Avoidance of the Real Objective’s Tie Locus

Prove whether the trajectory of the Riemannian trust-region solver for the Real Normal Procrustes Problem always avoids the tie locus at which the two blockwise objective branches coincide.

Background

The reduced real objective is a pointwise maximum of two smooth branch expressions, so it can fail to be differentiable on the tie locus. Random initialization avoids this set with probability one under the paper’s analytic-measure argument, but that does not establish that subsequent optimization iterates cannot reach it. The authors explicitly leave this trajectory-avoidance question unproved.