deletions | additions       diff --git a/The_Lagrangian_is_begin_multline__.tex b/The_Lagrangian_is_begin_multline__.tex  index 8b568db..fd5138f 100644  --- a/The_Lagrangian_is_begin_multline__.tex  +++ b/The_Lagrangian_is_begin_multline__.tex 
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The For the sake of briefness we will omit the full  Lagrangian is expression here.  To completely define $E^0(\hat{g}, \hat{c}; \phi, \psi)$ we will write down the KKT conditions.  The first order optimality conditions are  \begin{multline}  L(\phi; \psi) = \sum_u \hat{c}_u g_u \nabla_{g_u}L &= -\hat{c}_u  - \sum_{j\in N} \lambda_j\left(\sum_{i: ij\in L}p_{ij} \lambda_j  + \sum_{u\in j} g_u \omega^l_u  - \sum_{u \in j} d_u \right) \omega^u_u = 0  \\ \nabla_{d_k}L &= \bar{c}  + \sum_{(i,j)\in L} \left[\chi_{ij}( p_{ij} - B_{ij}(\theta_j - \theta_i)) + \sigma_{ij}(p_{ij} - p^{max}_{ij}) \right] + \\ \lambda_j  + \sum_{i\in U} \left[\omega_u(g_u \nu^l_k  - \hat{g}_u) - \nu_u g_u \right] \nu^u_k = 0  \end{multline}  The complementarity conditions