# Properties of the Quadratic Transformation of Dual Variables
## 6 Solving Test Problems
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{\(\Delta\psi\)} & \multirow{2}{*}{\(\Delta f\)} & \multicolumn{2}{c}{**Number of**} & \multicolumn{2}{c}{**Number of**} \\  & & & **Iterations** & **of Function and** \\  & & & & **Gradient** \\ \hline \multicolumn{5}{c}{Function LP\_R\_Y2 \(mb=75\), \(ma=10\), \(md=10\)} \\ sub & 4.850 \(\times\)\(10^{-1}\) & 3.150 \(\times\)\(10^{-4}\) & 429 & 860 \\ Shor & 4.850 \(\times\)\(10^{-1}\) & 3.128 \(\times\)\(10^{-5}\) & 341 & 655 \\ yv & 4.850 \(\times\)\(10^{-1}\) & 2.136 \(\times\)\(10^{-5}\) & 122 & 248 \\ MsgrPPR & 4.850 \(\times\)\(10^{-1}\) & 1.912 \(\times\)\(10^{-7}\) & 167 & 420 \\ BFGS & 4.850 \(\times\)\(10^{-1}\) & 1.163 \(\times\)\(10^{-5}\) & 170 & 372 \\ \multicolumn{5}{c}{Function LP\_R\_Y1 \(mb=75\), \(ma=10\), \(md=10\)} \\ sub & 4.850 \(\times\)\(10^{-1}\) & 3.967 \(\times\)\(10^{-3}\) & 1016 & 2059 \\ Shor & 4.850 \(\times\)\(10^{-1}\) & 1.649 \(\times\)\(10^{-5}\) & 566 & 1141 \\ yv & 4.850 \(\times\)\(10^{-1}\) & 1.444 \(\times\)\(10^{-5}\) & 393 & 773 \\  & & \multicolumn{2}{c}{Function L\_Y2 \(mb=75\), \(ma=10\), \(md=10\)} \\ sub & 1.957 \(\times\)\(10^{-11}\) & 4.027 \(\times\)\(10^{-4}\) & 348 & 703 \\ Shor & 2.678 \(\times\)\(10^{-11}\) & 1.657 \(\times\)\(10^{-5}\) & 468 & 881 \\ yv & 3.674 \(\times\)\(10^{-11}\) & 1.159 \(\times\)\(10^{-5}\) & 95 & 184 \\ MsgrPPR & 3.536 \(\times\)\(10^{-11}\) & 1.361 \(\times\)\(10^{-5}\) & 204 & 445 \\ BFGS & 1.440 \(\times\)\(10^{-12}\) & 3.770 \(\times\)\(10^{-6}\) & 159 & 370 \\ \multicolumn{5}{c}{Function L2\_Y1 \(mb=75\), \(ma=10\), \(md=10\)} \\ sub & 1.104 \(\times\)\(10^{-6}\) & 5.373 \(\times\)\(10^{-3}\) & 544 & 1097 \\ Shor & 6.602 \(\times\)\(10^{-11}\) & 1.385 \(\times\)\(10^{-5}\) & 580 & 1199 \\ yv & 8.253 \(\times\)\(10^{-11}\) & 3.199 \(\times\)\(10^{-5}\) & 369 & 710 \\ \multicolumn{5}{c}{Function 1/\(\chi\)\_Y2 \(mb=75\), \(ma=10\), \(md=10\)} \\ sub & 2.684 \(\times\)\(10^{-13}\) & 5.858 \(\times\)\(10^{-9}\) & 207 & 417 \\ Shor & 2.118 \(\times\)\(10^{-13}\) & 1.804 \(\times\)\(10^{-8}\) & 55 & 114 \\ yv & 1.320 \(\times\)\(10^{-15}\) & 2.850 \(\times\)\(10^{-9}\) & 64 & 125 \\ MsgrPPR & 1.651 \(\times\)\(10^{-16}\) & 8.098 \(\times\)\(10^{-11}\) & 101 & 207 \\ BFGS & 8.277 \(\times\)\(10^{-13}\) & 3.196 \(\times\)\(10^{-8}\) & 34 & 74 \\ Function & 1/\(\chi\)\_Y1 \(mb=75\), \(ma=10\), \(md=10\) & & \\ sub & 1.611 \(\times\)\(10^{-7}\) & 1.764 \(\times\)\(10^{-5}\) & 214 & 430 \\ Shor & 1.076 \(\times\)\(10^{-8}\) & 2.246 \(\times\)\(10^{-6}\) & 210 & 450 \\ yv & 6.973 \(\times\)\(10^{-8}\) & 2.496 \(\times\)\(10^{-6}\) & 106 & 214 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of calculations for dimensions \(n=1000\), \(m=100\) with \(mb=75\). from the set of constraints \(-x+a\leq 0\) for Q. Each of the problems is specified by the name of the function from Table 1 and the parameters (N_F, nx,ny, mb, ma, md). To solve problems (11), (31), we used the subgradient methods: sub [43], Shor [32], Orty [42], and yv [41] implemented with the one-dimensional search from [41; 42; 43]. To solve problem (31), we used additional methods for minimizing smooth functions: the Polak-Ribiere-Polyak conjugate gradient method (Msgr_PPR) [34] and the quasi-Newtonian method BFGS [44; 45; 46; 47]. The following conclusions have been made:  
1. When varying the number \(mb\) of active constraints, no important differences in accuracy and number of function and gradient evaluations were found. 2. In the case of a modulus transformation of dual variables, due to manifestations of the degeneracy of the function, the calculation accuracy by the multi-step method \(sub\) is significantly inferior compared to the method \(yv\) with a change in the space metric of the RSM. 3. Due to the low dimensionality of the dual problem, the quality of the solution is somewhat lower for a modulus transformation of the dual variables than for the quadratic transformation for RSM with a change in the space metric. 4.