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Splitting Algorithms for Distributionally Robust Optimization

Luis Briceño-Arias, Sergio López-Rivera, Emilio Vilches

Abstract

In this paper, we provide different splitting methods for solving distributionally robust optimization problems in cases where the uncertainties are described by discrete distributions. The first method involves computing the proximity operator of the supremum function that appears in the optimization problem. The second method solves an equivalent monotone inclusion formulation derived from the first-order optimality conditions, where the resolvents of the monotone operators involved in the inclusion are computable. The proposed methods are applied to solve the Couette inverse problem with uncertainty and the denoising problem with uncertainty. We present numerical results to compare the efficiency of the algorithms.

Splitting Algorithms for Distributionally Robust Optimization

Abstract

In this paper, we provide different splitting methods for solving distributionally robust optimization problems in cases where the uncertainties are described by discrete distributions. The first method involves computing the proximity operator of the supremum function that appears in the optimization problem. The second method solves an equivalent monotone inclusion formulation derived from the first-order optimality conditions, where the resolvents of the monotone operators involved in the inclusion are computable. The proposed methods are applied to solve the Couette inverse problem with uncertainty and the denoising problem with uncertainty. We present numerical results to compare the efficiency of the algorithms.

Paper Structure

This paper contains 14 sections, 13 theorems, 112 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Notation:
  3. Proximity of a supremum function
  4. Algorithms
  5. A proximal algorithm
  6. Distributed forward-backward method
  7. Forward-backward with subspaces
  8. Davis-Yin's Formulation
  9. Dual method
  10. Applications
  11. Couette inverse problem
  12. Denoising
  13. Numerical experiments
  14. Conclusions

Key Result

Proposition 2.1

Let $\{f_{i}\}_{i=1}^{N}\subset \Gamma_{0}(\mathcal{H})$ and let $\mathcal{P}\subset\Delta_{N}$ be a nonempty closed convex set. Let $\mathbf{x}\in\mathcal{H}^{N}$ and $\lambda>0$. Then the following hold.

Theorems & Definitions (24)

  • Remark 1.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 14 more