deletions | additions       diff --git a/Let_s_denote_by_W__.tex b/Let_s_denote_by_W__.tex  index e13135e..d70d79e 100644  --- a/Let_s_denote_by_W__.tex  +++ b/Let_s_denote_by_W__.tex 
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We also introduce the demand linkage matrix $W^d$ and branch matrix same way $W^l$.  The Lagrangian is  \begin{multline}  c^T g + \lambda^T (W^l p - W^g g - W^d d) + \sigma^{+} \sigma^{+T}  (p - p^+) - \simga^{-}(p \sigma^{-T}(p  - p^{-}) +\\ v^{+}^T(g - \hat{g})  \end{multline}