (right?).

Problem: You are given a set of 16 vectors, $\{\mathbf{a}^i\}_{i=1}^{16}$, and matrices $D_1$ and $D_2$ (both of dimension $[10\times6]$. Let the following be true:

  • $\sum_{i=1}^{16} \mathbf{a}^i = \mathbf{1} \in \mathbb{R}^6\;,$
  • $\mathbf{a}^i \geq \mathbf{0}\;,\forall i\;,$ (i.e. all elements positive),
  • $(D_1 + D_2)\mathbf{1} = \mathbf{1} \in \mathbb{R}^{10}\;.$

Then, find vectors $\{\mathbf{b}^i\}_{i=1}^{16}$ and $\{\mathbf{c}^i\}_{i=1}^{16}$ that minimize the functional: $$ \min \sum_{i,j=1}^{16} \biggl\vert \dfrac{\pi}{2} - \cos^{-1}\biggl(\dfrac{\mathbf{d}^i\cdot \mathbf{d}^j}{||\mathbf{d}^i||||\mathbf{d}^j||}\biggr)\biggr\vert\;, $$ where $\mathbf{d}^i := \left(\begin{array}{c} \mathbf{b}^i\\ \mathbf{c}^i \end{array}\right)$, and such that they satisfy the following constraints:

  • $\mathbf{c}^i - D_1 \mathbf{b}^i = D_2 \mathbf{a}^i\;,\quad \forall i\in \{1,2,\dots,16\}\;,$
  • $\sum_{i=1}^{24} \mathbf{c}^i = \mathbf{1} \in \mathbb{R}^{10}\;,$
  • $\sum_{i=1}^{24} \mathbf{b}^i = \mathbf{1} \in \mathbb{R}^6\;,$
  • $\mathbf{c}^i \geq \mathbf{0}\;,\forall i\;,$ (i.e. all elements of each vector positive),
  • $\mathbf{b}^i \geq \mathbf{0}\;,\forall i\;,$ (i.e. all elements of each vector positive).

Questions

  • Could anyone please give me any pointers on how I could go about tackling this problem? I have no idea about any methods I could use to solve this problem computationally.
  • Any comments on the objective functional I have defined? I was also thinking of $ \min \sum_{i,j=1}^{16} \biggl(\dfrac{\pi}{2} - \cos^{-1}\biggl(\dfrac{\mathbf{d}^i\cdot \mathbf{d}^j}{||\mathbf{d}^i||||\mathbf{d}^j||}\biggr)\biggr)^2\;, $ but thought that they would be equivalent. Any thoughts? Could I do better than these two?
  • Is there a way to devise a functional to drive them towards linear independence?

Also, this is NOT homework.

Thanks a lot.

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