Abstract:

In terms of the Slater condition of the Bouligand and Clarke tangent derivatives of the objective multifunction $\Psi$, this paper mainly studies the stability of error bound of $\Psi$ at a point $\bar{x}$ with respect to an ordering cone $C$. It is proved that the Slater condition of the Bouligand tangent derivative of $\Psi$ at $\bar{x}$ with respect to $C$ is always stable with respect to all small calm perturbations. Based on this result, we prove that the Slater condition of the Bouligand tangent derivative of $\Psi$ at $\bar{x}$ with respect to $C$ is a sufficient condition for $\Psi$ to have a stable error bound at $\bar{x}$ with respect to $C$ when $\Psi$ undergoes small calm and regular perturbations. These results extend the corresponding ones given by Zheng [Math Oper Res, 2022, 47(4): 3282--3303] from the vector-valued to the set-valued case. As applications, some sufficient conditions are provided for a convex progress to have a stable global error bound with respect to an ordering cone.

Key words: error bound, tangent derivative, Slater condition, convex progress