The Risk Quadrangle in Optimization: An Overview with Recent Results and Extensions

TL;DR

This paper reviews and extends the Risk Quadrangle framework for risk management and optimization, introducing new functionals and theoretical axioms, with applications in finance, machine learning, and statistics.

Contribution

It provides a comprehensive synthesis of recent theoretical advancements and practical extensions of the Risk Quadrangle, including new functionals and relaxed axioms.

Findings

New quadrangles like superquantile and expectile offer novel risk measures.

Extended theorems to a more general framework with relaxed axioms.

Applications demonstrated in portfolio optimization, regression, and classification.

Abstract

This paper revisits and extends the 2013 development by Rockafellar and Uryasev of the Risk Quadrangle (RQ) as a unified scheme for integrating risk management, optimization, and statistical estimation. The RQ features four stochastics-oriented functionals -- risk, deviation, regret, and error, along with an associated statistic, and articulates their revealing and in some ways surprising interrelationships and dualizations. Additions to the RQ framework that have come to light since 2013 are reviewed in a synthesis focused on both theoretical advancements and practical applications. New quadrangles -- superquantile, superquantile norm, expectile, biased mean, quantile symmetric average union, and $\varphi$-divergence-based quadrangles -- offer novel approaches to risk-sensitive decision-making across various fields such as machine learning, statistics, finance, and PDE-constrained optimization. The theoretical contribution comes in axioms for ``subregularity'' relaxing ``regularity'' of the quadrangle functionals, which is too restrictive for some applications. The main RQ theorems and connections are revisited and rigorously extended to this more ample framework. Examples are provided in portfolio optimization, regression, and classification, demonstrating the advantages and the role played by duality, especially in ties to robust optimization and generalized stochastic divergences.