deletions | additions       diff --git a/For_each_unit_we_define__.tex b/For_each_unit_we_define__.tex  index c484840..2fe3c47 100644  --- a/For_each_unit_we_define__.tex  +++ b/For_each_unit_we_define__.tex 
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The Lagrangian is  \begin{multline}  L = \sum_u \hat{c}_u g_u + \sum_{j\in N} \lambda_j\left(\sum_{i: ij\in L}p_{ij} + \sum_{u\in j} g_u - \sum_{u \in j} d_u \right) \\  + \sum_{(i,j)\in L} \left[\chi_{ij}( p_{ij} - B_{ij}(\theta_j - \theta_i)) + \sigma_{ij}(p_{ij} - p^{max}_{ij} \right] + \\  + \sum_{i\in U} \left[\nu_u(g_u - \hat{g}_u) - \omega_u g_u \right]  \end{multline}  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} + \lambda_j + \nu_u - \omega_u &= 0, \quad \forall u\in U \\  \nabla_{p_{ij}}L = \lambda_j + \chi_{ij} + \sigma_{ij} &= 0, \quad \forall (i,j) \in L \\  \nabla_{\theta{j}}L =   \nu_u g_u &= 0 \\  \omega_u (\hat{g}_u-g_u) &= 0\\  \sigma_{ij}(p_{ij} - p_{ij}^{max}) &= 0 \\  \sigma, \nu, \omega & \geq 0  \end{align*}  For each unit we define the profit, having the LMP for the unit defined by \lambda_u $\lambda_u  = \lambda_j \lambda_j$  we define the profit as \begin{equation}  \pi_u = g_u(\lambda_u - c_u)  \end{equation}