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Optimization under rare events: scaling laws for linear chance-constrained programs

Jose Blanchet, Joost Jorritsma, Bert Zwart

Abstract

We consider a class of chance-constrained programs in which profit needs to be maximized while enforcing that a given adverse event remains rare. Using techniques from large deviations and extreme value theory, we show how the optimal value scales as the prescribed bound on the violation probability becomes small and how convex programs emerge in the limit. We use our results to analyze the performance of existing popular approaches in the rare-event regime. We show that the popular CVaR and sample approximations have optimality properties under light-tailed assumptions on the randomness, while they behave sub-optimal in a heavy-tailed setting. Our results are derived using large deviations theory, extreme value theory, process techniques, and random set theory.

Optimization under rare events: scaling laws for linear chance-constrained programs

Abstract

We consider a class of chance-constrained programs in which profit needs to be maximized while enforcing that a given adverse event remains rare. Using techniques from large deviations and extreme value theory, we show how the optimal value scales as the prescribed bound on the violation probability becomes small and how convex programs emerge in the limit. We use our results to analyze the performance of existing popular approaches in the rare-event regime. We show that the popular CVaR and sample approximations have optimality properties under light-tailed assumptions on the randomness, while they behave sub-optimal in a heavy-tailed setting. Our results are derived using large deviations theory, extreme value theory, process techniques, and random set theory.
Paper Structure (14 sections, 8 theorems, 57 equations, 1 figure)

This paper contains 14 sections, 8 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem description
  3. Light tails
  4. Heavy tails
  5. Example for light-tailed distributions
  6. Example and Numerical Experiments
  7. Concluding comments
  8. Proofs
  9. Proof of Theorem \ref{['thm:lt-ccp']}
  10. Proof of Theorem \ref{['thm:lt-cvar']}
  11. Proof of Theorem \ref{['thm:lt-scen']}
  12. Proof of Theorem \ref{['thm:ht-ccp']}
  13. Proof of Theorem \ref{['thm:ht-cvar']}
  14. Proof of Theorem \ref{['thm:ht-scen']}

Key Result

Proposition 1

The set $\{b: I(b) \geq 1\}$ is convex, compact, and possesses a non-empty interior. The following asymptotics hold as $r\to\infty$,

Figures (1)

  • Figure 1: A detailed explanation of the figures is given when Figure \ref{['fig:panel']} is first introduced. The left vertical axis shows the exact chance constrained solution (represented as VaR) compared to the CVaR relaxation, the scenario approach and the large deviations approximation; the right vertical axis shows the ratios of CVaR/VaR solutions which converge to the theoretical limit (dotted line) as $\delta\downarrow0$. The figures on the right can be seen as a continuation of the figures on the left as the values of $\delta$ decrease.

Theorems & Definitions (10)

  • Definition 1: Asymptotic optimality
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 2
  • proof