Coordinate Shadows of Semidefinite and Euclidean Distance Matrices

We consider the projected semidefinite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semidefinite and Euclidean distance completion problems. We classify when these sets are closed and use the boundary structure of these two sets to elucidate the Krislock--Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the “minimal cones” of these problems admit combinatorial characterizations. As a by-product, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.