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I encountered a linear optimization problem which looks like the following:
Let $W^* = \{w^*_{ij}\}$ be an $n\times n$ real matrix with $w^*_{ij}\geq 0, \forall i, j$.
$$\underset{W_{n\times n}}{argmin \hspace{1mm}} 1^T W 1\\ s.t. W\geq 0, \\ (W - W^T)1 = (W^* - {W^{*}}^T)1$$where $1$ is an $n \times 1$ matrix with entries all equal to 1 and $W\geq 0$ simply means entries are non-negative.
I am not sure how to approach this problem as the optimization techniques I learned all deal with vectors.
Also, for my problem, I think the result will be equally valid if I set the problem to be: $$\underset{W_{n\times n}}{argmin \hspace{1mm}} ||W||_1 = \sum_i^n\sum_j^n |w_{ij}|\\ s.t. (W - W^T)1 = (W^* - {W^{*}}^T)1$$
I am wondering if this will simplify the problem in some way?

Ian Yang
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  • I would say that the left side of your constraint $(W - W^T)1$ is $$\sum_{j=1}^n w_{kj}-\sum_{i=1}^n w_{ik} , , \forall , k\in {1,2,\ldots, n }$$ – callculus42 Dec 27 '20 at 17:19
  • @callculus You are absolutely right. That's actually my initial setup, but I thought the matrix version might be easier to deal with. I am not sure tho... – Ian Yang Dec 28 '20 at 10:22
  • It is up to you what you prefer. It depends on your personell prefrerence which presentation you can comprehend better. – callculus42 Dec 28 '20 at 12:18
  • @callculus Oh I see what you are saying. Thank you very much. – Ian Yang Dec 30 '20 at 02:29

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