Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures

TL;DR

This paper investigates the role of qualification conditions, including classical and quasi-relative interior conditions, in deriving subdifferential formulae and dual representations for convex risk measures using conjugate duality theory.

Contribution

It highlights the importance of various qualification conditions, especially the quasi-relative interior, in convex risk measure analysis and duality.

Findings

Qualification conditions are crucial for subdifferential formulae.

Classical interiority and quasi-relative interior conditions are both effective.

The paper clarifies the role of these conditions in convex risk measure duality.

Abstract

A fruitful idea, when providing subdifferential formulae and dual representations for convex risk measures, is to make use of the conjugate duality theory in convex optimization. In this paper we underline the outstanding role played by the qualification conditions in the context of different problem formulations in this area. We show that not only the meanwhile classical generalized interiority point ones come here to bear, but also a recently introduced one formulated by means of the quasi-relative interior.