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Risk-Adaptive Approaches to Stochastic Optimization: A Survey

Johannes O. Royset

TL;DR

The survey presents risk measures as a unifying framework for stochastic optimization under uncertainty, tracing developments from finance to engineering and machine learning. It emphasizes superquantiles (CVaR) as central, with equivalent formulas, convex optimization properties, and broad applicability; it also develops duality, distributional robustness connections, and algorithmic strategies. The work highlights practical implications for reliability-based design, fairness, and surrogate modeling, while outlining extensions to dynamic, multi-stage, and reliability contexts and identifying key open problems. Overall, the paper offers a comprehensive foundation and roadmap for risk-adaptive optimization across finite-dimensional and various applied settings.

Abstract

Uncertainty is prevalent in engineering design, data-driven problems, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount the spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.

Risk-Adaptive Approaches to Stochastic Optimization: A Survey

TL;DR

The survey presents risk measures as a unifying framework for stochastic optimization under uncertainty, tracing developments from finance to engineering and machine learning. It emphasizes superquantiles (CVaR) as central, with equivalent formulas, convex optimization properties, and broad applicability; it also develops duality, distributional robustness connections, and algorithmic strategies. The work highlights practical implications for reliability-based design, fairness, and surrogate modeling, while outlining extensions to dynamic, multi-stage, and reliability contexts and identifying key open problems. Overall, the paper offers a comprehensive foundation and roadmap for risk-adaptive optimization across finite-dimensional and various applied settings.

Abstract

Uncertainty is prevalent in engineering design, data-driven problems, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount the spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.
Paper Structure (32 sections, 17 theorems, 107 equations, 6 figures)

This paper contains 32 sections, 17 theorems, 107 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Decision Making under Uncertainty
  3. Engineering Design and Control
  4. Stochastic Optimization in Statistics and Machine Learning
  5. Other Application Areas
  6. Measures of Risk
  7. Connections with Utility Theory
  8. Superquantiles
  9. Equivalent Formulas
  10. Superquantiles in Optimization Models
  11. Algorithms for Superquantile Minimization
  12. Estimating Superquantiles
  13. Risk and Regret
  14. Desirable Properties
  15. Risk Minimization
  16. ...and 17 more sections

Key Result

Theorem 3.1

(equivalent formulas for superquantiles). For $\alpha\in (0,1)$ and an integrable random variable $\hbox{\boldmath $\xi$}$ with cumulative distribution function $P$ and quantile function $Q$, the following hold: where the $\alpha$-tail distribution is defined as having $P^{[\alpha]}(\xi) = \max\{ 0, P(\xi)-\alpha\}/(1-\alpha)$ as its cumulative distribution function. Thus, any of these formulas c

Figures (6)

  • Figure 1: Outline of electrical substation (reproduced from ByunDeoliveiraRoyset.22).
  • Figure 2: Low- and high-fidelity estimates of lift force (blue asterisks) produced by a hydrofoil from Example \ref{['eHydrofoil']}BonfiglioRoyset.19 and five lines representing surrogates with varying degree of conservativeness.
  • Figure 3: Probability density functions $p_1$ and $p_2$ for random variables $f(\hbox{\boldmath $\xi$},1)$ and $f(\hbox{\boldmath $\xi$},-1)$, respectively.
  • Figure 4: Quantile $Q(0.9)$ (solid line) and superquantile $\bar{Q}(0.9)$ (dashed) of $f(\hbox{\boldmath $\xi$},x)$ in Example \ref{['eQuantvsSuperquant']} as functions of $x$.
  • Figure 5: Histogram of 100000 outcomes of the cumulative loss due to earthquake damage in Vancouver.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 3.1
  • Example 3.2
  • ...and 36 more