## 2 Materials and Methods
### Some Theory to Justify the Choice of \(\varepsilon\)
Specifically, the function \(C\) maps \(p_{j}\) to \(p_{k}\) by permuting indices without changing the set of uniformly selected multipliers. For example, suppose there are three variables and \(p_{j}\) perturbs the coefficients by multiplying \(c_{1}\) by 0.9998, \(c_{2}\) by 1.0007, and \(c_{3}\) by 0.9993. Then, an example of a permuted perturbation would be instead to multiply \(c_{1}\) by 1.0007, to multiply \(c_{2}\) by 0.9993 and \(c_{3}\) by 0.9998. For \(i=1,\ldots n\), we have two cases:  
1. If \(x_{i}\) is the \(r\)th variable \(x_{jr}\) in \(V_{j}\), then \(C\) maps the multiplier of \(c_{i}=c_{jr}\) to be instead the multiplier of \(c_{kr}\). 2. If \(x_{i}\) is the \(s\)th variable absent from \(V_{j}\), then \(C\) maps the multiplier of \(c_{i}\) to be instead the multiplier for the coefficient of the \(s\)th variable absent from \(V_{k}\), preserving the value of the perturbed coefficient. Since the mapping \(C\) preserves the values of the coefficients and of the objective function in the perturbed problem, it follows from Lemma 1 that \(C\) preserves optima of the perturbed problem. Since \(C\) preserves the values of the variables and any permutation of the dimensions is a one-to-one correspondence on the set of dimensions, it follows that \(C\) is a one-to-one correspondence between \(M_{j}\) and \(M_{k}\). We are also interested in the probability that each possible optimum of the original problem appears as an optimum of a perturbed problem. Suppose there are \(v\) variables that appear in the objective function, each corresponding to a distinct term. Thus, it follows that the space of perturbations is isomorphic to the hypercube \(\mathcal{Q}:=[1-\varepsilon,1+\varepsilon]^{v}\). The hypercube representation is reminiscent of the work of [31] on multiple optima for binary programs, though there exist a few notable differences. Their work focused exclusively on the vertices of the hypercube, each of which represented a different solution; \(0\) for the non-selected variables and \(1\) for the selected variables. By contrast, our interest extends to both the boundary and interior of \(\mathcal{Q}\), where \(\mathcal{Q}\) instead represents perturbations to the original objective function, rather than solutions. For \(\mathcal{Q}\), \"boundary\" refers to any point at which any one of the dimensions takes on either of the values \(1-\varepsilon\) or \(1+\varepsilon\), and \"interior\" refers to all other points. Whether a perturbation is on the boundary or in the interior does not affect the operation of MORSE. In our development and testing, we found it useful to test perturbations on the boundary to gain confidence in the correctness and robustness of the MORSE code. Employing the hypercube formalism, one may ask:  
**Question 1**. : _For any perturbation, which solutions remain optimal?_  
**Question 2**. : _Under what conditions on the perturbation \(p\in\mathcal{Q}\) is the solution to the perturbed problem for \(\mathrm{p}\) unique?_  
**Question 3**. : _For any optimum \(x\) to the original, unperturbed problem, does there exist a perturbation \(p\), such that if one applies \(p\), then \(x\) remains an optimum to the perturbed problem? (We gave an affirmative answer under some assumptions in Theorem 3; it would be desirable to determine if these assumptions can be weakened. )_  
1. _If so, we can define_ \(V(x)\) _as the volume within the hypercube of perturbations that preserve the optimality of_ \(x\)_. Can one determine the upper and lower bounds on_ \(V(x)\)_?_  
Our intuition, which we tested empirically via the solution pool method, suggests that for the majority of perturbations, the solution to the perturbed problem is unique for the variables constrained to be integers. However, there must be boundary regions of zero volume at which two or more solutions are tied. In Theorem 4, we answered Question 3(a) affirmatively for selection problems.