# Bundle Enrichment Method for Nonsmooth Difference of Convex Programming Problems
## 6 Numerical Experiments
### Results
#### 6.3.1 Results with One Starting Point  
Table 1 presents the best value of the objective function obtained by solvers using one starting point. We can see that the BEM-DC algorithm is sufficiently accurate in finding local solutions. Except for Problem 9, it finds global minimizers of DC optimization problems in all other cases. Other solvers fail in one or several problems. More specifically, ASM-DC, AggSub, and DCA fail to find solutions in Problem 4 with \(n=200\). Furthermore, AggSub fails in Problems 5 and 14 with \(n=5,10,50,100,200\); PBDC fails in Problem 12 with \(n=50,100,200\) and in Problem 13; DCA fails in Problem 8; PBMDC fails in Problems 7, 12, 13 and also fails in Problem 14 with \(n=100,200\); and finally, SDCA-LP fails in Problems 1, 8, 9, 13 and also fails in Problem 12 with \(n=10,50,100,200\).  
In Table 2, we report the number of function and subgradient evaluations required by the algorithms. These numbers are computed as an average of the number of function and subgradient evaluations of DC components. Note that in SDCA-LP, the number of function and subgradient evaluations are the same. The results show that, in general, PBMDC requires the least number of function and subgradient evaluations among all algorithms. These numbers are similar for PBDC and SDCA-LP. We can see that the computational effort required by the BEM-DC is reasonable and in some instances it is similar to that of by PBMDC, PBDC, and SDCA-LP. Here, Problem 4 with \(n=50,100,200\) is an exception. Three other solvers, ASM-DC, AggSub, and DCA also require a reasonable computational effort in most cases; however, in general, they use (in some cases significantly) more function and subgradient evaluations than BEM-DC PBMDC, PBDC, and SDCA-LP.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Prob.** & \(n\) & **BEM-DC** & **ASM-DC** & **AggSub** & **PBDC** & **DCA** & **PBMDC** & **SDCA-LP** \\ \hline \hline
1 & 2 & 2.00000 & 2.00000 & 2.00000 & 2.00000 & 2.00000 & 2.00000 & – \\
2 & 2 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 1.00000 & 0.00000 & 0.00000 \\
3 & 4 & 0.00000 & 0.00001 & 0.00001 & 0.00000 & 0.00000 & 0.00000 & 2.00000 \\
4 & 2 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\
4 & 5 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\
4 & 10 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\
4 & 50 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\
4 & 100 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Best values of the objective functions obtained by solvers (one starting point).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Prob.** & \(n\) & **BEM-DC** & **ASM-DC** & **AggSub** & **PBDC** & **DCA** & **PBMDC** & **SDCA-LP** \\  & & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) \\ \hline  
\ \hline \end{tabular}
\end{table}
Table 2: Number of function and subgradient evaluations required by solvers (one starting point).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Prob.** & \(n\) & **BEM-DC** & **ASM-DC** & **AggSub** & **PBDC** & **DCA** & **PBMDC** & **SDCA-LP** \\  & & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) & \(N_{g}\) & \(N_{f}\) \\ \hline
1 & 2 & 105 & 25 & 189 & 52 & 162 & 75 & 22 &