March  2022, 12(1): 159-212. doi: 10.3934/naco.2021057

Distributionally Robust Optimization: A review on theory and applications

1. 

Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, USA

2. 

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning 110819, China

* Corresponding author: Zheming Gao

Received  February 2021 Revised  June 2021 Published  March 2022 Early access  November 2021

In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.

Citation: Fengming Lin, Xiaolei Fang, Zheming Gao. Distributionally Robust Optimization: A review on theory and applications. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 159-212. doi: 10.3934/naco.2021057
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Figure 1.  Total publications per year for DRO references
Figure 2.  Sum of times cited per year for DRO references
Table 1.  Examples of $ \phi $-functions, their adjoints and $ \phi $-divergence [4]
Divergence $ \phi(t) $ $ \tilde{\phi}(t) $ $ \phi(t), t\geq 0 $ $ D_{\phi}(P_1 || P_2) $
Kullback-Leibler $ \phi_{KL} $ $ \phi_{B} $ $ t\log t -t+1 $ $ \int_{\Xi}\log(\frac{dP_1}{dP_2})dP_1 $
Burg entropy $ \phi_{B} $ $ \phi_{KL} $ $ -\log t +t-1 $ $ \int_{\Xi}\log(\frac{dP_2}{dP_1})dP_2 $
$ J $-divergence $ \phi_{J} $ $ \phi_{J} $ $ (t-1) \log t $ $ \int_{\Xi}\log(\frac{dP_1}{dP_2})(dP_1-dP_2) $
$ \chi^2 $-divergence $ \phi_{\chi^2} $ $ \phi_{M\chi^2} $ $ \frac{1}{t} (t-1)^2 $ $ \int_{\Xi}\frac{(dP_1-dP_2)^2}{dP_1} $
Modified $ \chi^2 $-divergence $ \phi_{M\chi^2} $ $ \phi_{\chi^2} $ $ (t-1)^2 $ $ \int_{\Xi}\frac{(dP_1-dP_2)^2}{dP_2} $
Variation distance $ \phi_{V} $ $ \phi_{V} $ $ |t-1| $ $ \int_{\Xi} |dP_1-dP_2| $
Hellinger distance $ \phi_{H} $ $ \phi_{H} $ $ (\sqrt{t}-1)^2 $ $ \int_{\Xi}(\sqrt{dP_1}-\sqrt{dP_2})^2 $
Divergence $ \phi(t) $ $ \tilde{\phi}(t) $ $ \phi(t), t\geq 0 $ $ D_{\phi}(P_1 || P_2) $
Kullback-Leibler $ \phi_{KL} $ $ \phi_{B} $ $ t\log t -t+1 $ $ \int_{\Xi}\log(\frac{dP_1}{dP_2})dP_1 $
Burg entropy $ \phi_{B} $ $ \phi_{KL} $ $ -\log t +t-1 $ $ \int_{\Xi}\log(\frac{dP_2}{dP_1})dP_2 $
$ J $-divergence $ \phi_{J} $ $ \phi_{J} $ $ (t-1) \log t $ $ \int_{\Xi}\log(\frac{dP_1}{dP_2})(dP_1-dP_2) $
$ \chi^2 $-divergence $ \phi_{\chi^2} $ $ \phi_{M\chi^2} $ $ \frac{1}{t} (t-1)^2 $ $ \int_{\Xi}\frac{(dP_1-dP_2)^2}{dP_1} $
Modified $ \chi^2 $-divergence $ \phi_{M\chi^2} $ $ \phi_{\chi^2} $ $ (t-1)^2 $ $ \int_{\Xi}\frac{(dP_1-dP_2)^2}{dP_2} $
Variation distance $ \phi_{V} $ $ \phi_{V} $ $ |t-1| $ $ \int_{\Xi} |dP_1-dP_2| $
Hellinger distance $ \phi_{H} $ $ \phi_{H} $ $ (\sqrt{t}-1)^2 $ $ \int_{\Xi}(\sqrt{dP_1}-\sqrt{dP_2})^2 $
Table 2.  Examples of $ \phi $-divergence, their conjugates and DRO counterparts [4,138]
Divergence $ \phi^*(s) $ DRO Counterpart
Kullback-Leibler $ e^s-1 $ Convex program
Burg entropy $ -\log (1-s), s< 1 $ Convex program
$ J $-divergence No closed form Convex program
$ \chi^2 $-divergence $ 2-2\sqrt{1-s}, s<1 $ SOCP
Modified $ \chi^2 $-divergence $ \left\{ \begin{array}{ll} -1, & s<-2\\s+\frac{s^2}{4},&s \geq -2 \end{array}\right. $ SOCP
Variation distance $ \left\{ \begin{array}{ll} -1,&s\leq-1 \\s,&-1\leq s \leq 1 \end{array}\right. $ LP
Hellinger distance $ \frac{s}{s-1}, s<1 $ SOCP
Divergence $ \phi^*(s) $ DRO Counterpart
Kullback-Leibler $ e^s-1 $ Convex program
Burg entropy $ -\log (1-s), s< 1 $ Convex program
$ J $-divergence No closed form Convex program
$ \chi^2 $-divergence $ 2-2\sqrt{1-s}, s<1 $ SOCP
Modified $ \chi^2 $-divergence $ \left\{ \begin{array}{ll} -1, & s<-2\\s+\frac{s^2}{4},&s \geq -2 \end{array}\right. $ SOCP
Variation distance $ \left\{ \begin{array}{ll} -1,&s\leq-1 \\s,&-1\leq s \leq 1 \end{array}\right. $ LP
Hellinger distance $ \frac{s}{s-1}, s<1 $ SOCP
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