On feasibility problems with spectral constraints

We study matrix feasibility problems "find X in K intersect C" where K is a closed convex matrix set and C = {X : sigma(X) in S} is defined by a convex constraint S on the ordered singular values. Using the classical spectral transfer identity, projection onto C reduces to an SVD plus a small quadratic program. For seven natural polyhedral S we embed this projector in a plain alternating projection (AP) loop. We experiment with two concrete families of K - linear constraints (an affine subspace intersected with an entrywise box) and ellipsoidal constraints (a non-centered anisotropic Frobenius ellipsoid) - although the method applies equally to more general convex constraints. The experiments expose three regimes: rapid feasibility, slow tail convergence, or informative infeasibility plateaus.