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Robust Optimization of Rank-Dependent Models with Uncertain Probabilities

Guanyu Jin, Roger J. A. Laeven, Dick den Hertog

TL;DR

The paper tackles optimization of rank-dependent risk measures under distributional ambiguity defined by φ-divergences in a discrete setting. It develops reformulations to rank-independent robust counterparts, establishes conic representability for concave distortions, and proposes two convergent algorithms: a cutting-plane method and a piecewise-linear distortion approach. The authors provide explicit epigraph/conjugate representations for canonical distortion/divergence pairs and demonstrate tight bounds on optimal values in robust newsvendor and portfolio problems. This work enables tractable, provably convergent optimization for non-linear-in-probabilities evaluations, with practical implications for inventory and financial decisions under ambiguity.

Abstract

This paper studies distributionally robust optimization for a rich class of risk measures with ambiguity sets defined by $φ$-divergences. The risk measures are allowed to be non-linear in probabilities, are represented by Choquet integrals possibly induced by a probability weighting function, and encompass many well-known examples. Optimization for this class of risk measures is challenging due to their rank-dependent nature. We show that for various shapes of probability weighting functions, including concave, convex and inverse $S$-shaped, the robust optimization problem can be reformulated into a rank-independent problem. In the case of a concave probability weighting function, the problem can be reformulated further into a convex optimization problem that admits explicit conic representability for a collection of canonical examples. While the number of constraints in general scales exponentially with the dimension of the state space, we circumvent this dimensionality curse and develop two types of algorithms. They yield tight upper and lower bounds on the exact optimal value and are formally shown to converge asymptotically. This is illustrated numerically in a robust newsvendor problem and a robust portfolio choice problem.

Robust Optimization of Rank-Dependent Models with Uncertain Probabilities

TL;DR

The paper tackles optimization of rank-dependent risk measures under distributional ambiguity defined by φ-divergences in a discrete setting. It develops reformulations to rank-independent robust counterparts, establishes conic representability for concave distortions, and proposes two convergent algorithms: a cutting-plane method and a piecewise-linear distortion approach. The authors provide explicit epigraph/conjugate representations for canonical distortion/divergence pairs and demonstrate tight bounds on optimal values in robust newsvendor and portfolio problems. This work enables tractable, provably convergent optimization for non-linear-in-probabilities evaluations, with practical implications for inventory and financial decisions under ambiguity.

Abstract

This paper studies distributionally robust optimization for a rich class of risk measures with ambiguity sets defined by -divergences. The risk measures are allowed to be non-linear in probabilities, are represented by Choquet integrals possibly induced by a probability weighting function, and encompass many well-known examples. Optimization for this class of risk measures is challenging due to their rank-dependent nature. We show that for various shapes of probability weighting functions, including concave, convex and inverse -shaped, the robust optimization problem can be reformulated into a rank-independent problem. In the case of a concave probability weighting function, the problem can be reformulated further into a convex optimization problem that admits explicit conic representability for a collection of canonical examples. While the number of constraints in general scales exponentially with the dimension of the state space, we circumvent this dimensionality curse and develop two types of algorithms. They yield tight upper and lower bounds on the exact optimal value and are formally shown to converge asymptotically. This is illustrated numerically in a robust newsvendor problem and a robust portfolio choice problem.

Paper Structure

This paper contains 38 sections, 25 theorems, 201 equations, 5 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Setup and Notation
  4. Rank-Dependent Evaluation
  5. TEXT-Divergence Ambiguity Sets
  6. Problem Formulations, Terminology and Assumptions
  7. Further Notation
  8. Robust Counterpart of Rank-Dependent Models
  9. Reformulation into the Robust Counterpart
  10. Conic Representability of the Robust Counterpart
  11. Solving the Robust Problem I: A Cutting-Plane Method
  12. The Cutting-Plane Algorithm
  13. Robust Optimization with a Rank-Dependent Evaluation in the Constraint
  14. Choquet Expected Utility
  15. Solving the Robust Problem II: Piecewise-Linear Approximation
  16. ...and 23 more sections

Key Result

Theorem 1

Let $h:[0,1]\to [0,1]$ be a concave distortion function. Then, for all $(\mathbf{a},c)\in \mathbb{R}^{n_a+1}$, we have that rob_rho_constr is satisfied if and only if where $\mathcal{U}_{\phi,h}(\mathbf{p})$ is a convex composite uncertainty set given by As a special case of interest, nom_rho_constr is satisfied if and only if where $M_h(\mathbf{p})$ is the set induced by $h$:

Figures (5)

  • Figure 1: Single-item newsvendor problem. This figure displays the worst-case evaluation $\mathrm{WC}(\alpha_0)\triangleq\sup_{\mathbf{q}\in \mathcal{D}_{\phi}(\mathbf{p},r(n))}\mathrm{CVaR}_{1-\alpha_0}(.)$ under the robust and nominal solutions, for a range of values of $r(n) = \chi^2_{2,0.95}/(2n)$ and $\alpha_0= 0.4, 0.3, 0.2, 0.1$.
  • Figure 2: Single-item newsvendor problem (continued): This figure displays, for each sampled probability vector $\mathbf{q}$ from the KL-divergence uncertainty set for $n=50$, the corresponding $\mathrm{CVaR}_{1-\alpha_0}(.)$ evaluation with $\alpha_0=0.4$ of the robust and nominal solutions.
  • Figure 3: Multi-item newsvendor problem. This figure displays the worst-case evaluation $\mathrm{WC}(\alpha_0)\triangleq\sup_{\mathbf{q}\in \mathcal{D}_{\phi}(\mathbf{p},r(n))}\mathrm{CVaR}_{1-\alpha_0}(.)$ under the robust and nominal solutions, for a range of values of $r(n) = \chi^2_{2,0.95}/(2n)$ and $\alpha_0=0.9, 0.8, 0.4, 0.3$.
  • Figure 4: Prelec's distortion. This figure displays Prelec's distortion function \ref{['Prelec']} and its upper and lower piecewise-linear approximations (dashed) for $\alpha=0.6,\, 0.75$. The approximation error is set to $\epsilon = 0.003$.
  • Figure 5: Projections of the uncertainty set $\mathcal{U}_{\phi,h}(\mathbf{p})$ in \ref{['uncertaintyset']} on the coordinates $(\bar{q}_1,\bar{q}_2)$, for $\mathbf{p}=(1/3,1/3,1/3)$ and $r=\frac{1}{n}\chi^2_{0.95,2}$, plotted for a range of values of the sample size $n$. We choose the modified chi-squared divergence function $\phi(t)=(t-1)^2$ and the second dual moment distortion function $h(p)=1-(1-p)^2$. As $n$ approaches $0$, we observe that the uncertainty set grows and the projection eventually approaches the entire probability simplex in $\mathbb{R}^2$, the case in which the decision-maker is completely ambiguous w.r.t. $\mathbf{p}$.

Theorems & Definitions (58)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Example 1
  • Lemma 3
  • Theorem 3
  • ...and 48 more