deletions | additions       diff --git a/section_Bi_level_formulation_subsection__.tex b/section_Bi_level_formulation_subsection__.tex  index b57de82..7bd524b 100644  --- a/section_Bi_level_formulation_subsection__.tex  +++ b/section_Bi_level_formulation_subsection__.tex 
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\sum_{i: ij\in L}p_{ij} &= \sum_{u\in j} g_u - \sum_{u \in j} d_u &&\forall j\in N &| &\lambda_j\\  p_{ij} &= B_{ij}(\theta_j - \theta_i)&&\forall (i,j) &|& \chi_{ij} \\  p_{ij} - p_{ji} &\leq p^{max}_{ij},&& \forall (i,j) \in L &| &\sigma_{ij} \\  0&\leq g_u \leq \hat{g}_u &&\forall u & | & v_u, w_u \nu_u, \omega_u  \end{align*} In order to model Nash Equilibrium we have to solve similar inner problems but with different parameters.  The system described above is linear, so KKT conditions are sufficient for global optimum.  Hence, we can refer to this problem as to the equations system with inner variables and outer parameters.  If we denote it as $E(\hat{g}, \hat{c}; g, p, \lambda, \chi, \sigma, \nu, \omega)$ than required subproblems used to filter the Nash Equilibrium are formulated as $E(\hat{g}^{(s,p)}, \hat{c}^{(s,p)}; g^{sp}, p^{sp}, \lambda^{sp}, \chi^{sp}, \sigma^{sp}, \nu^{sp}, \omega^{sp})$, where   \begin{description}  \item[$\hat{g}^{(s,p)}$] is equal to current $\hat{g}_u$ value for $u\notin p$ and fixed on strategy level for others;  \item[$\hat{c}^{(s,p)}$] ditto for bidded cost;  \item[$(\cdot)^{sp}$] is inner variable for this system (independent from originals).  \end{description}