Theorem 8, p. 30 — if the quadratic eigenvalue multiplicity satisfies k ≥ m + 1, then conv(𝒟 ∩ H) = 𝒟_SDP ∩ H and Opt = Opt_SDP

Consider a QCQP $\inf\{q_0(x) : q_i(x)\le 0\ (i\in[\![m_I]\!]),\ q_i(x)=0\ (i\in[\![m_I+1,m]\!])\}$ with $m\ge 1$ constraints, epigraph $\mathcal D$, and Shor SDP relaxation with optimal value $\mathrm{Opt}_{\mathrm{SDP}}$ and projected epigraph $\mathcal D_{\mathrm{SDP}}$. Suppose Assumption 1 holds: the QCQP is feasible and some $\gamma^*$ with $\gamma^*_i\ge 0$ on the inequality constraints has $A_0+\sum_i\gamma^*_iA_i\succ 0$. Define the hyperplane

If the quadratic eigenvalue multiplicity $k$ of the QCQP (Definition 3) satisfies $k\ge m+1$, then

The theorem says that, with one copy of symmetry fewer than Theorem 7 requires, the SDP relaxation is still exact in value and its optimal face is the convex hull of the QCQP's optimal epigraph points.

Formalization Note $\mathrm{Opt}$ and $\mathrm{Opt}_{\mathrm{SDP}}$ are EReal infima, and $H$ is defined from the EReal value, so the value identity is stated as a separate conjunct. The quadratic eigenvalue multiplicity is the largest $k$ (IsGreatest), as in Definition 3. Constraints are indexed by Fin m with inequalities first.