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Minimizing risk measures with applications in network traffic engineering

Ashish Chandra, Mohit Tawarmalani

TL;DR

The paper tackles risk-aware decision-making under uncertainty by introducing an integrated quantile function-based measure, $\mathbb{E}_{\alpha-\gamma}[\mathbb{T}(x)]$, and formulating a two-stage bilinear optimization problem (P) that encompasses VaR and CVaR as special cases. It develops both under- and over-estimation techniques via first-level RLT relaxations and CVaR-based surrogates, plus an alternating-minimization scheme to tighten bounds. The authors prove key theoretical relations to integrated quantile functions and demonstrate the approach on a network traffic engineering case study with multiple failure scenarios, showing tighter estimates and practical performance improvements. This work provides a versatile, quantile-centered framework for robust network design and risk-sensitive optimization under discrete uncertainty.

Abstract

This paper presents a novel two-stage optimization framework designed to model integrated quantile functions, which leads to the formulation of a bilinear optimization problem (P). A specific instance of this framework offers a new approach to minimizing the Value-at-risk (Var) and the Conditional Value-at-risk (CVar), thus providing a broader perspective on risk assessment and optimization. We investigate various convexification techniques to under- and over-estimate the optimal value of (P), resulting in new and tighter lower- and upper-convex estimators for the Var minimization problems. Furthermore, we explore the properties and implications of the bilinear optimization problem (P) in connection to the integrated quantile functions. Finally, to illustrate the practical applications of our approach, we present computational comparisons in the context of real-life network traffic engineering problems, demonstrating the effectiveness of our proposed framework.

Minimizing risk measures with applications in network traffic engineering

TL;DR

The paper tackles risk-aware decision-making under uncertainty by introducing an integrated quantile function-based measure, , and formulating a two-stage bilinear optimization problem (P) that encompasses VaR and CVaR as special cases. It develops both under- and over-estimation techniques via first-level RLT relaxations and CVaR-based surrogates, plus an alternating-minimization scheme to tighten bounds. The authors prove key theoretical relations to integrated quantile functions and demonstrate the approach on a network traffic engineering case study with multiple failure scenarios, showing tighter estimates and practical performance improvements. This work provides a versatile, quantile-centered framework for robust network design and risk-sensitive optimization under discrete uncertainty.

Abstract

This paper presents a novel two-stage optimization framework designed to model integrated quantile functions, which leads to the formulation of a bilinear optimization problem (P). A specific instance of this framework offers a new approach to minimizing the Value-at-risk (Var) and the Conditional Value-at-risk (CVar), thus providing a broader perspective on risk assessment and optimization. We investigate various convexification techniques to under- and over-estimate the optimal value of (P), resulting in new and tighter lower- and upper-convex estimators for the Var minimization problems. Furthermore, we explore the properties and implications of the bilinear optimization problem (P) in connection to the integrated quantile functions. Finally, to illustrate the practical applications of our approach, we present computational comparisons in the context of real-life network traffic engineering problems, demonstrating the effectiveness of our proposed framework.

Paper Structure

This paper contains 13 sections, 12 theorems, 30 equations, 6 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proposed risk measure: Rationale and design
  3. Interpreting $\alpha - \gamma$ expectation: $\mathbb{E}_{\alpha-\gamma}[\cdot]$
  4. Properties of $\alpha-\gamma$ expectation
  5. Minimizing $\alpha - \gamma$ expectation
  6. Minimizing value-at-risk ($\text{VaR}$)
  7. Underestimation for minimizing $\alpha - \gamma$ expectation
  8. Overestimation for minimizing $\alpha - \gamma$ expectation
  9. Case study: Minimizing value-at-risk
  10. Network traffic engineering problem
  11. Network traffic engineering - Optimization framework
  12. Computational Evaluation on Network traffic engineering
  13. Conclusions

Key Result

Proposition 1

Given $x\in \mathcal{X}$ and probability levels $\alpha, \gamma$ satisfying Assumption assumption:alpha_gamma, for the random variable $\mathbb{T}(x)$ we have $\text{VaR}_{\gamma}\mathbb{T}(x) = \lim_{\alpha \uparrow {\gamma}} \mathbb{E}_{\alpha - \gamma}[\mathbb{T}(x)]$.

Figures (6)

  • Figure 1: Computing $\mathbb{E}_{0-1}[\mathbb{T}(x)]$ via $F^{-1}(\cdot)$
  • Figure 2: Interpreting $\mathbb{E}_{p-p'}[\mathbb{T}(x)]$ via Integrated Quantile Function of $\mathbb{T}(x)$
  • Figure 3: Shaded region: $(\gamma - \alpha)\mathbb{E}_{\alpha-\gamma}[\mathbb{T}(x)]$
  • Figure 4: Proposition \ref{['proposition_Min_var_relation']} using Integrated Quantile Function for $\mathbb{T}(x)$
  • Figure 5: Lemma \ref{['lemma_convex_combin_E']} and Corollary \ref{['lemma_E_monotonic']} using $\text{I}_{\mathbb{T}(x)}(\cdot)$
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Lemma 1
  • Corollary 1
  • Remark 2
  • Lemma 2
  • Proposition 3
  • Corollary 2
  • ...and 7 more