Rademacher rounding for a semidefinite relaxation ($\hat{x}=V{\Lambda }^{1/2}\xi ,x=\hat{x}/{\max }_i|a_i^T\hat{x}|$)

Use an orthogonal eigendecomposition $X=V\Lambda V^T$ and a vector $\xi$ of independent Rademacher random variables. Because $\Lambda$ is diagonal and ${\xi }_j^2=1$, every sign vector gives $\Vert \hat{x}{\Vert }_2^2=\mathrm{tr}X$, without taking an expectation. If the scaling denominator $M$ is positive, then $x$ satisfies every slab constraint and $\Vert x{\Vert }_2^2=\mathrm{tr}X/M^2$. Under spanning constraints and positive trace, $M>0$ automatically. A nonzero $\hat{x}$ with $M=0$ instead certifies an unbounded direction.

For spanning slab normals, apply the maximum of finitely many Rademacher linear forms bound to $u_i=(V{\Lambda }^{1/2}{)}^T{a}_i$, whose squared norms are $a_i^{T}X{a}_i\le 1$. Some sign vector has scaling denominator squared at most $2\log (2m)$. Rademacher rounding for a semidefinite relaxation then produces a feasible vector of squared norm at least the displayed fraction of the SDP optimum. This is an existence guarantee from the probabilistic method.