# Properties of the Quadratic Transformation of Dual Variables
## 7 Solving Applied Problem
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\Delta\psi\) & \(\Delta f\) & \
\begin{tabular}{c} **Number of** \\ **Terations** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **of Function and** \\ **Gradient** \\ \end{tabular} \\ \hline \multicolumn{4}{c}{Function LP\_R\_Y1 _mb_ = 50, _ma_ = 25, _md_ = 25} \\ sub & \(5.050\times 10^{-1}\) & \(6.822\times 10^{-5}\) & 2497 & 4998 \\ Shor & \(5.050\times 10^{-1}\) & \(1.832\times 10^{-4}\) & 4300 & 9000 \\ Orty & \(5.050\times 10^{-1}\) & \(8.383\times 10^{-4}\) & 1360 & 2374 \\ yv & \(5.050\times 10^{-1}\) & \(3.862\times 10^{-4}\) & 1751 & 3234 \\ \multicolumn{4}{c}{Function L2\_Y2 _mb_ = 50, _ma_ = 25, _md_ = 25} \\ sub & \(1.915\times 10^{-8}\) & \(4.225\times 10^{-4}\) & 154 & 313 \\ Shor & \(6.302\times 10^{-12}\) & \(1.926\times 10^{-5}\) & 2165 & 4259 \\ Orty & \(4.738\times 10^{-12}\) & \(8.416\times 10^{-6}\) & 361 & 741 \\ yv & \(3.793\times 10^{-13}\) & \(1.865\times 10^{-6}\) & 165 & 330 \\ MsgrPPR & \(1.000\times 10^{-15}\) & \(3.116\times 10^{-8}\) & 414 & 897 \\ BFGS & \(1.359\times 10^{-12}\) & \(3.216\times 10^{-5}\) & 124 & 272 \\ \multicolumn{4}{c}{Function L2\_Y1 _mb_ = 50, _ma_ = 25, _md_ = 25} \\ sub & \(2.569\times 10^{-10}\) & \(8.985\times 10^{-5}\) & 2577 & 5157 \\ Shor & \(2.541\times 10^{-9}\) & \(2.008\times 10^{-4}\) & 4317 & 9058 \\ Orty & \(4.461\times 10^{-10}\) & \(1.431\times 10^{-4}\) & 2516 & 4331 \\ yv & \(4.310\times 10^{-10}\) & \(1.019\times 10^{-4}\) & 2024 & 3708 \\ \multicolumn{4}{c}{Function 1/\(\chi\_\)Y2 _mb_ = 50, _ma_ = 25, _md_ = 25} \\ sub & \(1.200\times 10^{-15}\) & \(2.118\times 10^{-11}\) & 117 & 240 \\ Shor & \(1.279\times 10^{-13}\) & \(2.000\times 10^{-8}\) & 46 & 101 \\ Orty & \(2.271\times 10^{-13}\) & \(1.907\times 10^{-7}\) & 99 & 199 \\ yv & \(3.357\times 10^{-13}\) & \(1.299\times 10^{-7}\) & 80 & 144 \\ MsgrPPR & \(1.375\times 10^{-13}\) & \(1.243\times 10^{-8}\) & 53 & 110 \\ BFGS & \(4.152\times 10^{-13}\) & \(3.220\times 10^{-8}\) & 59 & 118 \\ \multicolumn{4}{c}{Function 1/\(\chi\_\)Y1 _mb_ = 50, _ma_ = 25, _md_ = 25} \\ sub & \(2.023\times 10^{-6}\) & \(6.824\times 10^{-5}\) & 1122 & 2249 \\ Shor & \(6.456\times 10^{-4}\) & \(1.764\times 10^{-4}\) & 1407 & 2951 \\ Orty & \(1.790\times 10^{-4}\) & \(6.067\times 10^{-5}\) & 284 & 523 \\ yv & \(4.916\times 10^{-5}\) & \(2.513\times 10^{-6}\) & 392 & 712 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._terms of the wind speed and direction. It can be seen that the pollution zone with an excess of the MPC (inside the black line) covers the living sector closest to the industrial site, where compliance with MPC is required. To select points for the constraints (27) on the concentration, an additional calculation was performed to cover the living sector with 300 calculated points, which will be the set of quota points. To solve the problem in the selected quota points, the influence coefficients \(A_{\bar{\mu}}\) are calculated, which form the matrix \(A\). All elements of the \(b_{j}\) vector in this example are equal to MPC, but their individual values can be set. The vectors \(a\) and \(d\) of constraints (28) on the solution search area \(x\) are given as follows: the vector \(d\) is a set of emissions from all 1085 sources for which the initial calculation was carried out, and the vector \(a=0.1d\). Thus, the MPE for each source cannot exceed its existing emission, and the allowable emission reduction at the source cannot be more than 90%. The problem was solved by various methods. The coefficients of socio-economic significance [57] were chosen to be the same and were given above, and, in this example, these are the components of the vector \(\mathbb{C}\). The ERA-AIR software package includes modules for calculating source emission reductions (necessary to achieve MPC) using methods [58; 59]. The first of them [58] enables one to find a unique solution to the system of inequalities (41), but it is not an optimization one.