deletions | additions       diff --git a/The_complementary_slackness_conditions_introduce__.tex b/The_complementary_slackness_conditions_introduce__.tex  index 2fd74aa..9969770 100644  --- a/The_complementary_slackness_conditions_introduce__.tex  +++ b/The_complementary_slackness_conditions_introduce__.tex 
...
The complementary complementarity  slackness conditions introduce nonlinearity on the second level of the problem, so in order to keep the problem linear we couldeither  use binary linearization of it. As an alternative alternative,  since the problem is linear linear,  the strong duality condition could be used first to replace complementarity and then binarization linearization  (for equations with outer variables). The strong duality conditions is as follows  \begin{multline} 
...
+ \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) + \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}  Note: There are four terms that are non-linear: one is  the $\omega \hat{g}$ could be linearized using the same techniques as in part I.  All variations of $X (1 - b_{ij})$ could be linearized by adding equation $X\leq (1 - b_{ij})M$ \hat{g}$,  and replacing all term three terms  with $X$. $b_{ij}$.  So by By  adding the following conditions \begin{align}  \mu_{ij} &\leq M (1 - b_{ij}), \\  \rho^u_{ij} &\leq M(1 - b_{ij}), \\ 
...
\sum_{ij\in L^1}\left[ -\mu_{ij}M + \rho^u_{ij}2M + p^u_{ij} \sigma^u_{ij} - p^{l}_{ij} \sigma^l_{ij} \right ]  \end{multline}  The $\omega \hat{g}$ expression is linearized using the techniques described in section 1.  Let $\hat{g}_u = \sum a_i x_i^u + a_0$