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Distributionally robust multi-item newsvendor problems with multimodal demand distributions

Abstract

We present a risk-averse multi-dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed. The demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description. We assume that the newsvendor hedges against distributional ambiguity by minimizing the worst-case risk of the order portfolio over all distributions that are compatible with the given modality information. We demonstrate that the resulting distributionally robust optimization problem is \(\mathrm{NP}\)-hard but admits an efficient numerical solution in quadratic decision rules. This approximation is conservative and computationally tractable. Moreover, it achieves a high level of accuracy in numerical tests. We further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments.

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References

  1. 1.

    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Financ. 9(3), 203–228 (1999)

  2. 2.

    Bandi, C., Bertsimas, D.: Tractable stochastic analysis in high dimensions via robust optimization. Math. Program. 134(1), 23–70 (2012)

  3. 3.

    Bartezzaghi, E., Verganti, R., Zotteri, G.: Measuring the impact of asymmetric demand distributions on inventories. Int. J. Prod. Econ. 60–61, 395–404 (1999)

  4. 4.

    Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Manag. Sci. 59(2), 341–357 (2012)

  5. 5.

    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

  6. 6.

    Bertsimas, D., Doan, X.V., Natarajan, K., Teo, C.-P.: Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35(3), 580–602 (2010)

  7. 7.

    Bertsimas, D., Iancu, D., Parrilo, P.: A hierarchy of near-optimal policies for multistage adaptive optimization. IEEE Trans. Automat. Contr. 56(12), 2809–2824 (2011)

  8. 8.

    Choi, S., Ruszczyński, A., Zhao, Y.: A multiproduct risk-averse newsvendor with law-invariant coherent measures of risk. Oper. Res. 59(2), 346–364 (2011)

  9. 9.

    Delage, E.: Distributionally Robust Optimization in Context of Data-driven Problems. PhD thesis, Stanford University, Palo Alto, CA (2009)

  10. 10.

    Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58, 595–612 (2010)

  11. 11.

    Dupačová, J.: Stress testing via contamination. In: Marti, K., et al. (eds.) Coping with Uncertainty: Modeling and Policy Issues, Lecture Notes in Economics and Mathematical Systems, vol. 58, pp. 29–46. Springer, Berlin (2006)

  12. 12.

    Dupačová, J., Kopa, M.: Robustness in stochastic programs with risk constraints. Ann. Oper. Res. 200(1), 55–74 (2012)

  13. 13.

    Dupačová, J.: The minimax approach to stochastic programming and an illustrative approach. Stochastics 20(1), 73–88 (1987)

  14. 14.

    Dupačová, J. (as Žáčková): On minimax solutions of stochastic linear programming problems. Časopis pro pěstování matematiky 91(4), 423–430 (1966)

  15. 15.

    Ellsberg, D.: Risk, ambiguity, and the Savage axioms. Q. J. Econ. 75(4), 643–669 (1961)

  16. 16.

    Gallego, G., Moon, I.: The distribution free newsboy problem: review and extensions. J. Oper. Res. Soc. 44(8), 825–834 (1993)

  17. 17.

    Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18(2), 141–153 (1989)

  18. 18.

    Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4), 902–917 (2010)

  19. 19.

    Gorissen, B.L., den Hertog, D.: Robust counterparts of inequalities containing sums of maxima of linear functions. Eur. J. Oper. Res. 227(1), 30–43 (2012)

  20. 20.

    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)

  21. 21.

    Hadley, G., Whitin, T.M.: Analysis of Inventory Systems. Prentice-Hall International Series in Management. Prentice-Hall, Englewood Cliffs, NJ (1963)

  22. 22.

    Isii, K.: The extrema of probability determined by generalized moments (i) bounded random variables. Ann. Inst. Stat. Math. 12(2), 119–134 (1960)

  23. 23.

    Kök, A.G., Fisher, M.: Demand estimation and assortment optimization under substitution: methodology and application. Oper. Res. 55(6), 1001–1021 (2007)

  24. 24.

    G. Kök, A., Fisher, M., Vaidyanathan, R.: Assortment planning: review of literature and industry practice. In: Agrawal, B., Smith, S.A. (eds.) Retail Supply Chain Management, pp. 99–154. Springer, Berlin (2009)

  25. 25.

    Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference (2004)

  26. 26.

    Mahajan, S., van Ryzin, G.: Retail inventories and consumer choice. In: Tayur, S., Ganesham, R., Magasine, M. (eds.) Quantitative Methods in Supply Chain Management, pp. 201–209. Kluwer Publishers, Amsterdam (1998)

  27. 27.

    Natarajan, K., Sim, M., Uichanco, J.: Asymmetry and ambiguity in newsvendor models. Technical report, National University of Singapore (2008)

  28. 28.

    Natarajan, K., Sim, M., Uichanco, J.: Tractable robust expected utility and risk models for portfolio optimization. Math. Financ. 20(4), 695–731 (2010)

  29. 29.

    Ogryczak, W., Ruszczyński, A.: Dual stochastic dominance and related mean-risk models. SIAM J. Optim. 13(1), 60–78 (2002)

  30. 30.

    Pardo, L.: Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. CRC Press, Taylor & Francis Group, Boca Raton (2006)

  31. 31.

    Perakis, G., Roels, G.: Regret in the newsvendor model with partial information. Oper. Res. 56(1), 188–203 (2008)

  32. 32.

    Pólik, I., Terlaky, T.: A survey of the S-lemma. SIAM Rev. 49(3), 371–418 (2007)

  33. 33.

    Popescu, I.: A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Math. Oper. Res. 30(3), 632–657 (2005)

  34. 34.

    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2002)

  35. 35.

    Scarf, H.E.: A min-max solution to an inventory problem. In: Arrow, K.J., Karlin, S., Scarf, H.E. (eds.) Studies in Mathematical Theory of Inventory and Production, pp. 201–209. Stanford University Press, Stanford (1958)

  36. 36.

    Shapiro, A.: On duality theory of conic linear problems. In: Goberna, M.A., Lopez, M.A. (eds.) Semi-Infinite Programming: Recent Advances, pp. 135–165. Kluwer Academic Publishers, Dordrecht (2001)

  37. 37.

    Shapiro, A., Ahmed, S.: On a class of minimax stochastic programs. SIAM J. Optim. 14(1), 1237–1249 (2004)

  38. 38.

    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)

  39. 39.

    Shapiro, A., Kleywegt, A.J.: Minimax analysis of stochastic problems. Optim. Methods Softw. 17(3), 523–542 (2002)

  40. 40.

    Signorelly, S., Heskett, J.L.: Benetton. In: Harvard Business School Case # 685020 (1985)

  41. 41.

    Steinberg, D.: Computation of Matrix Norms with Applications to Robust Optimization. Master’s thesis, Technion-Israel Institute of Technology (2005)

  42. 42.

    Sun, P., Freund, R.M.: Computation of minimum-volume covering ellipsoids. Oper. Res. 52(5), 690–706 (2004)

  43. 43.

    Tallis, G.M.: Elliptical and radial truncation in normal populations. Ann. Math. Stat. 34(3), 940–944 (1963)

  44. 44.

    Vaagen, H., Wallace, S.W.: Product variety arising from hedging in the fashion supply chains. Int. J. Prod. Econ. 114(2), 431–455 (2008)

  45. 45.

    Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49(1), 580–595 (2007)

  46. 46.

    Wang, Z., Glynn, P.W., Ye, Y.: Likelihood robust optimization for data-driven newsvendor problems. Technical report, Stanford University (2009)

  47. 47.

    Wiesemann, W., Kuhn, D., Sim, M.: Distributionally robust convex optimization. Working paper (2013)

  48. 48.

    Zhu, S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(6), 1155–1168 (2009)

  49. 49.

    Zuluaga, L., Peña, J.: A conic programming approach to generalized Tchebycheff inequalities. Math. Oper. Res. 30(2), 369–388 (2005)

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Acknowledgments

We are indebted to Karthik Natarajan for valuable discussions on the topic of this paper. We also thank two anonymous reviewers whose comments led to substantial improvements of this paper. This research was supported by EPSRC under Grant EP/I014640/1.

Author information

Correspondence to Grani A. Hanasusanto.

Appendix: Extremal distributions

Appendix: Extremal distributions

In order to facilitate targeted stress tests for different order portfolios in Sect. 5, we describe here a method to construct extremal distributions for the worst-case CVaR problem (8). The general recipe is to first solve the SDP (18) and to substitute the resulting optimal \(\beta ^*\) into (17). An extremal distribution that achieves the supremum in (8) can then be inferred from the dual solution of the SDP (17) by adapting a procedure for generic worst-case expectation problems with piecewise linear objective functions and without support information; see [6, 10]. We extend this procedure so that it can handle support information. Our construction of the extremal distribution will rely on the following corollary of [45, Section 2.1].

Proposition 6.1

Let \(\Xi =\{\varvec{\xi }\in \mathbb R^n:[\varvec{\xi }^\intercal \;1]\mathbf W[\varvec{\xi }^\intercal \;1]^\intercal \le 0 \}\) be the ellipsoid specified by the matrix

$$\begin{aligned} \mathbf W= \left[ \begin{array}{c@{\quad }c} \varvec{\Lambda }^{-1} &{} -\varvec{\Lambda }^{-1}\varvec{\nu } \\ -(\varvec{\Lambda }^{-1}\varvec{\nu })^\intercal &{} \varvec{\nu }^\intercal \varvec{\Lambda }^{-1}\varvec{\nu }-\delta ^2 \end{array}\right] \end{aligned}$$

with \(\varvec{\Lambda } \in \mathbb S^{n}\), \(\varvec{\Lambda }\succ \varvec{0}\), \(\varvec{\nu }\in \mathbb R^n\) and \(\delta \in \mathbb R\). If

$$\begin{aligned} \left[ \begin{array}{l@{\quad }l} \mathbf Z &{} \varvec{z} \\ \varvec{z}^\intercal &{} 1 \end{array}\right] \succcurlyeq \varvec{0}\quad \text {and}\quad \left<\mathbf W, \left[ \begin{array}{l@{\quad }l} \mathbf Z &{} \varvec{z} \\ \varvec{z}^\intercal &{} 1 \end{array}\right] \right>\le 0\,, \end{aligned}$$

then one can construct a discrete distribution supported on \(\Xi \) with mean \(\varvec{z}\) and covariance matrix \(\mathbf Z-\varvec{z}\varvec{z}^\intercal \).

Proof

This proposition strenghtens a result proved in [45, Section 2.1] in the special case when \(\Xi \) is an ellipsoid. The covariance matrix of the discrete distribution constructed in [45] is only upper bounded (in a positive semidefinite sense) by but not necessarily equal to \(\mathbf Z-\varvec{z}\varvec{z}^\intercal \).

By using an eigenvalue decomposition, we can factorize \(\mathbf Z-\varvec{z}\varvec{z}^\intercal \) as \(\sum _{i=1}^n\varvec{w}_i\varvec{w}_i^\intercal \). Thus, we have

$$\begin{aligned} \left\langle \mathbf {W}, \left[ \begin{array}{l@{\quad }l} \mathbf {Z} &{} \varvec{z}\\ \varvec{z}^\intercal &{} 1 \end{array}\right] \right\rangle \le 0 \!\!\iff \!\! \varvec{z}^\intercal \varvec{\Lambda }^{-1}\varvec{z} \!+\! \sum _{i=1}^n\varvec{w}_i^\intercal \varvec{\Lambda }^{-1}\varvec{w}_i \!-\! 2\varvec{\nu }\varvec{\Lambda }^{-1}\varvec{z} \!+\! \varvec{\nu }^\intercal \varvec{\Lambda }^{-1}\varvec{\nu } \!-\! \delta ^2 \le 0\,.\nonumber \\ \end{aligned}$$
(33)

Assume first that \(\varvec{w}_i=\varvec{0}\) for all \(i=1,\dots ,n\). Then, (33) implies that \(\varvec{z}\in \Xi \). The Dirac distribution that concentrates unit mass at \(\varvec{z}\) is thus supported on \(\Xi \), and it has mean \(\varvec{z}\) and covariance matrix \(\mathbf Z-\varvec{z}\varvec{z}^\intercal =\varvec{0}\). Assume now that \(\varvec{w}_{i}\ne \varvec{0}\) for all \(i\in \mathcal {N}\), where \(\mathcal {N}\) is a nonempty subset of \(\{1,\dots ,n\}\). As \({\varvec{\Lambda }}^{-1}\succ \varvec{0}\), we have \(\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i> 0\) if \(i\in {\mathcal {N}}\); \(=0\) otherwise. Next, we define the constants

$$\begin{aligned}&s =-(\varvec{z}^\intercal {\varvec{\Lambda }}^{-1}\varvec{z} + \sum _{i\in {\mathcal {N}}}\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i - 2\varvec{\nu }{\varvec{\Lambda }}^{-1}\varvec{z} + \varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2) \ge 0\,,\\&t =-\sum _{i\in {\mathcal {N}}}\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i < 0\,. \end{aligned}$$

For each \(i\in {\mathcal {N}}\) we construct the following univariate quadratic function in \(\omega \).

$$\begin{aligned} f_i(\omega )&= \left\langle \mathbf {W}, \left[ \begin{array}{c@{\quad }c} (\varvec{z}+\omega \varvec{w}_i)(\varvec{z}+\omega \varvec{w}_i)^\intercal &{} \varvec{z}+\omega \varvec{w}_i \\ (\varvec{z}+\omega \varvec{w}_i)^\intercal &{} 1 \end{array}\right] \right\rangle + s \\&= (\varvec{z} +\omega \varvec{w}_i)^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z} +\omega \varvec{w}_i) - 2\varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}+\omega \varvec{w}_i) + \varvec{\nu }^{\intercal } {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2 + s\\&= \varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i\omega ^2+ 2\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}\!-\!\varvec{\nu })\omega \!+\!\varvec{z}^\intercal {\varvec{\Lambda }}^{-1}\varvec{z} \!-\! 2{\varvec{\nu }}{\varvec{\Lambda }}^{-1}\varvec{z} \!+\!\varvec{\nu }^\intercal {\varvec{\Lambda }}^{-1}\varvec{\nu } - \delta ^2 + s\\&= \varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i\omega ^2+ 2\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}(\varvec{z}-\varvec{\nu })\omega +t \end{aligned}$$

Since \(\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i>0\) and \(t<0\), the quadratic function \(f_i(\omega )\) has two real roots \(\omega _i>0\) and \(\omega _{i+n}<0\). By construction, as \(f_i(\omega _i)=f_i(\omega _{i+n})=0\) and \(s\ge 0\), the points \(\varvec{z}+\omega _i\varvec{w}_i\) and \(\varvec{z}+\omega _{i+n}\varvec{w}_i\) are contained in \(\Xi \). This allows us to define a discrete probability distribution \(\mathbb P\) supported on \(\Xi \) with \(2|{\mathcal {N}}|\) atoms at \(\varvec{z}+\omega _i\varvec{w}_i\) and \(\varvec{z}+\omega _{i+n}\varvec{w}_{i}\), \(i\in {\mathcal {N}}\), and with

$$\begin{aligned} \left. \begin{array}{ll} &{}\mathbb P(\{\varvec{z}+\omega _i\varvec{w}_i\}) \!=\! \kappa _i \!=\! \dfrac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1-\omega _i/\omega _{i+n})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}\\ &{}\mathbb {P}(\{\varvec{z}+\omega _{i+n}\varvec{w}_{i}\})\!=\! \kappa _{i+n} \!=\!-\kappa _i\frac{\omega _i}{\omega _{i+n}}\!=\! \dfrac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1\!-\!\omega _{i+n}/\omega _{i})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}} \end{array}\right\} ~\forall i\in {\mathcal {N}}. \end{aligned}$$

Since \(\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i>0\) and \(\omega _i/\omega _{i+n}<0\), we have \(\kappa _i>0\) for all \(i\in {\mathcal {N}}\). Next, observe that

$$\begin{aligned} \sum \limits _{i\in {\mathcal {N}}} \kappa _i+\kappa _{i+n}=\sum _{i\in {\mathcal {N}}}\frac{[(1-\omega _{i+n}/\omega _{i}) + (1-\omega _i/\omega _{i+n})]\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_i}{(1-\omega _i/\omega _{i+n})(1-\omega _{i+n}/\omega _{i})\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}=1\,. \end{aligned}$$

Thus \(\mathbb {P}\) constitutes a valid discrete probability distribution. Furthermore, the first- and second-order moments of \(\mathbb {P}\) can be expressed as

$$\begin{aligned} \mathbb {E}_\mathbb {P}[\tilde{\varvec{\xi }}]&= \sum _{i\in {\mathcal {N}}}\kappa _i(\varvec{z}\!+\!\omega _i\varvec{w}_i)+\kappa _{i+n}(\varvec{z}+\omega _{i+n}\varvec{w}_i)=\varvec{z} + \sum _{i\in {\mathcal {N}}}(\kappa _i\omega _i+\kappa _{i+n}\omega _{i+n})\varvec{w}_i\,,\\ \mathbb {E}_\mathbb {P}[\tilde{\varvec{\xi }}\tilde{\varvec{\xi }}^\intercal ]&= \varvec{z} \varvec{z}^\intercal \!+\!\sum _{i\in {\mathcal {N}}}(\kappa _i\omega _i\!+\!\kappa _{i+n}\omega _{i+n})(\varvec{z}\varvec{w}_i^\intercal \!+\!\varvec{w}_i\varvec{z}^\intercal ) \!+\!\sum _{i\!\in \!{\mathcal {N}}} (\kappa _i\omega _i^2\!+\!\kappa _{i+n}\omega _{i+n}^2)\varvec{w}_i\varvec{w}_i^\intercal . \end{aligned}$$

For all \(i\in {\mathcal {N}}\) we have

$$\begin{aligned} \kappa _i\omega _i+\kappa _{i+n}\omega _{i+n}=\kappa _i\omega _i-\kappa _i(\omega _i/\omega _{i+n})\omega _{i+n}=0 \end{aligned}$$

and

$$\begin{aligned} \kappa _i\omega _i^2+\kappa _{i+n}\omega _{i+n}^2&= \frac{\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_i}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_{i'}}\left( \frac{\omega _i^2\omega _{i+n}}{\omega _{i+n}-\omega _i}+\frac{\omega _i\omega _{i+n}^2}{\omega _{i}-\omega _{i+n}}\right) \\&= \frac{t}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1}\varvec{w}_{i'}}\left( \frac{\omega _i}{\omega _{i+n}-\omega _i}-\frac{\omega _{i+n}}{\omega _{i+n}-\omega _{i}}\right) \\&= \frac{-t}{\sum _{i'\in {\mathcal {N}}}\varvec{w}_{i'}^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_{i'}}=1\,, \end{aligned}$$

where the second equality follows from the relation \(\omega _i\omega _{i+n}=t/(\varvec{w}_i^\intercal {\varvec{\Lambda }}^{-1} \varvec{w}_i)\). Thus, \(\mathbb P\) has all the desired properties, that is, \(\mathbb E_\mathbb P[\tilde{\varvec{\xi }}]=\varvec{z}\), \(\mathbb E_\mathbb P[\tilde{\varvec{\xi }}\tilde{\varvec{\xi }}^\intercal ]=\mathbf Z\) and \(\mathbb P(\tilde{\varvec{\xi }}\in \Xi )=1\).

\(\square \)

To avoid clutter in the subsequent derivation, we introduce the notational shorthands

$$\begin{aligned} \varvec{w}_0 \!=\! \varvec{0},\quad w_0 = 0,\quad \varvec{w}_k \!=\! \varvec{b} - \mathcal {I}_k(\varvec{h}),\quad w_k \!=\! (\varvec{d} \!+\! \mathcal {I}_k(\varvec{h}))^\intercal \varvec{x} - \beta ^* \quad \forall k=1,\dots ,2^n\,. \end{aligned}$$

Using these definitions, we can replace \(\max (0,L(\varvec{x}, \varvec{\xi }) - \beta ^*)\) with \(\max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \varvec{\xi } + w_k\), which facilitates a streamlined reformulation of the SDP (17).

$$\begin{aligned} \begin{array}{l@{\quad }l} \inf &{} \displaystyle \sum \limits _{j=1}^mp_j\langle \varvec{\Omega }_j, \mathbf {M}_j \rangle \\ \mathrm{s.t.}&{} \mathbf M_j \in \mathbb {S}^{n+1},\quad \gamma _{{jk}} \in \mathbb {R}_+ \quad \forall j=1,\dots ,m,~k=0,\dots ,2^n \\ &{} \mathbf {M}_j + \gamma _{{jk}} \mathbf {W}_j \succcurlyeq \left[ \begin{array}{c@{\quad }c} \varvec{0} &{} \frac{1}{2}\varvec{w}_k \\ \frac{1}{2}\varvec{w}_k^\intercal &{} w_k \end{array}\right] \;~\begin{array}{l} \forall j=1,\dots ,m\\ \forall k=0,\dots ,2^n \end{array} \end{array} \end{aligned}$$

The dual of the above problem is given by

$$\begin{aligned} \begin{array}{l@{\quad }l} \sup &{} \displaystyle \sum \limits _{j=1}^m\displaystyle \sum \limits _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}_{jk} + w_kz_{jk} \\ \mathrm{s.t.}&{} z_{jk} \in \mathbb {R},\quad \varvec{z}_{jk} \in \mathbb {R}^n, \quad \mathbf {Z}_{jk} \in \mathbb {S}^n\quad \forall j=1,\dots ,m,~ k=0,\dots ,2^n \\ &{} \displaystyle \sum \limits _{k=0}^{2^n} \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] = p_j{\varvec{\Omega }}_j \quad \forall j=1,\dots ,m\\ &{} \begin{array}{l@{\quad }l} \left\langle \mathbf {W}_j, \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] \right\rangle \le 0,\quad \left[ \begin{array}{c@{\quad }c} \mathbf {Z}_{jk} &{} \varvec{z}_{jk} \\ \varvec{z}_{jk}^\intercal &{} z_{jk} \end{array}\right] \succcurlyeq \mathbf 0 \end{array} \quad \forall j=1,\dots ,m,~ k=0,\dots ,2^n \,. \end{array} \end{aligned}$$
(34)

Strong duality holds as the primal minimization problem is strictly feasible. This implies that the optimal value of (34) is equal to that of the worst-case expectation problem (9). We now denote by \(\mathbf Z_{jk}^*\), \(\varvec{z}_{jk}^*\), and \(z_{jk}^{*}\) for \(j=1,\ldots , m\) and \(k=0,\ldots , 2^n\) an optimal solution of the SDP (34). Assume first that all \(z_{jk}^{*}\) are strictly positive. For any fixed \(j\) and \(k\) we thus construct the positive semidefinite matrix

$$\begin{aligned} \left[ \begin{array}{c@{\quad }c}\mathbf Z_{jk}^*/z_{jk}^{*} &{} \varvec{z}_{jk}^*/z_{jk}^{*} \\ (\varvec{z}_{jk}^*/z_{jk}^{*})^\intercal &{} 1 \end{array}\right] \succcurlyeq \varvec{0}\,, \end{aligned}$$

which satisfies, together with \(\mathbf W_j\), the conditions of Proposition 6.1. Hence, we can construct a discrete distribution \(\mathbb P_{jk}^*\) supported in \(\Xi _j\) with mean \(\varvec{z}_{jk}^*/z_{jk}^{*}\) and covariance matrix \(\mathbf Z_{jk}^*/z_{jk}^*-\varvec{z}_{jk}^*/z_{jk}^{*}(\varvec{z}_{jk}^*/z_{jk}^{*})^\intercal \). In the remainder we will argue that \(\mathbb P^*=\sum _{j=1}^m\sum _{k=0}^{2^n}z_{jk}^{*} \mathbb P_{jk}^*\) is the sought extremal distribution.

From the equality constraints in (34) it is clear that \(\mathbb P^*\) is feasible in (9). Thus,

$$\begin{aligned} \sum _{j=1}^m\sum _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}^*_{jk} + w_kz^*_{jk}&= \sup \limits _{\mathbb P\in {\mathcal {P}}}\mathbb E_{\mathbb P}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&\ge \mathbb {E}_{\mathbb {P}^*}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&= \sum \limits _{j=1}^m\sum \limits _{\ell =0}^{2^n} z^*_{j\ell }\mathbb E_{\mathbb P^*_{j\ell }}\left[ \max _{k=0,\dots ,2^n} \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k\right] \\&\ge \sum \limits _{j=1}^m\sum \limits _{k=0}^{2^n} z^*_{jk} \mathbb E_{\mathbb P^*_{jk}}[ \varvec{w}_k^\intercal \tilde{\varvec{\xi }} + w_k] =\sum \limits _{j=1}^m\sum \limits _{k=0}^{2^n} \varvec{w}_k^\intercal \varvec{z}^*_{jk} + w_kz^*_{jk}, \end{aligned}$$

where the first equality follows from strong duality. As all inequalities in the above expression must be binding, \(\mathbb P^*\) is indeed extremal in (9).

Assume now that \(z_{jk}^{*}=0\) for some \(j\) and \(k\). The positive semidefiniteness constraint in (34) then implies that \(\varvec{z}_{jk}^*=\varvec{0}\), while the trace constraint implies \(\langle {\varvec{\Lambda }}_j^{-1},\mathbf {Z}_{jk}\rangle \le 0\), see also the definition of \(\mathbf W_j\) in (13). Our assumption \({\varvec{\Lambda }}_j^{-1}\succ \varvec{0}\) thus implies that \(\mathbf Z_{jk}^*=\varvec{0}\). This argument suggests that all pairs of indices \(j\) and \(k\) for which \(z_{jk}^{*}=0\) can be ignored altogether. In general, an extremal distribution for the worst-case CVaR problem (8) can thus be constructed as

$$\begin{aligned} \mathbb P^*=\sum \limits _{j=1}^m\sum \limits _{\mathop {z_{jk}^{*} \ne 0}\limits ^{k=0}}^{2^n}z_{jk}^{*} \mathbb P_{jk}^*\,. \end{aligned}$$

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Hanasusanto, G.A., Kuhn, D., Wallace, S.W. et al. Distributionally robust multi-item newsvendor problems with multimodal demand distributions. Math. Program. 152, 1–32 (2015). https://doi.org/10.1007/s10107-014-0776-y

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Mathematics Subject Classification

  • 90C15
  • 90C22