david/opt-split-ideas

In order to prove the correctness of the algorithm, it would be sufficient to show that if some \(\alpha^+,\alpha^-\) in the lemma give positive \(\alpha^{up},\alpha^{down}\), then there exists a splitting \(\alpha^+,\alpha^-\) such that \(\alpha^+_{ij} \alpha^-_{ij}=0\) for each \(i,j\). In order to show this, one could show that if \(\alpha^+_{ij} = x\) and \(\alpha^-_{ij}=y<x\), then replacing these entries with \(x-y, 0\) still gives a valid splitting.

The first step is to show that replacing any entry in the formula for \(\alpha^{down}\) by zero maintains positivity.