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Nonlinear Distributionally Robust Optimization

Mohammed Rayyan Sheriff, Peyman Mohajerin Esfahani

TL;DR

A G-derivative based Frank–Wolfe (FW) algorithm is proposed for generic nonlinear optimization problems in probability spaces and its convergence under the proposed notion of smoothness in a completely norm-independent manner is established.

Abstract

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative-based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the $entropic$ and $variance$ risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and "sufficient statistic", compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.

Nonlinear Distributionally Robust Optimization

TL;DR

A G-derivative based Frank–Wolfe (FW) algorithm is proposed for generic nonlinear optimization problems in probability spaces and its convergence under the proposed notion of smoothness in a completely norm-independent manner is established.

Abstract

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative-based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the and risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and "sufficient statistic", compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.
Paper Structure (52 sections, 20 theorems, 159 equations, 7 figures, 1 algorithm)

This paper contains 52 sections, 20 theorems, 159 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Distributionally Robust Optimization (DRO)
  3. Frank-Wolfe (FW) algorithm
  4. Contributions
  5. Worst-case Nonlinear Risk Measures
  6. Derivatives of Risk Measures
  7. Gateaux directional derivatives
  8. Smoothness
  9. Connection to the existing notions of smoothness
  10. Concave quadratic lower bound via smoothness
  11. Frechet derivatives
  12. Smoothness with F-derivatives
  13. Comparison with G-derivatives
  14. Challenges with F-derivatives
  15. G-derivatives in non-flat spaces
  16. ...and 37 more sections

Key Result

Lemma 3.2

Suppose that the risk measure is regular, i.e., $R ( \mathds{P} ) = r (\mathds{E}_{\mathds{P}}[L(\xi)] )$ with $\nabla r$ denoting the gradient of function $r$. Then, for any $\mathds{P}, \mathds{Q} \in \mathcal{P}$, we have

Figures (7)

  • Figure 1: Pictorial representation of the risk surface and its directional derivative.
  • Figure 2: Concave-quadratic lower bound via smoothness.
  • Figure 3: FW-oracle.
  • Figure 4: Convergence plots for Algorithm \ref{['algo:NDRO-min-max']} with $K(\varepsilon) = 350$, applied to \ref{['eq:er-risk-portfolio-selection']}.
  • Figure 5: Convergence plots for Algorithm \ref{['algo:NDRO-min-max']} with $K(\varepsilon) = 350$, applied to \ref{['eq:er-risk-simulation-problem-with-reg']}.
  • ...and 2 more figures

Theorems & Definitions (54)

  • Definition 2.1: Regular risk (RR) measures & sufficient statistic
  • Example 2.2: RR-examples
  • Definition 3.1: G-derivative
  • Lemma 3.2: Regular G-derivatives
  • Example 3.3: Regular G-derivatives
  • Proposition 3.4: G-derivative: properties
  • Remark 3.5: Optimality conditions
  • Definition 3.6: G-smoothness
  • Lemma 3.7: Regular G-smoothness
  • Example 3.8: Regular G-smoothness
  • ...and 44 more