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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.

Use the spectral theorem for real symmetric matrices to write $X^{\ast }=V\Lambda V^T$, with $V$ orthogonal and $\Lambda$ diagonal and nonnegative. For any sign vector, ${\xi }_j^2=1$, soThis is an exact identity for every sign choice. Put $M(\xi )={\max }_i|a_i^T\hat{x}(\xi )|$. When $M>0$, the Rademacher rounding for a semidefinite relaxation givesHence $x(\xi )$ is feasible. In the finite spanning case from part (a), $\mathrm{tr}{X}^{\ast }>0$ makes $\hat{x}\ne 0$, and the spanning condition then gives $M>0$. If a nonzero $\hat{x}$ instead had $M=0$, it would be an unbounded feasible direction rather than a vector to divide by zero.

Randomized rounding converts a solution of a continuous relaxation of a integer programming problem into a random discrete candidate. For fractional binary variables $x_i\in [0,1]$, independent Bernoulli random variables $X_i$ with $P(X_i=1)=x_i$ preserve a linear objective in expectation: $\mathbb{E}{\sum }_i{c}_i{X}_i={\sum }_i{c}_i{x}_i$. They do not automatically preserve every constraint.

For example, $x_1=x_2=1/2$ satisfies $x_1+x_2\le 1$, but independent rounding chooses both items with probability $1/4$. A feasibility argument, rejection, repair, or a dependent rounding construction is therefore needed before this becomes an approximation algorithm. The objective bound and the feasibility guarantee are separate parts of its proof.

Gaussian hyperplane rounding uses a shared random hyperplane to produce a feasible sign vector for a semidefinite relaxation of binary quadratic optimization. Rademacher rounding for a semidefinite relaxation instead uses random signs in its construction. In both cases the distribution is chosen so that an expected objective bound can be related to the relaxation value.