Geometric characterizations of Lipschitz stability for convex optimization problems
Tran T. A. Nghia
TL;DR
This work develops geometric, computable characterizations of tilt stability and Lipschitz stability for convex optimization problems of the form $\min_x f(x)+g(x)$, where $f$ is smooth and $g$ is convex (possibly nonpolyhedral). By imposing a quadratic growth condition for $g$ on a targeted set $\mathcal{M}$ and establishing relative approximations of $\partial g$ by $\mathcal{M}$, the authors derive a clean condition $\ker \nabla^2 f(\bar{x})\cap 𝔗_{\mathcal{M}}=\{0\}$ that characterizes tilt stability via a second-subderivative framework, avoiding the need to compute generalized Hessians. They then show that tilt stability implies Lipschitz stability of the solution map $S(b,\mu)$, with a stronger message for nonpolyhedral regularizers such as the $\ell_1/\ell_2$ group Lasso and the nuclear norm: if the solution map is single-valued, Lipschitz stability follows automatically. The paper provides explicit geometric conditions for group Lasso and nuclear-norm minimization, including a strong sufficient condition and, in the nuclear-norm case, a nondegeneracy-based equivalence to uniqueness. Together, these results offer a practical, second-order-analytic route to certify stability and sensitivity in important convex optimization problems.
Abstract
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.