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Distributionally Robust Optimization: A review on theory and applications

  • * Corresponding author: Zheming Gao

    * Corresponding author: Zheming Gao
Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • 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.


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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 $
     | Show Table
    DownLoad: CSV

    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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