Theorem 3, p. 23 — polyhedral Γ and 0 ∉ Proj_{𝒱(ℱ)}{b(γ) : γ ∈ ℱ} on semidefinite faces: SDP optimizers lie in 𝒟, Opt = Opt_SDP

Consider a QCQP with $m\ge1$ constraints and suppose:

1. Assumption 1: the QCQP is feasible and some $\gamma^*$ with $\gamma^*_i\ge 0$ for $i\in[\![m_I]\!]$ has $A(\gamma^*)\succ 0$;
2. Assumption 3: $\Gamma$ is polyhedral;
3. for every semidefinite face $\mathcal F$ of $\Gamma$,

Then every optimizer $(x^*,t^*)\in\arg\min_{(x,t)\in\mathcal D_{\mathrm{SDP}}}2t$ satisfies $(x^*,t^*)\in\mathcal D$. In particular,

So under these conditions the SDP relaxation is exact, and the $x$-block of any optimal lifted matrix of the SDP is an optimal solution of the QCQP.

Formalization Note "$(x^*,t^*)\in\arg\min 2t$" is stated as membership in $\mathcal D_{\mathrm{SDP}}$ together with $2t^*\le 2t$ for every $(x,t)\in\mathcal D_{\mathrm{SDP}}$. Existence of an optimizer is not assumed, and the value identity is stated as a separate conclusion in EReal. The standing assumption $m\ge 1$ of the paper (p. 1) is a hypothesis. Faces are nonempty, and the projection is the Euclidean orthogonal projection.