Theorem 1, p. 13 — for polyhedral Γ with dim 𝒱(ℱ) ≥ aff dim{b(γ) : γ ∈ ℱ} + 1 on every semidefinite face, conv(𝒟) = 𝒟_SDP and Opt = Opt_SDP

Consider the QCQP (1) with $m\ge1$ constraints, its epigraph $\mathcal D$ and the projected epigraph $\mathcal D_{\mathrm{SDP}}$ of its standard SDP relaxation. Suppose Assumption 1 (primal feasibility and some $\gamma^*\in\mathbb R^m$, nonnegative on the inequality constraints, with $A(\gamma^*)\succ0$) and Assumption 3 ($\Gamma$ is polyhedral) hold. Suppose furthermore that every semidefinite face $\mathcal F$ of $\Gamma$ satisfies

Then

This is the main convex hull result of §4: under the stated conditions the Shor relaxation describes the convex hull of the epigraph exactly, and in particular its value equals the QCQP's value.

Formalization Note. Constraints are indexed by $i\in\{0,\dots,m-1\}$ (Fin m), constraint $i$ being the paper's constraint $i+1$; it is an inequality exactly when $i<m_I$. Faces are nonempty (the empty face imposes nothing under the page's convention $\operatorname{aff\,dim}\emptyset=-1$). $\mathrm{Opt}$ and $\mathrm{Opt}_{\mathrm{SDP}}$ are infima in EReal. The hypotheses are exactly those of the page together with the paper's standing assumption $m\ge1$ (p. 1); Assumption 2 is not assumed, since it follows from Assumption 3.