Error Bounds for a Class of Cone-Convex Inclusion Problems
Nguyen Quang Huy, Nguyen Huy Hung, Nguyen Van Tuyen, Hoang Ngoc Tuan
TL;DR
This paper addresses error bounds for cone-convex inclusion problems of the form $h(x)\in K$ in finite dimensions, with a smooth cone $K$ and a continuously differentiable, $K$-concave $h$. The authors prove that a local error bound around a reference point is equivalent to the Abadie constraint qualification (ACQ) in a neighborhood, thereby extending known polyhedral results to the non-polyhedral smooth cone setting. They further characterize conditions under which a global error bound exists for the affine case $h(x)=A x+b$, identifying precise criteria linked to the image of $A$ and the cone $K$, and they provide a counterexample showing ACQ does not guarantee a global bound. Additional results include a detailed development of geometric properties of smooth cones and their relation to regular and strictly convex cones, offering insights relevant to second-order, $p$- and circular cones. These results have implications for convergence analysis and sensitivity in optimization problems using smooth-cone constraints.
Abstract
In this paper, we investigate error bounds for cone-convex inclusion problems in finite-dimensional settings of the form $f(x)\in K$, where $K$ is a smooth cone and $f$ is a continuously differentiable and $K$-concave function. We show that local error bounds for the inclusion can be characterized by the Abadie constraint qualification around the reference point. In the case where $f$ is an affine function, we precisely identify the conditions under which the inclusion admits global error bounds. Additionally, we derive some properties of smooth cones, as well as regular cones and strictly convex cones.