# Strong Convexity for Risk-Averse Two-Stage Models with Fixed Complete Linear Recourse

## TL;DR

This paper extends the understanding of strong convexity in two-stage risk-averse models with linear recourse, providing conditions for various risk measures and implications for stability and optimization algorithms.

## Contribution

It introduces the concept of partial strong convexity and derives verifiable conditions for strong convexity in models with distortion risk measures.

## Key findings

- Conditions for strong convexity in models with CVaR and distortion risk measures.
- Implications for stability under probability measure perturbations.
- Relevance for convergence rates in stochastic optimization algorithms.

## Abstract

This paper generalizes results concerning strong convexity of two-stage mean-risk models with linear recourse to distortion risk measures. Introducing the concept of (restricted) partial strong convexity, we conduct an in-depth analysis of the expected excess functional with respect to the decision variable and the threshold parameter. These results allow to derive sufficient conditions for strong convexity of models building on the conditional value-at-risk due to its variational representation. Via Kusuoka representation these carry over to comonotonic and distortion risk measures, where we obtain verifiable conditions in terms of the distortion function. For stochastic optimisation models, we point out implications for quantitative stability with respect to perturbations of the underlying probability measure. Recent work in \cite{Ba14} and \cite{WaXi17} also gives testimony to the importance of strong convexity for the convergence rates of modern stochastic subgradient descent algorithms and in the setting of machine learning.