Error Bounds for a Class of Cone-Convex Inclusion Problems

TL;DR

This paper studies error bounds for cone-convex inclusion problems, characterizing local bounds via constraint qualifications and identifying conditions for global bounds in affine cases, with insights into cone properties.

Contribution

It provides new characterizations of error bounds for cone-inclusion problems and conditions for their global validity, extending understanding of cone properties.

Findings

Local error bounds characterized by Abadie constraint qualification

Global error bounds identified for affine functions

Properties of smooth, regular, and strictly convex cones derived

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.