A note on alternating projections for ill-posed semidefinite feasibility problems

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of \(\mathcal {O}\Big (k^{-\frac{1}{2^{d+1}-2}}\Big )\), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of \(\mathcal {O}\Big (\frac{1}{\sqrt{k}}\Big )\).