Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems
Zhou Wei, Michel Thera, Jen-Chih Yao
TL;DR
This work develops a directional-derivative framework to characterize the stability of error bounds for convex inequality constraint systems, covering both a single convex inequality and semi-infinite constraint systems. It shows that local stability corresponds to a nonzero minimal directional derivative over the unit sphere, while global stability corresponds to a strictly positive infimum of the distance from the origin to the subdifferential on the positive-side region; for semi-infinite systems, stability is achievable under a shared linear perturbation of all component functions and appropriate index-set conditions. The results connect stability to solving convex optimization problems on the unit sphere and provide precise equivalences and necessary conditions, including a constraint qualification for global stability. Overall, the paper offers a unified, derivative-based approach to stability analysis that can guide both theoretical investigation and computational verification in convex inequality systems.
Abstract
In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimun of the directional derivative at a reference point over the sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequality constraint systems is, to some degree, equivalent to solving the convex optimization/minimization problems (defined by directional derivatives) over the sphere.