\succeq& 0 \qquad \text{using} \qquad B^\dagger = \sum_{\alpha\beta\gamma}t^2_{\alpha\beta\gamma}a^\dagger_\alpha a^\dagger_\beta a_\gamma + \sum_\mu s_\mu a^\dagger_\mu~.\end{aligned}$$ These are matrix-positivity constraints on respectively three-particle and two-particle-one-hole space, and are referred to as three-index constraints. Enforcing these conditions usually greatly improves the accuracy of the results, but at a considerable computational cost.

v2DM can now be formulated as the following constrained optimization problem: $$\min_{\Gamma} \mathrm{Tr}~\Gamma H^{(2)}~,$$ on the condition that $$\begin{aligned}
\mathrm{Tr}~\Gamma &=& \frac{N(N-1)}{2}~,\\
\mathcal{L}(\Gamma) &\succeq& 0 \qquad \forall \mathcal{L} \in \{\mathcal{P,Q,G},\mathcal{T}_1,\mathcal{T}_2\}~.\end{aligned}$$ Because the set of 2DM’s over which the optimization is performed is too large, one obtains a variational lower bound to the energy, complementary to variational wave-function techniques where one finds an upper bound.

This problem is an instance of the class of semidefinite programs (SDP). This is a well studied optimization problem for which many algorithms have been developed. A number of these algorithms have been tailored to the specific case of v2DM, and they can be divided into two classes. On the one hand there are the interior-point methods [@nakata_first; @primal_dual] which are very robust, but have a slow computational scaling of about $\mathcal{O}(M^{12})$, regardless of when two- and three-index conditions are imposed, where $M$ is the dimension of single-particle 