## 3 Results
In addition to testing MORSE on MIPLIB instances, we tested on instances from MadHitter, an algorithm that identifies gene targets for experimental CAR-T, conjugated antibody, and coated nanoparticle therapies [1]. The algorithm takes single-cell tumor data and non-tumor data for a set of patients and returns the smallest set of target genes that kills at least a specified _lower bound_ percentage of each patient's tumor cells (for example, \(\geq\)80%) while killing at most a specified _upper bound_ percentage of each patient's non-tumor cells (for example, \(\leq\)10%). The single-cell datasets were reduced to focus on the 1269 genes encoding cell surface receptors, which may serve as targets for one or multiple \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MIPLIB** & **PopulateLim** & **Number of Distinct** & **Number of Distinct** & **Number of** \\
**Instance** & **Setting** & **Optima (CPLEX)** & **Optima (SCIP)** & **Missed Optima** \\ \hline khb05250 & 51 & 1 & 51 & 50 \\ khb05250 & 102 & 1 & 51 & 50 \\ khb05250 & 204 & 1 & 51 & 50 \\ khb05250 & 408 & 1 & 51 & 50 \\ khb05250 & 816 & 1 & 51 & 50 \\ khb05250 & 1632 & 1 & 51 & 50 \\ khb05250 & 3264 & 1 & 51 & 50 \\ khb05250 & 6528 & 1 & 51 & 50 \\ khb05250 & 13,056 & 1 & 51 & 50 \\ khb05250 & 26,112 & 1 & 51 & 50 \\ set1 al & 2 & 1 & 2 & 1 \\ set1c1 & 2 & 1 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tested MIPLIB instances for which the CPLEX populate function did not find all optimal solutions. Figure 6: Number of distinct optima vs. run number for MIPLIB instance vpm1. While we cannot find all optima in the allotted 1000 runs, we find 998 distinct optima with MORSE and 573 distinct optima with the uniform weights method. of the experimental therapies listed above [1]. The objective is to minimize the number of treatments that a patient must receive. In a second layer of optimization for a cohort of multiple patients, we optimize the total number of distinct gene targets used, such that the number of targets for each patient is optimal. Intuitively, if a single patient has multiple sets of gene targets of the same optimal size, we select the optimal set that best permits us to reuse the same targets for other patients in the cohort; the exact ILP formulation is given in [1]. Figure 8: Number of Distinct Optima vs. Run Number for MIPLIB instance pigeon-10. With MORSE, we find 10 distinct optima in 23 runs, while 70 runs are needed to find 10 distinct optima using the uniform weights method. Figure 7: Number of distinct optima vs. run number for MIPLIB instance stein15. With MORSE, we find 264 distinct optima in the allotted 500 runs, while we find one optimum with the uniform weights method. For each feasible instance (dataset, lower bound, and upper bound permutation for which an optimum exists), we executed 1000 runs for both MORSE (in which each gene is randomly assigned a weight in the range \([1-\varepsilon,1+\varepsilon]\)) and the uniform weights method (in which each gene has a uniform weight of 1). Since all solutions found were composed of fewer than 100 targets, \(\varepsilon=0.01\) was used for all instances. This value of \(\varepsilon\) was chosen such that the maximum magnitude of the sum of perturbations, given by \(\varepsilon\)-(_number of targets in solution_), did not exceed 1. Just as we did for the MIPLIB instances, we measured the number of distinct optima found as a function of run number. Distinct solutions were defined as solutions that differed globally; we did not account for differences in local (patient) variables. For example, consider _global optimum A_, given by [G1, G2, G3], with corresponding local optima as follows: _patient 1_ receives the treatments [G1, G2] and _patient 2_ receives the treatments [G2, G3]. Next, consider global optimum B, also given by [G1, G2, G3], with corresponding local optima as follows: _patient 1_ receives the treatments [G1, G3], and _patient 2_ receives the treatments [G1, G2].