Asymptotic separability in sensitivity analysis

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Attributing Effects to a Cluster-Randomized Get-Out-the-Vote Campaign (2009, unknown): “This separable optimization is simpler than direct minimization of z(a) subject to 0 ≤ a ≤ A and a′1 = a, or “joint optimization” (Gastwirth et al. 2000), and unlike joint optimization it is always computationally feasible.”
On Sensitivity Value of Pair-Matched Observational Studies (2017, background): “When there are multiple controls for each treated observation, the theoretical analysis becomes more difficult as there is no closed form solution of the bounding variable TΓ, though it is possible to find the asymptotic normal distribution of TΓ (Gastwirth et al., 2000).”
Design sensitivity in observational studies (2004, background): “As shown in Gastwirth et al. (2000, § 3·1), in matched set i, the largest null expectation of Wk+1 j=1 Z ij q ij under (1) is k Ci = max aµ{1, . . . ,k} qWaj=1 qi(j)+C Wk+1j=a+1 qi(j)a+C(k+1−a) r .”