Singularity degree of non-facially exposed faces

TL;DR

This paper introduces the concept of singularity degree for faces of cones and establishes its equivalence to the number of facial reduction steps in conic optimization, extending previous complexity results to broader cases.

Contribution

It defines the singularity degree for non-facially exposed faces and proves its equivalence to facial reduction steps, generalizing existing results to arbitrary singularity degrees.

Findings

Singularity degree equals the number of facial reduction steps.

Frameworks of chordal graphs have at most one stress matrix level.

Results apply to linear images of cones and their facial structures.

Abstract

In this paper, we study the facial structure of the linear image of a cone. We define the singularity degree of a face of a cone to be the minimum number of steps it takes to expose it using exposing vectors from the dual cone. We show that the singularity degree of the linear image of a cone is exactly the number of facial reduction steps to obtain the minimal face in a corresponding primal conic optimization problem. This result generalizes the relationship between the complexity of general facial reduction algorithms and facial exposedness of conic images under a linear transform by Drusvyatskiy, Pataki and Wolkowicz to arbitrary singularity degree. We present our results in the original form and also in its nullspace form. As a by-product, we show that frameworks underlying a chordal graph have at most one level of stress matrix.