On the $p$-order Semismoothness of the Metric Projection onto Slices of the Positive Semidefinite Cone

The metric projection onto the positive semidefinite (PSD) cone is strongly semismooth, a property that guarantees local quadratic convergence for many powerful algorithms in semidefinite programming. In this paper, we investigate whether this essential property holds for the metric projection onto an affine slice of the PSD cone, which is the operator implicitly used by many algorithms that handle linear constraints directly. Although this property is known to be preserved for the second-order cone, we conclusively demonstrate that this is not the case for the PSD cone. Specifically, we provide a constructive example that for any $p > 0$, there exists an affine slice of a PSD cone for which the metric projection operator fails to be $p$-order semismooth. This finding establishes a fundamental difference between the geometry of the second-order cone and the PSD cone and necessitates new approaches for both analysis and algorithm design for linear semidefinite programming problems.