# A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software
## Appendix C.2.1 Algorithmic Innovations in DECIS
### Computational Results for DECIS  
We use DECIS to solve the random and SAPHIR test instances. The number of iterations and walltime for different solution methodologies are shown in Table 14. Two initialization strategies are tested on Benders decomposition: (U) where the initial first-stage candidate solution is 0, and (EV+U) where BD is employed to solve the EV (expected value) problem. The EV optimal solution is then used as a starting point for the stochastic instance. Iter-EV and Iter-U stand for the number of iterations required to solve the EV and stochastic problem, respectively. A stopping optimality gap of \(1\times 10^{-5}\) is considered. DECIS-CPLEX (12.7.0) uses primal simplex in both the MP and subproblems in Benders decomposition. DECIS-MINOS (5.6) is used in the quadratic MP and linear subproblems in regularized decomposition.  
To exemplify the effects of the constant penalty term on the performance of regularized decomposition, two \(\rho\) values, 1 and 10, are tested. From Table 14, it can be observed that regularized decomposition may significantly reduce the number of iterations, and thus the solution time of the overall decomposition algorithm. In addition, stronger penalization  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multirow{2}{*}{**Scenarios**} & \multicolumn{2}{c}{**Benders (U)**} & \multicolumn{3}{c}{**Benders (EV+U)**} & \multicolumn{3}{c}{**RD \(-\) 1 (U)**} & \multicolumn{3}{c}{**RD \(-\) 10 (U)**} \\ \cline{3-10}  & & **Iter** & **Time** & **Iter-EV** & **Iter-U** & **Time** & **Iter** & **Time** & **Iter** & **Time** \\ \hline \multirow{4}{*}{rand0} & 2000 & 82 & 29.72 & 31 & 77 & 27.98 & 50 & 13.17 & 72 & 18.30 \\  & 4000 & 71 & 53.77 & 35 & 58 & 48.49 & 42 & 22.11 & 58 & 30.16 \\  & 6000 & 105 & 112.36 & 47 & 106 & 120.96 & 58 & 40.6 & 85 & 58.83 \\  & 8000 & 121 & 170.25 & 38 & 111 & 155.61 & 59 & 54.23 & 102 & 91.64 \\  & 10,000 & 229 & 410.76 & 40 & 213 & 389.04 & 110 & 133.2 & 135 & 163.31 \\ \hline \multirow{4}{*}{rand1} & 2000 & 391 & 459.29 & 91 & 384 & 448.74 & 120 & 264.43 & 255 & 551.64 \\  & 4000 & 488 & 1051.82 & 87 & 487 & 1031.35 & 117 & 448.65 & 296 & 1175.35 \\  & 6000 & 396 & 1269.56 & 118 & 363 & 1158.85 & 100 & 533.02 & 146 & 781.95 \\  & 8000 & 443 & 1763.46 & 100 & 436 & 1688.43 & 106 & 679.39 & 153 & 1004.85 \\  & 10,000 & 449 & 2356.12 & 115 & 437 & 2353.02 & 113 & 983.68 & 193 & 1736.57 \\ \hline \multirow{4}{*}{rand2} & 2000 & 885 & 3213.08 & 125 & 870 & 3225.03 & 142 & 1147.62 & 265 & 2620.33 \\  & 4000 & 411 & 2784.49 & 136 & 405 & 2786.91 & 93 & 1696.08 & 212 & 3879.52 \\  & 6000 & 496 & 5470.71 & 165 & 520 & 5764.87 & 132 & 4196.52 & 223 & 6981.10 \\  & 8000 & 457 & 6151.33 & 173 & 459 & 6277.49 & 97 & 3631.94 & 140 & 5224.19 \\  & 10,000 & - & - & - & - & - & - & - & - & - \\ \hline \multirow{4}{*}{Saphir} & 50 & 167 & 362.21 & 163 & 80 & 317.21 & - & - & - & - \\  & 100 & 151 & 568.44 & 151 & 83 & 539.73 & - & - & - & - \\ \cline{1-1}  & 200 & - & - & - & - & - & - & - & - & - \\ \cline{1-1}  & 500 & 138 & 1357.83 & 109 & 73 & 917.47 & - & - & - & - \\ \cline{1-1}  & 1000 & - & - & - & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 14: Computational results for DECIS.  
might increase the number of iterations, as it restricts the movement of the first-stage candidate to be close to the best incumbent solution. Furthermore, this methodology might present numerical issues, such as bad scaling in the master problem, which makes the algorithm stop without closing the optimality gap. For instance, regularized decomposition fails to solve the whole set of SAPHIR problems.  
Using the (EV+U) initialization can accelerate the convergence of Benders decomposition. In 14 of 17 instances where BD converged, (EV+U) had fewer iterations than the (U) strategy, as well as less solution time. The reduction in the iteration number alleviates the time spent computing an appropriate starting point.