deletions | additions       diff --git a/To_simplify_notation_we_will__.tex b/To_simplify_notation_we_will__.tex  index ceae336..8dc15aa 100644  --- a/To_simplify_notation_we_will__.tex  +++ b/To_simplify_notation_we_will__.tex 
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To simplify notation we will use $\lambda_u \stackrel{def}= \lambda_j, u\in j.$  The KKT conditions for basic system are as follows  \begin{align*}  \nabla_{g_u}L = -\hat{c_u} + \hat{c_u} -  \lambda_u+ \nu_u  - \nu_u +  \omega_u &= 0, \quad \forall u\in U \\ \nabla_{p_{ij}}L = \lambda_j -\lambda_j  + \chi_{ij} + \sigma_{ij} &= 0, \quad \forall (i,j) \in L \\ \nabla_{\theta{j}}L = \sum_{j:(i,j)\in J}\chi_{ij}B_{ij} - \sum_{j:(j,i)\in L} \chi_{ji}B_{ji} = 0 \\  \nu_u g_u &= 0 \\  \omega_u (\hat{g}_u-g_u) &= 0\\  \sigma_{ij}(p_{ij} - p_{ij}^{max}) &= 0 \\