\[optim\_con\]Finding the optimal constraint
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![\[bisection\] Illustration of the self-consistent eigenvalue problem (\[sc\_eig\]). The solution is found when the lowest eigenvalue of $\Sigma(\epsilon)$ ($\lambda_0(\Sigma)$) equals $\epsilon$. The black vertical lines are poles in $\Sigma(\epsilon)$ and correspond to the precalculated eigenvalues of the $\mathcal{T}_1$ matrix on the cluster.](bisection.pdf)

In Sections \[eT1\] and \[eT2\] we derived new constraints which include and extend the three-index constraints imposed on local clusters on the lattice. These new constraints depend on states defined outside of the cluster, giving rise to additional complications. In this Section we explore a method to choose these states, optimizing the class of positive Hamiltonians generated by Eqs. (\[ecT1\_red\]) and (\[ecT2\_red\]) to make the constraints as strict as possible. In what follows we limit ourselves to the extended constraints with only one index outside of the cluster, [*i.e.*]{} for the $e\mathcal{T}_1$ there is only the dependence on the single-particle state $p$, whereas the $e\mathcal{T}_2'$ depends on the single-hole state $h$ and the single-particle state $p$.

The problem we discuss is the following: given a 2DM $\Gamma$ from some previous calculation, for what state $p$, outside of the cluster, is the $e\mathcal{T}_1(\Gamma)$ constraint maximally violated. We have chosen to define the most violated 