deletions | additions       diff --git a/For_the_sake_of_briefness__.tex b/For_the_sake_of_briefness__.tex  index 6bcdaf5..771acfe 100644  --- a/For_the_sake_of_briefness__.tex  +++ b/For_the_sake_of_briefness__.tex 
...
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} \begin{align}  \nabla_{g_u}L &= \hat{c}_u - \lambda_j - \omega^l_u + \omega^u_u = 0 \\  \nabla_{d_k}L &= -\bar{c} + \lambda_j - \nu^l_k + \nu^u_k = 0 \\  \nabla_{\theta_j} &= -\sum_{i}\mu_{ij}B_{ij} = 0 \\  \nabla_{p_{ij}} &= -\lambda_j + \mu_{ij} -\sigma_{ij}^l + \sigma_{ij}^u = 0 \\  \nabla_{b_{ij}} &= -\mu_{ij}M - \sigma^l_{ij} + \sigma_{ij}^u + 2M\rho_{ij}^u= 0 \\  \nabra_{r_{ij}} \nabla_{r_{ij}}  &= -\mu_{ij} - \rho^l_{ij} + \rho^u_{ij} = 0 \end{multline} \end{align}  The complementarity conditions