Geometric characterizations of Lipschitz stability for convex optimization problems

In this paper, we mainly study tilt stability and Lipschitz stability of convex optimization problems. Our characterizations are geometric and fully computable in many important cases. As a result, we apply our theory to the group Lasso problem and the nuclear norm minimization problem and reveal that the Lipschitz stability of the solution mapping in these problems is automatic whenever the solution mapping is single-valued.