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2022, Volume 12, Issue 1: 159-212. Doi: 10.3934/naco.2021057
This issue Previous Article An active set solver for constrained $ H_\infty $ optimal control problems with state and input constraints Next Article

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

    * Corresponding author: Zheming Gao
Received:  February 2021
Revised:  June 2021
Early access:  November 2021
Published:  March 2022
Abstract / Introduction Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
    Abstract

  • 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.

    Mathematics Subject Classification: Primary: 90-02, 90C17, 90C47; Secondary: 90C90.

    Citation:
    shu

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  • Figure 1.  Total publications per year for DRO references

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    Figure 2.  Sum of times cited per year for DRO references

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    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 $
     | Show Table
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    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
     | Show Table
    DownLoad: CSV
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