# A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software
## 7 Conclusions  
We presented a summary of the state-of-the-art methodologies for the solution of two-stage linear stochastic problems. First, we introduced the mathematical formulation of such programs and highlighted features in their structure, which enable the development of decomposition algorithms. These methodologies are classified in two groups: node-based decomposition and scenario-based decomposition. For two-stage stochastic programs with continuous recourse, we summarized Benders decomposition, which partitions the problem according to its time structure. BD may present computational problems, which can be alleviated by reducing the cost of each iteration, and/or decreasing the number of iterations. Scenario decomposition methodologies are popular alternatives in the case of (mixed) integer recourse. The progressive hedging algorithm and dual decomposition relax the nonanticipativity restrictions and provide the user with valid bounds. Our numerical results \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Instances** & **Time** & **LB** & **UB** & **GAP [\%]** \\ \hline SSLPPRF\_5\_25\_100\_1 & 697.74 & \(-\)74,005.8 & \(-\)74,005.8 & 0 \\ SSLPPRF\_5\_25\_100\_2 & 1021.76 & \(-\)72,675 & \(-\)72,675 & 0 \\ SSLPPRF\_5\_25\_100\_3 & 673.04 & \(-\)75,666.9 & \(-\)75,666.9 & 0 \\ \hline SSLPPRF\_5\_50\_100\_1 & 3071.82 & 138,900 & 138,900 & 0 \\ SSLPPRF\_5\_50\_100\_2 & 9083.39 & 163,931 & 163,931 & 0 \\ SSLPPRF\_5\_50\_100\_3 & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of DSP with DW Branch-and- Bound on SSLPRF instances. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Instances** & **Scenarios** & **Time** & **Solution** \\ \hline SSLP5\_25 & 50 & 1.70 & \(-\)121.6 \\  & 100 & 3.16 & \(-\)127.37 \\ \hline  & 50 & 103.9 & \(-\)364.64 \\ SSLP10\_50 & 100 & 101.62 & \(-\)364.191 \\  & 500 & 1110.64 & \(-\)349.136 \\  & 1000 & 3021.83 & \(-\)351.711 \\ \hline  & 5 & 11.22 & \(-\)262.4 \\ SSLP15\_45 & 10 & 151.79 & \(-\)260.5 \\  & 15 & 377.65 & \(-\)253.601 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of DSP with DW-based branch-and-bound on SSLP instances. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Instances** & **Scenarios** & **Time** & **LB** & **UB** & **GAP [\%]** \\ \hline  & 200 & 417.08 & 1834.57 & 1834.57 & 0 \\ DCAP233 & 300 & 512.97 & 1644.25 & 1644.25 & 0 \\  & 500 & 898.05 & 1834.57 & 1834.57 & 0 \\ \hline  & 200 & 370.68 & 2322.5 & 2322.5 & 0 \\ DCAP243 & 300 & 525.91 & 2559.19 & 2559.19 & 0 \\  & 500 & 1523.61 & 2167.4 & 2167.4 & 0 \\ \hline  & 200 & 949.64 & 1060.75 & 1060.75 & 0 \\ DCAP332 & 300 & 1379.98 & 1252.77 & 1252.77 & 0 \\  & 500 & - & 1588.660 & 1589.203 & 0.03 \\ \hline  & 200 & 2732.97 & 1619.61 & 1619.61 & 0 \\ DCAP342 & 300 & 1970.75 & 2067.5 & 2067.5 & 0 \\  & 500 & 3958.25 & 1904.74 & 1904.74 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of DSP with DW Branch-and- Bound on DCAP instances. show that the performance of PH is strongly affected by the constant penalty multiplier. Furthermore, its performance and the quality of the approximate solution may be enhanced by grouping the scenarios in large bundles (or scenario sets). We also tested the dual decomposition algorithm with the DSP package. The computational results show that DSP is able to provide a tight lower bound on the instances that we tested. However, the optimality gaps can be as large as 10%, relative to the upper bound from literature. Therefore, for those tested instances, future effort should be focused on developing more advanced heuristics to improve the best feasible solution. Moreover, we tested the newest DSP implementation, which incorporates a Dantzig-Wolfe decomposition-based branch-and-bound algorithm to close the optimality gap. In most of the instances, it is able to retrieve a global solution in the given walltime limit. We highlight that its computational performance depends on the strength of the dual decomposition.