deletions | additions       diff --git a/The_complementarity_slackness_conditions_introduce__.tex b/The_complementarity_slackness_conditions_introduce__.tex  index d1247b3..6e71e5c 100644  --- a/The_complementarity_slackness_conditions_introduce__.tex  +++ b/The_complementarity_slackness_conditions_introduce__.tex 
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\begin{multline}  -\bar{c}\sum_k d_k + \sum_u \hat{c}_u g_u = \sum_u \omega^u_u \hat{g}_u + \sum_k \nu^u_k \hat{d}_k \\  + \sum_{ij\in L^0}\left[ \sigma^u_{ij} p^u_{ij} -\sigma^l_{ij} p^l_{ij}\right] + \\  \sum_{ij\in L^1}\left[ \mu_{ij}M(b_{ij}-1) -\mu_{ij}M(1 - b_{ij})  + \rho^u_{ij}2M(1 - b_{ij}) + p^u_{ij} \sigma^u_{ij} b_{ij} - p^{l}_{ij} \sigma^l_{ij} b_{ij} \right ] \end{multline}  There are four terms that are non-linear: one is the $\omega \hat{g}$, and three terms with $b_{ij}$.  All of them could be linearized using the techniques described in the first part.  Let us denote by $\dot{x}$ the special variable that replaces $x$ $xb$  in equations and is bounded by \begin{align*}  x&\leq \dot{x} + M(1-b)\leq M \\  0&\leq \dot{x}\leq x 
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By adding the following conditions  \begin{align}  \mu_{ij} 0\leq\dot{\mu}_{ij}  &\leq M (1 - b_{ij}), \mu_{ij},  \\ \rho^u_{ij} 0\leq\dot{\rho^u_{ij}}  &\leq M(1 - b_{ij}), \rho_{ij},  \\ \sigma^{u,l}_{ij} 0\leq\dot\sigma^{u,l}_{ij}  &\leq Mb_{ij}, \sigma^{u,l}_{ij}\\  \mu_{ij}\leq \dot{\mu}_{ij} + b_{ij}M \leq M\\  \rho^u_{ij} &\leq\dot\rho^u_{ij} + b_{ij}M\leq M \\  \sigma^l_{ij} &\leq\dot\sigma^l_{ij} + b_{ij}M\leq M \\  \sigma^u_{ij} &\leq\dot\sigma^u_{ij} + (1-b_{ij})M\leq M  \end{align}  we get  \begin{multline}