Generalized power cones: optimal error bounds and automorphisms

TL;DR

This paper develops tight error bounds for generalized power cones using facial residual functions, enabling better understanding of solution reliability and cone automorphisms.

Contribution

It introduces a novel framework for error bounds in generalized power cones, including applications to their algebraic structure and automorphism groups.

Findings

Error bounds are tight and applicable without constraint qualifications.

The automorphism group dimension for generalized power cones is computed.

Identification of self-dual, irreducible, nonhomogeneous, perfect generalized power cones.

Abstract

Error bounds are a requisite for trusting or distrusting solutions in an informed way. Until recently, provable error bounds in the absence of constraint qualifications were unattainable for many classes of cones that do not admit projections with known succinct expressions. We build such error bounds for the generalized power cones, using the recently developed framework of one-step facial residual functions. We also show that our error bounds are tight in the sense of that framework. Besides their utility for understanding solution reliability, the error bounds we discover have additional applications to the algebraic structure of the underlying cone, which we describe. In particular we use the error bounds to compute the dimension of the automorphism group for the generalized power cones, and to identify a set of generalized power cones that are self-dual, irreducible, nonhomogeneous, and perfect