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Globalized distributionally robust optimization with multi core sets

Yueyao Li, Chenglong Bao, Wenxun Xing

TL;DR

A globalized distributionally robust optimization framework with multiple core sets (MGDRO) to handle the complicated situation when the uncertain data is multimodal, and results turn out that the MGDRO models greatly outperform traditional DRO models and other multimodal models.

Abstract

It is essential to capture the true probability distribution of uncertain data in the distributionally robust optimization (DRO). The uncertain data presents multimodality in numerous application scenarios, in the sense that the probability density function of the uncertain data has two or more modes (local maximums). In this paper, we propose a globalized distributionally robust optimization framework with multiple core sets (MGDRO) to handle the multimodal data. This framework captures the multimodal structure via a penalty function composed of the minimum distances from the random vector to all core sets. Under some assumptions, the MGDRO model can be reformulated as tractable semi-definite programs for both moment-based and metric-based ambiguity sets. We applied the MGDRO models to a multi-product newswendor problem with multimodal demands. The numerical results turn out that the MGDRO models outperform traditional DRO models and other multimodal models greatly.

Globalized distributionally robust optimization with multi core sets

TL;DR

A globalized distributionally robust optimization framework with multiple core sets (MGDRO) to handle the complicated situation when the uncertain data is multimodal, and results turn out that the MGDRO models greatly outperform traditional DRO models and other multimodal models.

Abstract

It is essential to capture the true probability distribution of uncertain data in the distributionally robust optimization (DRO). The uncertain data presents multimodality in numerous application scenarios, in the sense that the probability density function of the uncertain data has two or more modes (local maximums). In this paper, we propose a globalized distributionally robust optimization framework with multiple core sets (MGDRO) to handle the multimodal data. This framework captures the multimodal structure via a penalty function composed of the minimum distances from the random vector to all core sets. Under some assumptions, the MGDRO model can be reformulated as tractable semi-definite programs for both moment-based and metric-based ambiguity sets. We applied the MGDRO models to a multi-product newswendor problem with multimodal demands. The numerical results turn out that the MGDRO models outperform traditional DRO models and other multimodal models greatly.
Paper Structure (13 sections, 4 theorems, 41 equations, 1 figure, 6 tables)

This paper contains 13 sections, 4 theorems, 41 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. MGDRO models and reformulations
  3. Introduction to MGDRO models
  4. Reformulations of MGDRO models
  5. MGDRO model with moment-based ambiguity set
  6. MGDRO model with metric-based ambiguity set
  7. Application in multi-product newsvendor problem
  8. Multi-product newsvendor problem
  9. Numerical experiments
  10. Comparison of different models with moment-based ambiguity set
  11. Comparison between moment-based and metric-based ambiguity set
  12. Comparison with other multimodal ambiguity set
  13. Conclusions

Key Result

Lemma 2.1

Let $f(\cdot,\bm{x})$ be a closed proper concave function in $\mathbb{R}^p$ for all $\bm{x} \in \mathbb{R}^n$ and $\phi:\mathbb{R}^p \times \mathbb{R}^p \to \mathbb{R}$ be a closed, jointly convex, and nonnegative function for which $\phi(\bm{\xi},\bm{\xi})=0$ for all $\bm{\xi} \in \mathbb{R}^p$. Le if and only if there exist $\bm{v},\bm{w} \in \mathbb{R}^p$ satisfying the single inequality

Figures (1)

  • Figure 1: Scatter plot of sample points with multimodal distribution

Theorems & Definitions (6)

  • Lemma 2.1: Ben-Tal et al. gro Theorem 1
  • Lemma 2.2: Delage and Ye dro:ye Lemma 1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof