# Bundle Enrichment Method for Nonsmooth Difference of Convex Programming Problems
## 6 Numerical Experiments
### Results
#### 6.3.3 Results for Large-Scale Problems  
Tables 4 and 5 provide the results obtained by different algorithms for three large-scale problems from [36]: \(P_{L1},P_{L2},\) and \(P_{L3}\). More specifically, Table 4 contains the best objective function values found by each solver, and Table 5 displays the number of function and subgradient evaluations. As we mentioned above, the number of function and subgradient evaluations are the same for SDCA-LP. Only two algorithms, BEM-DC and ASM-DC, can find approximate solutions to all three problems. AggSub fails in Problem \(P_{L3}\), PBDC in Problem \(P_{L1}\), DCA in Problems \(P_{L1}\) and \(P_{L3}\), PBMDC in Problem \(P_{L1}\), and SDCA-LP in Problems \(P_{L1}\) and \(P_{L2}\). Summarizing results from Table 4, we can conclude that the BEM-DC is the most accurate among all algorithms for solving large-scale problems used in numerical experiments. \begin{table}
\begin{tabular}{l r r r r r r r r r r r r r r} \hline \hline
**Prob.** & \(n\) & **BEM-DC** & \multicolumn{2}{c}{**ASM-DC**} & \multicolumn{2}{c}{**AggSub**} & \multicolumn{2}{c}{**PBDC**} & \multicolumn{2}{c}{**DCA**} & \multicolumn{2}{c}{**PBMDC**} & \multicolumn{2}{c}{**SDCA-LP**} \\  & & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) & \(N_{f}\) & \(N_{\xi}\) \\ \hline
12 & 50 & 739 & 156 & 3015 & 771 & 1411 & 694 & – & – & 99 & 97 & – & – & – \\
12 & 100 & 1032 & 173 & 6998 & 2519 & 1239 & 907 & – & – & 117 & 115 & – & – & – \\
12 & 200 & 2757 & 174 & 15,302 & 5090 & 2850 & 1397 & – & – & 338 & 337 & – & – & – \\
13 & 10 & 145 & 21 & 26 & 13 & 70 & 38 & – & – & 20 & 19 & – & – & – \\
14 & 2 & 19 & 4 & 50 & 23 & 71 & 34 & 9 & 5 & 56 & 54 & 11 & 16 & 29 \\
14 & 5 & 145 & 26 & 206 & 73 & 226 & 100 & 109 & 105 & 63 & 61 & 22 & 31 & 58 \\
14 & 10 & 249 & 59 & 357 & 107 & 322 & 155 & 6169 & 6151 & 61 & 60 & 34 & 50 & 107 \\
14 & 50 & 1168 & 439 & 672 & 211 & 1653 & 790 & 991 & 966 & 16 & 14 & 59 & 89 & 280 \\
14 & 100 & 2273 & 708 & 1779 & 549 & – & – & 780 & 740 & 18 & 16 & – & – & 552 \\
14 & 200 & 2591 & 724 & 1746 & 541 & – & – & 983 & 961 & 28 & 26 & – & – & 619 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{l r r r r r r r} \hline \hline \(n\) & 2 & 3 & 4 & 5 & 10 & 50 & 100 \\ \hline
1 starting point & 0.000 & 0.000 & 0.001 & 0.012 & 0.009 & 0.672 & 16.328 \\ \hline
20 starting points & 0.000 & 0.000 & 0.002 & 0.018 & 0.616 & 14.613 & 42.461 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: CPU time (in seconds) required by BEM-DC. Figure 1: Performance profiles using the number of function evaluations (20 starting points). Figure 2: Performance profiles using the number of subgradient evaluations (20 starting points). Results presented in Table 5 show that among all algorithms, BEM-DC uses the least number of function and subgradient evaluations for solving Problem \(P_{L1}\), and PBMDC requires the least computational effort for solving Problems \(P_{L2}\) and \(P_{L3}\). Although the BEM-DC can find accurate solutions for Problems \(P_{L2}\) and \(P_{L3}\), it nevertheless requires more--and in the case of Problem \(P_{L3}\), significantly more--function and subgradient evaluations than other solvers. All other algorithms also require reasonable computational effort in problems where they succeeded to find solutions.