  In this paper, we mainly study solution uniqueness of some convex
optimization problems. Our characterizations of solution uniqueness are in
terms of the radial cone. This approach allows us to know when a unique
solution is a strong solution or even a tilt-stable one without checking
second-order information. Consequently, we apply our theory to low-rank
optimization problems. The radial cone is fully calculated in this case and
numerical experiments show that our characterizations are sharp.
