## 3 Results  
We verified the correctness of the MORSE method and tested its performance empirically on the 20 MIPLIB instances listed in Table 1. For the first set of large-scale tests, in parallel, we executed 100 runs with MORSE and 100 Gurobi runs with homogeneous (unchanged) weights and collected one optimal solution for each run. For each problem instance, we assigned a distinct random seed to each run to maintain the reproducibility of the results, ensuring that the runs were independent and shared the same initial conditions. Runs were sorted using (as the sort key) either the random weights seed (for MORSE runs) or the Gurobi random generator seed (for homogeneous weights runs). From the two side-by-side orderings of homogeneous weights runs and MORSE runs, we aggregated lists of optima by taking one optimum from each run. Each list may contain duplicates because the runs were executed in parallel, in contrast to the inherently sequential approach of [31,32] described in Section 1. We further ordered the paired lists of optima in three distinct ways:  
1. _Seed-based:_ Keeping the random number generator in seed-based ordering, as outlined above. 2. _Random shuffle:_ Randomly shuffling the optima collected using the above _seed-based_ order. 3. _Greedy for diversity:_ Randomly choosing a solution from the seed-based list to be the first solution reported. Then, for each next solution chosen, choosing the solution that maximizes the average pairwise Hamming distance with all previously selected solutions. Our first test measured diversity as quantified by the Hamming distances (see Definition 1) for binary and integer variables between pairs of optima selected. To account for potential variations in integer representations, we rounded each binary/integer variable to the nearest integer. For each instance, we built a plot of average pairwise Hamming distance normalized by the total binary solution size (\(y\)-axis) vs. solution number (\(x\)-axis) for both MORSE and uniform weights for each of the three methods described above. For every sorting method, the average pairwise Hamming distance between optima found with MORSE exceeded the pairwise Hamming distance of optima found with the uniform weights method at almost all run numbers (see Figures 1-5 below). The shaded gray area surrounding the lines gives the 95% confidence interval. \begin{table}
\begin{tabular}{c c c} \hline
**MIPLIB Instance** & **Most Recent** & \multicolumn{2}{c}{**Number of**} \\  & **MIPLIB Version** & **Distinct Optima** \\ \hline set1 al & 2.0 & 2 \\ set1 cl & 2.0 & 2 \\ p0201 & 3.0 & * 4 \\ bell3a & 3.0 & 5 \\ mod008 & 3.0 & * 6 \\ p0033 & 3.0 & * 9 \\ lp41 & 2.0 & 24 \\ vpm2 & 4.0 & 33 \\ khb05250 & 3.0 & 51 \\ stein9 & 2.0 & 54 \\ stein45 & 3.0 & * 70 \\ supportcase14 & 6.0 & 256 \\ supportcase16 & 6.0 & 256 \\ stein15 & 3.0 & 315 \\ stein27 & 3.0 & * 2106 \\ harp2 & 5.0 & \(\geq\)1 \\ pigeon-10 & 6.0 & \(\geq\)6574 \\ app2-2 & 6.0 & \(\geq\)41,819 \\ noswot & 6.0 & \(\geq\)48,776 \\ vpm1 & 3.0 & \(\geq\)86,186 \\ \hline \end{tabular}
\end{table} 
Table 1: List of tested MIPLIB instances, the highest-numbered version of MIPLIB in which each instance appears, and the corresponding number of distinct optima. MIPLIB versions 2.0 and 3.0 were released in 1996, version 4.0 was released in 2003, version 5.0 was released in 2010, and version 6.0 was released in 2017. Distinct optima were enumerated using the PySCIPOpt count function [3]. Instances marked with * are also found in Table 1 of [4], and the number of optima claimed in that table is consistent with the number of optima found by count. Distinct optima counts marked with \(\geq\) give the number of optima enumerated by count within a 24 h runtime; if there is no \(\geq\), then enumeration was fully completed in runtime \(<\) 24 h. We did not perform full MORSE testing on any MIPLIB instance not listed in this table; some instances were excluded by the preliminary testing described in Section 2.6.