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.

翻译：本文重新审视并扩展了Rockafellar与Uryasev在2013年提出的风险四边形（RQ）框架，该框架作为整合风险管理、优化与统计估计的统一方案。RQ包含四个面向随机过程的泛函——风险、偏差、遗憾与误差，以及关联的统计量，并阐明了它们之间具有揭示性且在某些方面令人意想不到的相互关系与对偶性。本文综合评述了自2013年以来RQ框架的新增内容，重点关注理论进展与实际应用。新提出的四边形——超分位数四边形、超分位数范数四边形、分位数期望值四边形、有偏均值四边形、分位数对称平均并集四边形以及基于φ散度的四边形——为机器学习、统计学、金融学以及含偏微分方程约束优化等领域的风险敏感决策提供了新方法。理论贡献在于提出了“次正则性”公理，该公理弱化了四边形泛函的“正则性”要求——后者在某些应用中限制性过强。本文重新审视了RQ的核心定理与关联，并将其严格扩展至这一更广泛的框架中。通过投资组合优化、回归与分类等实例，展示了对偶性（尤其是与鲁棒优化和广义随机散度的联系）的优势与作用。