# On Risk-Averse Stochastic Semidefinite Programs with Continuous Recourse

## TL;DR

This paper introduces mean-risk models for stochastic semidefinite programs with continuous recourse, analyzing their structural properties and stability under distribution perturbations, with implications for computational approaches.

## Contribution

It develops a framework for risk-averse stochastic SDPs, exploring convexity, continuity, and stability, and presents extended formulations for finite discrete distributions.

## Key findings

- Mean-risk models exhibit convexity and Lipschitz continuity.
- Extended formulations lead to deterministic mixed-integer SDPs.
- Models are stable under distribution perturbations.

## Abstract

The vast majority of the literature on stochastic semidefinite programs (stochastic SDPs) with recourse is concerned with risk-neutral models. In this paper, we introduce mean-risk models for stochastic SDPs and study structural properties as convexity and (Lipschitz) continuity. Special emphasis is placed on stability with respect to changes of the underlying probability distribution. Perturbations of the true distribution may arise from incomplete information or working with (finite discrete) approximations for the sake of computational efficiency. We discuss extended formulations for stochastic SDPs under finite discrete distributions, which turn out to be deterministic (mixed-integer) SDPs that are (almost) block-structured for many popular risk measures.