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 & Optimization, 2022, 12 (1) : 159-212. doi: 10.3934/naco.2021057
References:
[1]

F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, 95 (2003), 3-51.  doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer Science & Business Media, 2006.  Google Scholar

[3]

I. E. Bardakci and C. M. Lagoa, Distributionally robust portfolio optimization, in 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE, (2019), 1526–1531. doi: 10.1007/s11579-019-00241-1.  Google Scholar

[4]

G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, in The Operations Research Revolution, INFORMS, (2015), 1–19. Google Scholar

[5]

A. Ben-TalD. den HertogA. De WaegenaereB. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), 341-357.   Google Scholar

[6] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009.  doi: 10.1515/9781400831050.  Google Scholar
[7]

A. Ben-TalE. HazanT. Koren and S. Mannor, Oracle-based robust optimization via online learning, Operations Research, 63 (2015), 628-638.  doi: 10.1287/opre.2015.1374.  Google Scholar

[8]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[9]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations Research Letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[10]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424.  doi: 10.1007/PL00011380.  Google Scholar

[11]

D. BertsimasD. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501.  doi: 10.1137/080734510.  Google Scholar

[12]

D. Bertsimas and M. S. Copenhaver, Characterization of the equivalence of robustification and regularization in linear and matrix regression, European Journal of Operational Research, 270 (2018), 931-942.  doi: 10.1016/j.ejor.2017.03.051.  Google Scholar

[13]

D. BertsimasX. V. DoanK. Natarajan and C.-P. Teo, Models for minimax stochastic linear optimization problems with risk aversion, Mathematics of Operations Research, 35 (2010), 580-602.  doi: 10.1287/moor.1100.0445.  Google Scholar

[14]

D. BertsimasV. Gupta and N. Kallus, Data-driven robust optimization, Mathematical Programming, 167 (2018), 235-292.  doi: 10.1007/s10107-017-1125-8.  Google Scholar

[15]

D. BertsimasV. Gupta and N. Kallus, Robust sample average approximation, Mathematical Programming, 171 (2018), 217-282.  doi: 10.1007/s10107-017-1174-z.  Google Scholar

[16]

D. Bertsimas and I. Popescu, Optimal inequalities in probability theory: A convex optimization approach, SIAM Journal on Optimization, 15 (2005), 780-804.  doi: 10.1137/S1052623401399903.  Google Scholar

[17]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[18]

D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Mathematical Programming, 107 (2006), 5-36.  doi: 10.1007/s10107-005-0677-1.  Google Scholar

[19]

D. BertsimasM. Sim and M. Zhang, Adaptive distributionally robust optimization, Management Science, 65 (2019), 604-618.   Google Scholar

[20]

H.-G. Beyer and B. Sendhoff, Robust optimization–a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3190-3218.  doi: 10.1016/j.cma.2007.03.003.  Google Scholar

[21]

C. Bhattacharyya, Second order cone programming formulations for feature selection, Journal of Machine Learning Research, 5 (2004), 1417-1433.   Google Scholar

[22]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd edition, Springer Publishing Company, Incorporated, 2011. doi: 10.1007/978-1-4614-0237-4.  Google Scholar

[23]

J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis and L. Rey-Bellet, $(f, \gamma) $-divergences: Interpolating between $ f $-divergences and integral probability metrics, arXiv: 2011.05953. Google Scholar

[24]

J. Blanchet, L. Chen and X. Y. Zhou, Distributionally robust mean-variance portfolio selection with wasserstein distances, arXiv: 1802.04885. Google Scholar

[25]

J. Blanchet, P. W. Glynn, J. Yan and Z. Zhou, Multivariate distributionally robust convex regression under absolute error loss, arXiv: 1905.12231. Google Scholar

[26]

J. Blanchet and Y. Kang, Sample out-of-sample inference based on wasserstein distance, arXiv: 1605.01340. doi: 10.1287/opre.2020.2028.  Google Scholar

[27]

J. Blanchet and Y. Kang, Semi-supervised learning based on distributionally robust optimization, Data Analysis and Applications 3: Computational, Classification, Financial, Statistical and Stochastic Methods, 5 (2020), 1-33.   Google Scholar

[28]

J. BlanchetY. Kang and K. Murthy, Robust wasserstein profile inference and applications to machine learning, Journal of Applied Probability, 56 (2019), 830-857.  doi: 10.1017/jpr.2019.49.  Google Scholar

[29]

J. Blanchet, K. Murthy and N. Si, Confidence regions in wasserstein distributionally robust estimation, reprint arXiv: 1906.01614. Google Scholar

[30]

V. I. Bogachev and A. V. Kolesnikov, The monge-kantorovich problem: achievements, connections, and perspectives, Russian Mathematical Surveys, 67 (2012), 785-890.  doi: 10.1070/rm2012v067n05abeh004808.  Google Scholar

[31]

F. BolleyA. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probability Theory and Related Fields, 137 (2007), 541-593.  doi: 10.1007/s00440-006-0004-7.  Google Scholar

[32]

M. Breton and S. El Hachem, Algorithms for the solution of stochastic dynamic minimax problems, Computational Optimization and Applications, 4 (1995), 317-345.  doi: 10.1007/BF01300861.  Google Scholar

[33]

M. Breton and S. El Hachem, A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems, Stochastics and Stochastic Reports, 53 (1995), 305-322.  doi: 10.1080/17442509508833994.  Google Scholar

[34]

G. C. Calafiore, Ambiguous risk measures and optimal robust portfolios, SIAM Journal on Optimization, 18 (2007), 853-877.  doi: 10.1137/060654803.  Google Scholar

[35]

G. C. Calafiore and L. El Ghaoui, On distributionally robust chance-constrained linear programs, Journal of Optimization Theory and Applications, 130 (2006), 1-22.  doi: 10.1007/s10957-006-9084-x.  Google Scholar

[36]

R. Chen and I. C. Paschalidis, A robust learning approach for regression models based on distributionally robust optimization, Journal of Machine Learning Research, 19 (2018), 1-48.   Google Scholar

[37]

Y. ChenQ. GuoH. SunZ. LiW. Wu and Z. Li, A distributionally robust optimization model for unit commitment based on kullback–leibler divergence, IEEE Transactions on Power Systems, 33 (2018), 5147-5160.   Google Scholar

[38]

J. ChengE. Delage and A. Lisser, Distributionally robust stochastic knapsack problem, SIAM Journal on Optimization, 24 (2014), 1485-1506.  doi: 10.1137/130915315.  Google Scholar

[39]

J. ChengR. Li-Yang ChenH. N. NajmA. PinarC. Safta and J.-P. Watson, Distributionally robust optimization with principal component analysis, SIAM Journal on Optimization, 28 (2018), 1817-1841.  doi: 10.1137/16M1075910.  Google Scholar

[40]

A. Cherukuri and J. Cortés, Cooperative data-driven distributionally robust optimization, IEEE Transactions on Automatic Control, 65 (2019), 4400-4407.   Google Scholar

[41]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio choice, in Handbook of the Fundamentals of Financial Decision Making: Part I, World Scientific, (2013), 365–373. Google Scholar

[42]

K. L. ClarksonE. Hazan and D. P. Woodruff, Sublinear optimization for machine learning, Journal of the ACM (JACM), 59 (2012), 1-49.  doi: 10.1145/2371656.2371658.  Google Scholar

[43]

A. R. Conn and N. I. Gould, An exact penalty function for semi-infinite programming, Mathematical Programming, 37 (1987), 19-40.  doi: 10.1007/BF02591681.  Google Scholar

[44]

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition, Wiley Online Library, Hoboken, 2012.  Google Scholar

[45]

G. B. Dantzig, Linear programming under uncertainty, Management Science, 50 (2004), 1764-1769.  doi: 10.1287/opre.50.1.42.17798.  Google Scholar

[46]

E. del BarrioE. Giné and C. Matrán, Central limit theorems for the wasserstein distance between the empirical and the true distributions, Annals of Probability, 27 (1999), 1009-1071.  doi: 10.1214/aop/1022677394.  Google Scholar

[47]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[48]

S. Dharmadhikari and K. Joag-Dev, Unimodality, Convexity, and Applications, Elsevier, 1988.  Google Scholar

[49]

T. DinhR. Fukasawa and J. Luedtke, Exact algorithms for the chance-constrained vehicle routing problem, Mathematical Programming, 172 (2018), 105-138.  doi: 10.1007/s10107-017-1151-6.  Google Scholar

[50]

X. V. DoanX. Li and K. Natarajan, Robustness to dependency in portfolio optimization using overlapping marginals, Operations Research, 63 (2015), 1468-1488.  doi: 10.1287/opre.2015.1424.  Google Scholar

[51]

N. DuY. Liu and Y. Liu, A new data-driven distributionally robust portfolio optimization method based on wasserstein ambiguity set, IEEE Access, 9 (2021), 3174-3194.   Google Scholar

[52]

C. DuanW. FangL. JiangL. Yao and J. Liu, Distributionally robust chance-constrained approximate ac-opf with wasserstein metric, IEEE Transactions on Power Systems, 33 (2018), 4924-4936.   Google Scholar

[53]

J. C. Duchi and H. Namkoong, Variance-based regularization with convex objectives, Journal of Machine Learning Research, 20 (2019), 1-68.   Google Scholar

[54]

J. C. Duchi and H. Namkoong, Learning models with uniform performance via distributionally robust optimization, arXiv: 1810.08750. doi: 10.1214/20-aos2004.  Google Scholar

[55]

J. Dupačová, Stochastic programming: Minimax approach, Encyclopedia Optimization, 5 (2001), 327-330.   Google Scholar

[56]

J. Dupačová, On minimax solutions of stochastic linear programming problems, Časopis Pro Pěstování Matematiky, 091 (1966), 423–430.  Google Scholar

[57]

J. Dupačová, The minimax approach to stochastic programming and an illustrative application, Stochastics, 20 (1987), 73-88.  doi: 10.1080/17442508708833436.  Google Scholar

[58]

J. Dupačová, Uncertainties in minimax stochastic programs, Optimization, 60 (2011), 1235-1250.  doi: 10.1080/02331934.2010.532214.  Google Scholar

[59]

L. El Ghaoui and H. Lebret, Robust solutions to least-squares problems with uncertain data, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 1035-1064.  doi: 10.1137/S0895479896298130.  Google Scholar

[60]

E. Erdoğan and G. Iyengar, Ambiguous chance constrained problems and robust optimization, Mathematical Programming, 107 (2006), 37-61.  doi: 10.1007/s10107-005-0678-0.  Google Scholar

[61]

P. M. Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations, Mathematical Programming, 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1.  Google Scholar

[62]

F. Farnia and D. Tse, A minimax approach to supervised learning, Advances in Neural Information Processing Systems, 29 (2016), 4240-4248.   Google Scholar

[63]

L. FauryU. TanielianE. DohmatobE. Smirnova and F. Vasile, Distributionally robust counterfactual risk minimization, Proceedings of the AAAI Conference on Artificial Intelligence, 34 (2020), 3850-3857.   Google Scholar

[64]

N. Fournier and A. Guillin, On the rate of convergence in wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.  Google Scholar

[65]

D. Fouskakis and D. Draper, Stochastic optimization: a review, International Statistical Review, 70 (2002), 315-349.   Google Scholar

[66]

C. FrognerS. ClaiciE. Chien and J. Solomon, Incorporating unlabeled data into distributionally robust learning, Journal of Machine Learning Research, 22 (2021), 1-46.   Google Scholar

[67]

V. GabrelC. Murat and A. Thiele, Recent advances in robust optimization: An overview, European Journal of Operational Research, 235 (2014), 471-483.  doi: 10.1016/j.ejor.2013.09.036.  Google Scholar

[68]

G. Gallego and I. Moon, The distribution free newsboy problem: review and extensions, Journal of the Operational Research Society, 44 (1993), 825-834.   Google Scholar

[69]

R. Gao, Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality, arXiv: 2009.04382. Google Scholar

[70]

R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with wasserstein distance, arXiv: 1604.02199. Google Scholar

[71]

R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with dependence structure, arXiv: 1701.04200. Google Scholar

[72]

R. Gao, L. Xie, Y. Xie and H. Xu, Robust hypothesis testing using wasserstein uncertainty sets, in NeurIPS, (2018), 7913–7923. Google Scholar

[73]

L. E. GhaouiM. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.  Google Scholar

[74]

S. Ghosal and W. Wiesemann, The distributionally robust chance-constrained vehicle routing problem, Operations Research, 68 (2020), 716-732.  doi: 10.1287/opre.2019.1924.  Google Scholar

[75]

A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, International Statistical Review, 70 (2002), 419-435.   Google Scholar

[76]

M. Á. Goberna and M. A. López, Semi-Infinite Programming: Recent Advances, Springer Science & Business Media, 2013. doi: 10.1007/0-387-26771-9_1.  Google Scholar

[77]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operations Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.  Google Scholar

[78]

I. J. Goodfellow, J. Shlens and C. Szegedy, Explaining and harnessing adversarial examples, arXiv: 1412.6572. Google Scholar

[79]

S. GuoH. Xu and L. Zhang, Probability approximation schemes for stochastic programs with distributionally robust second-order dominance constraints, Optimization Methods and Software, 32 (2017), 770-789.  doi: 10.1080/10556788.2016.1175003.  Google Scholar

[80]

V. Gupta, Near-optimal bayesian ambiguity sets for distributionally robust optimization, Management Science, 65 (2019), 4242-4260.   Google Scholar

[81]

S.-Å. Gustafson, Semi-infinite programming: Methods for linear problems, in Encyclopedia of Optimization, Springer, (2009), 3424–3429. Google Scholar

[82]

G. A. Hanasusanto and D. Kuhn, Conic programming reformulations of two-stage distributionally robust linear programs over wasserstein balls, Operations Research, 66 (2018), 849-869.  doi: 10.1287/opre.2017.1698.  Google Scholar

[83]

G. A. HanasusantoD. KuhnS. W. Wallace and S. Zymler, Distributionally robust multi-item newsvendor problems with multimodal demand distributions, Mathematical Programming, 152 (2015), 1-32.  doi: 10.1007/s10107-014-0776-y.  Google Scholar

[84]

G. A. HanasusantoV. RoitchD. Kuhn and W. Wiesemann, A distributionally robust perspective on uncertainty quantification and chance constrained programming, Mathematical Programming, 151 (2015), 35-62.  doi: 10.1007/s10107-015-0896-z.  Google Scholar

[85]

R. Hettich, A. Kaplan and R. Tichatschke, Semi-infinite programming: Numerical methods, in Encyclopedia of Optimization, Springer, (2009), 3429–3434. Google Scholar

[86]

R. Hettich and K. O. Kortanek, Semi-infinite programming: theory, methods, and applications, SIAM Review, 35 (1993), 380-429.  doi: 10.1137/1035089.  Google Scholar

[87]

Z. Hu and L. J. Hong, Kullback-leibler divergence constrained distributionally robust optimization, Available at Optimization Online. Google Scholar

[88]

K. HuangH. YangI. KingM. R. Lyu and L. Chan, The minimum error minimax probability machine, Journal of Machine Learning Research, 5 (2004), 1253-1286.   Google Scholar

[89]

G. Infanger, Planning under uncertainty solving large-scale stochastic linear programs, Technical report, Stanford University, 1992. Google Scholar

[90]

K. Isii, On sharpness of tchebycheff-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962), 185-197.  doi: 10.1007/BF02868641.  Google Scholar

[91]

R. Ji and M. A. Lejeune, Data-driven distributionally robust chance-constrained optimization with wasserstein metric, Journal of Global Optimization, (2020), 1–33. doi: 10.1007/s10898-020-00966-0.  Google Scholar

[92]

R. Jiang and Y. Guan, Data-driven chance constrained stochastic program, Mathematical Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7.  Google Scholar

[93]

R. Jiang and Y. Guan, Risk-averse two-stage stochastic program with distributional ambiguity, Operations Research, 66 (2018), 1390-1405.  doi: 10.1287/opre.2018.1729.  Google Scholar

[94]

R. Jiang, M. Ryu and G. Xu, Data-driven distributionally robust appointment scheduling over wasserstein balls, arXiv: 1907.03219. Google Scholar

[95]

P. Kall, S. W. Wallace and P. Kall, Stochastic Programming, Springer, 1994.  Google Scholar

[96]

Z. KangX. LiZ. Li and S. Zhu, Data-driven robust mean-cvar portfolio selection under distribution ambiguity, Quantitative Finance, 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057.  Google Scholar

[97]

L. V. Kantorovich, On the translocation of masses, Journal of Mathematical Sciences, 133 (2006), 1381-1382.  doi: 10.1007/s10958-006-0049-2.  Google Scholar

[98]

D. KlabjanD. Simchi-Levi and M. Song, Robust stochastic lot-sizing by means of histograms, Production and Operations Management, 22 (2013), 691-710.   Google Scholar

[99]

Ç. KoçyiğitG. IyengarD. Kuhn and W. Wiesemann, Distributionally robust mechanism design, Management Science, 66 (2020), 159-189.   Google Scholar

[100]

D. Kuhn, P. M. Esfahani, V. A. Nguyen and S. Shafieezadeh-Abadeh, Wasserstein distributionally robust optimization: Theory and applications in machine learning, in Operations Research & Management Science in the Age of Analytics, INFORMS, (2019), 130–166. doi: 10.3770/j.issn:2095-2651.2021.01.010.  Google Scholar

[101]

S. Kullback, Information Theory and Statistics, Courier Corporation, Mineola, 1997.  Google Scholar

[102]

G. Lanckriet, L. E. Ghaoui, C. Bhattacharyya and M. I. Jordan, Minimax probability machine, in Advances in neural information processing systems, (2001), 801–807. Google Scholar

[103]

G. R. LanckrietL. E. GhaouiC. Bhattacharyya and M. I. Jordan, A robust minimax approach to classification, Journal of Machine Learning Research, 3 (2002), 555-582.  doi: 10.1162/153244303321897726.  Google Scholar

[104]

H. J. Landau, Maximum entropy and the moment problem, Bulletin of the American Mathematical Society, 16 (1987), 47-77.  doi: 10.1090/S0273-0979-1987-15464-4.  Google Scholar

[105]

C. Lee and S. Mehrotra, A distributionally-robust approach for finding support vector machines, 2015. Google Scholar

[106]

S. Lee, H. Kim and I. Moon, A data-driven distributionally robust newsvendor model with a wasserstein ambiguity set, Journal of the Operational Research Society, (2020), 1–19. Google Scholar

[107]

D. Levy, Y. Carmon, J. C. Duchi and A. Sidford, Large-scale methods for distributionally robust optimization, Advances in Neural Information Processing Systems, 33. Google Scholar

[108]

A. S. Lewis and C. J. Pang, Lipschitz behavior of the robust regularization, SIAM Journal on Control and Optimization, 48 (2010), 3080-3104.  doi: 10.1137/08073682X.  Google Scholar

[109]

J. Y. Li and R. H. Kwon, Portfolio selection under model uncertainty: a penalized moment-based optimization approach, Journal of Global Optimization, 56 (2013), 131-164.  doi: 10.1007/s10898-012-9969-1.  Google Scholar

[110]

G. D. Lin, Recent developments on the moment problem, Journal of Statistical Distributions and Applications, 4 (2017), 1-17.   Google Scholar

[111]

Q. LinR. LoxtonK. L. TeoY. H. Wu and C. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010.  Google Scholar

[112]

J. LiuZ. ChenA. Lisser and Z. Xu, Closed-form optimal portfolios of distributionally robust mean-cvar problems with unknown mean and variance, Applied Mathematics & Optimization, 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y.  Google Scholar

[113]

Y. LiuR. Meskarian and H. Xu, Distributionally robust reward-risk ratio optimization with moment constraints, SIAM Journal on Optimization, 27 (2017), 957-985.  doi: 10.1137/16M106114X.  Google Scholar

[114]

F. Luo and S. Mehrotra, Decomposition algorithm for distributionally robust optimization using wasserstein metric with an application to a class of regression models, European Journal of Operational Research, 278 (2019), 20-35.  doi: 10.1016/j.ejor.2019.03.008.  Google Scholar

[115]

C. Lyu, K. Huang and H.-N. Liang, A unified gradient regularization family for adversarial examples, in 2015 IEEE International Conference on Data Mining, IEEE, (2015), 301–309.  Google Scholar

[116]

A. MajumdarG. Hall and A. A. Ahmadi, Recent scalability improvements for semidefinite programming with applications in machine learning, control, and robotics, Annual Review of Control, Robotics, and Autonomous Systems, 3 (2020), 331-360.   Google Scholar

[117]

S. Mehrotra and D. Papp, A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization, SIAM Journal on Optimization, 24 (2014), 1670-1697.  doi: 10.1137/130925013.  Google Scholar

[118]

S. Mehrotra and H. Zhang, Models and algorithms for distributionally robust least squares problems, Mathematical Programming, 146 (2014), 123-141.  doi: 10.1007/s10107-013-0681-9.  Google Scholar

[119]

R. O. Michaud, The markowitz optimization enigma: Is 'optimized' optimal?, Financial Analysts Journal, 45 (1989), 31-42.   Google Scholar

[120]

H. Mostafaei and S. Kordnourie, Probability metrics and their applications, Applied Mathematical Sciences, 5 (2011), 181-192.   Google Scholar

[121]

H. Namkoong and J. C. Duchi, Stochastic gradient methods for distributionally robust optimization with f-divergences, NIPS, 29 (2016), 2208-2216.   Google Scholar

[122]

K. NatarajanM. Sim and J. Uichanco, Tractable robust expected utility and risk models for portfolio optimization, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 20 (2010), 695-731.  doi: 10.1111/j.1467-9965.2010.00417.x.  Google Scholar

[123]

K. Natarajan and C.-P. Teo, On reduced semidefinite programs for second order moment bounds with applications, Mathematical Programming, 161 (2017), 487-518.  doi: 10.1007/s10107-016-1019-1.  Google Scholar

[124]

A. NemirovskiA. JuditskyG. Lan and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization, 19 (2009), 1574-1609.  doi: 10.1137/070704277.  Google Scholar

[125]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2007), 969-996.  doi: 10.1137/050622328.  Google Scholar

[126]

V. A. Nguyen, D. Kuhn and P. M. Esfahani, Distributionally robust inverse covariance estimation: The wasserstein shrinkage estimator, arXiv: 1805.07194. Google Scholar

[127]

V. A. Nguyen, S. Shafieezadeh-Abadeh, D. Kuhn and P. M. Esfahani, Bridging bayesian and minimax mean square error estimation via wasserstein distributionally robust optimization, arXiv: 1911.03539. Google Scholar

[128] A. B. Owen, Empirical Likelihood, CRC Press, 2001.   Google Scholar
[129] L. Pardo, Statistical Inference Based on Divergence Measures, CRC Press, 2018.   Google Scholar
[130]

G. C. Pflug and A. Pichler, Approximations for probability distributions and stochastic optimization problems, in Stochastic Optimization Methods in Finance and Energy, Springer, 2011,343–387. doi: 10.1007/978-1-4419-9586-5_15.  Google Scholar

[131]

R. R. Phelps, Lectures on Choquet's Theorem, Springer Science & Business Media, 2001. doi: 10.1007/b76887.  Google Scholar

[132]

A. B. PhilpottV. L. de Matos and L. Kapelevich, Distributionally robust sddp, Computational Management Science, 15 (2018), 431-454.  doi: 10.1007/s10287-018-0314-0.  Google Scholar

[133]

I. Popescu, A semidefinite programming approach to optimal-moment bounds for convex classes of distributions, Mathematics of Operations Research, 30 (2005), 632-657.  doi: 10.1287/moor.1040.0137.  Google Scholar

[134]

I. Popescu, Robust mean-covariance solutions for stochastic optimization, Operations Research, 55 (2007), 98-112.  doi: 10.1287/opre.1060.0353.  Google Scholar

[135]

K. PostekD. den Hertog and B. Melenberg, Computationally tractable counterparts of distributionally robust constraints on risk measures, SIAM Review, 58 (2016), 603-650.  doi: 10.1137/151005221.  Google Scholar

[136] M. A. Proschan and P. A. Shaw, Essentials of Probability Theory for Statisticians, CRC Press, 2018.   Google Scholar
[137]

H. RahimianG. Bayraksan and T. Homem-de Mello, Identifying effective scenarios in distributionally robust stochastic programs with total variation distance, Mathematical Programming, 173 (2019), 393-430.  doi: 10.1007/s10107-017-1224-6.  Google Scholar

[138]

H. Rahimian and S. Mehrotra, Distributionally robust optimization: A review, arXiv: 1908.05659. Google Scholar

[139]

M. Riis and K. A. Andersen, Applying the minimax criterion in stochastic recourse programs, European Journal of Operational Research, 165 (2005), 569-584.  doi: 10.1016/j.ejor.2003.09.033.  Google Scholar

[140] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.   Google Scholar
[141]

Y. RubnerC. Tomasi and L. J. Guibas, The earth mover's distance as a metric for image retrieval, International Journal of Computer Vision, 40 (2000), 99-121.  doi: 10.1007/3-540-46238-4_2.  Google Scholar

[142]

N. RujeerapaiboonD. Kuhn and W. Wiesemann, Robust growth-optimal portfolios, Management Science, 62 (2016), 2090-2109.   Google Scholar

[143]

N. RujeerapaiboonD. Kuhn and W. Wiesemann, Chebyshev inequalities for products of random variables, Mathematics of Operations Research, 43 (2018), 887-918.  doi: 10.1287/moor.2017.0888.  Google Scholar

[144]

L. Rüschendorf, Bounds for distributions with multivariate marginals, Lecture Notes-Monograph Series, (1991), 285–310. doi: 10.1214/lnms/1215459862.  Google Scholar

[145]

H. Scarf, A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production. Google Scholar

[146]

S. Shafieezadeh-AbadehD. Kuhn and P. M. Esfahani, Regularization via mass transportation, Journal of Machine Learning Research, 20 (2019), 1-68.   Google Scholar

[147]

S. Shafieezadeh AbadehP. M. Mohajerin Esfahani and D. Kuhn, Distributionally robust logistic regression, Advances in Neural Information Processing Systems, 28 (2015), 1576-1584.   Google Scholar

[148]

S. Shafieezadeh-Abadeh, V. A. Nguyen, D. Kuhn and P. M. Esfahani, Wasserstein distributionally robust kalman filtering, in Advances in Neural Information Processing Systems, vol. 31, Curran Associates, Inc., 2018, 8474–8483. Google Scholar

[149]

U. ShahamY. Yamada and S. Negahban, Understanding adversarial training: Increasing local stability of supervised models through robust optimization, Neurocomputing, 307 (2018), 195-204.   Google Scholar

[150]

S. Shalev-Shwartz and Y. Wexler, Minimizing the maximal loss: How and why, in International Conference on Machine Learning, PMLR, 2016,793–801. Google Scholar

[151]

C. Shang and F. You, Distributionally robust optimization for planning and scheduling under uncertainty, Computers & Chemical Engineering, 110 (2018), 53-68.   Google Scholar

[152]

A. Shapiro, On duality theory of conic linear problems, in Semi-infinite Programming, Springer, Boston, MA, 2001,135–165. doi: 10.1007/978-1-4757-3403-4_7.  Google Scholar

[153]

A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96.  doi: 10.1007/s10107-005-0680-6.  Google Scholar

[154]

A. Shapiro, Semi-infinite programming, duality, discretization and optimality conditions, Optimization, 58 (2009), 133-161.  doi: 10.1080/02331930902730070.  Google Scholar

[155]

A. Shapiro, Distributionally robust stochastic programming, SIAM Journal on Optimization, 27 (2017), 2258-2275.  doi: 10.1137/16M1058297.  Google Scholar

[156]

A. Shapiro, Tutorial on risk neutral, distributionally robust and risk averse multistage stochastic programming, European Journal of Operational Research, 288 (2020), 1-13.  doi: 10.1016/j.ejor.2020.03.065.  Google Scholar

[157]

A. Shapiro and S. Ahmed, On a class of minimax stochastic programs, SIAM Journal on Optimization, 14 (2004), 1237-1249.  doi: 10.1137/S1052623403434012.  Google Scholar

[158]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, 2nd edition, SIAM, Philadelphia, PA, 2014. Google Scholar

[159]

A. Shapiro and A. Kleywegt, Minimax analysis of stochastic problems, Optimization Methods and Software, 17 (2002), 523-542.  doi: 10.1080/1055678021000034008.  Google Scholar

[160]

K. S. ShehadehA. E. Cohn and R. Jiang, A distributionally robust optimization approach for outpatient colonoscopy scheduling, European Journal of Operational Research, 283 (2020), 549-561.  doi: 10.1016/j.ejor.2019.11.039.  Google Scholar

[161]

A. Sinha, H. Namkoong and J. Duchi, Certifying some distributional robustness with principled adversarial training, in International Conference on Learning Representations, 2018. Google Scholar

[162]

J. E. Smith, Generalized chebychev inequalities: theory and applications in decision analysis, Operations Research, 43 (1995), 807-825.  doi: 10.1287/opre.43.5.807.  Google Scholar

[163]

J. E. Smith and R. L. Winkler, The optimizer's curse: Skepticism and postdecision surprise in decision analysis, Management Science, 52 (2006), 311-322.   Google Scholar

[164]

A. M.-C. So, Moment inequalities for sums of random matrices and their applications in optimization, Mathematical Programming, 130 (2011), 125-151.  doi: 10.1007/s10107-009-0330-5.  Google Scholar

[165]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 21 (1973), 1154-1157.  doi: 10.1287/opre.22.4.892.  Google Scholar

[166] A. Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Observational Data, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511754081.  Google Scholar
[167]

G. Still, Optimization problems with infinitely many constraints, Buletinul tiinific al Universitatii Baia Mare, Seria B, Fascicola matematică-informatică, 18 (2002), 343–354.  Google Scholar

[168]

T. Strohmann and G. Z. Grudic, A formulation for minimax probability machine regression, in NIPS, Citeseer, 2002,769–776. Google Scholar

[169]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.  Google Scholar

[170]

S. Takriti and S. Ahmed, Managing short-term electricity contracts under uncertainty: A minimax approach, 2002. Google Scholar

[171]

B. P. Van Parys, P. M. Esfahani and D. Kuhn, From data to decisions: Distributionally robust optimization is optimal, Management Science, (2020), preprint. Google Scholar

[172]

B. P. Van ParysP. J. Goulart and D. Kuhn, Generalized gauss inequalities via semidefinite programming, Mathematical Programming, 156 (2016), 271-302.  doi: 10.1007/s10107-015-0878-1.  Google Scholar

[173]

B. P. Van ParysP. J. Goulart and M. Morari, Distributionally robust expectation inequalities for structured distributions, Mathematical Programming, 173 (2019), 251-280.  doi: 10.1007/s10107-017-1220-x.  Google Scholar

[174]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003.  Google Scholar

[175]

L. VandenbergheS. Boyd and K. Comanor, Generalized chebyshev bounds via semidefinite programming, SIAM Review, 49 (2007), 52-64.  doi: 10.1137/S0036144504440543.  Google Scholar

[176]

C. Villani, Optimal Transport: Old and New, vol. 338, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[177]

M. R. Wagner, Stochastic 0–1 linear programming under limited distributional information, Operations Research Letters, 36 (2008), 150-156.  doi: 10.1016/j.orl.2007.07.003.  Google Scholar

[178]

A. Wald, Statistical decision functions which minimize the maximum risk, Annals of Mathematics, 46 (1945), 265-280.  doi: 10.2307/1969022.  Google Scholar

[179]

A. Wald, Statistical decision functions, in Breakthroughs in Statistics, Springer, 1992,342–357. Google Scholar

[180]

C. WangR. GaoF. QiuJ. Wang and L. Xin, Risk-based distributionally robust optimal power flow with dynamic line rating, IEEE Transactions on Power Systems, 33 (2018), 6074-6086.   Google Scholar

[181]

C. WangR. GaoW. WeiM. Shafie-khahT. Bi and J. P. Catalao, Risk-based distributionally robust optimal gas-power flow with wasserstein distance, IEEE Transactions on Power Systems, 34 (2018), 2190-2204.   Google Scholar

[182]

S. Wang and Y. Yuan, Feasible method for semi-infinite programs, SIAM Journal on Optimization, 25 (2015), 2537-2560.  doi: 10.1137/140982143.  Google Scholar

[183]

Z. WangK. YouS. Song and Y. Zhang, Wasserstein distributionally robust shortest path problem, European Journal of Operational Research, 284 (2020), 31-43.  doi: 10.1016/j.ejor.2020.01.009.  Google Scholar

[184]

Z. WangP. W. Glynn and Y. Ye, Likelihood robust optimization for data-driven problems, Computational Management Science, 13 (2016), 241-261.  doi: 10.1007/s10287-015-0240-3.  Google Scholar

[185]

W. WiesemannD. Kuhn and B. Rustem, Robust markov decision processes, Mathematics of Operations Research, 38 (2013), 153-183.  doi: 10.1287/moor.1120.0566.  Google Scholar

[186]

W. WiesemannD. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[187]

L. A. Wolsey, Integer Programming, Wiley Online Library, 1998.  Google Scholar

[188]

D. Wozabal, Robustifying convex risk measures for linear portfolios: A nonparametric approach, Operations Research, 62 (2014), 1302-1315.  doi: 10.1287/opre.2014.1323.  Google Scholar

[189]

H. XuC. Caramanis and S. Mannor, Robustness and regularization of support vector machines, Journal of Machine Learning Research, 10 (2009), 1485-1510.   Google Scholar

[190]

H. XuC. Caramanis and S. Mannor, Robust regression and lasso, IEEE Transactions on Information Theory, 56 (2010), 3561-3574.  doi: 10.1109/TIT.2010.2048503.  Google Scholar

[191]

H. XuC. Caramanis and S. Mannor, A distributional interpretation of robust optimization, Mathematics of Operations Research, 37 (2012), 95-110.  doi: 10.1287/moor.1110.0531.  Google Scholar

[192]

H. XuY. Liu and H. Sun, Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods, Mathematical Programming, 169 (2018), 489-529.  doi: 10.1007/s10107-017-1143-6.  Google Scholar

[193]

M. XuS.-Y. Wu and J. Y. Jane, Solving semi-infinite programs by smoothing projected gradient method, Computational Optimization and Applications, 59 (2014), 591-616.  doi: 10.1007/s10589-014-9654-z.  Google Scholar

[194]

I. Yang, A convex optimization approach to distributionally robust markov decision processes with wasserstein distance, IEEE Control Systems Letters, 1 (2017), 164-169.   Google Scholar

[195]

I. Yang, Wasserstein distributionally robust stochastic control: A data-driven approach, IEEE Transactions on Automatic Control, (2020), 1-8.   Google Scholar

[196]

X. YangZ. Chen and J. Zhou, Optimality conditions for semi-infinite and generalized semi-infinite programs via lower order exact penalty functions, Journal of Optimization Theory and Applications, 169 (2016), 984-1012.  doi: 10.1007/s10957-016-0914-1.  Google Scholar

[197]

Y. Ye, Interior Point Algorithms: Theory and Analysis, John Wiley & Sons, 2011. doi: 10.1002/9781118032701.  Google Scholar

[198]

M. YildirimX. A. Sun and N. Z. Gebraeel, Sensor-driven condition-based generator maintenance scheduling-part i: Maintenance problem, IEEE Transactions on Power Systems, 31 (2016), 4253-4262.   Google Scholar

[199]

J. YueB. Chen and M.-C. Wang, Expected value of distribution information for the newsvendor problem, Operations Research, 54 (2006), 1128-1136.  doi: 10.1287/opre.1060.0318.  Google Scholar

[200]

Y. ZhangR. Jiang and S. Shen, Ambiguous chance-constrained binary programs under mean-covariance information, SIAM Journal on Optimization, 28 (2018), 2922-2944.  doi: 10.1137/17M1158707.  Google Scholar

[201]

Y. ZhangS. SongZ.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE transactions on intelligent transportation systems, 19 (2017), 1080-1090.   Google Scholar

[202]

A. ZhouM. YangM. Wang and Y. Zhang, A linear programming approximation of distributionally robust chance-constrained dispatch with wasserstein distance, IEEE Transactions on Power Systems, 35 (2020), 3366-3377.   Google Scholar

[203]

Z. ZhuJ. Zhang and Y. Ye, Newsvendor optimization with limited distribution information, Optimization Methods and Software, 28 (2013), 640-667.  doi: 10.1080/10556788.2013.768994.  Google Scholar

[204]

J. ZouS. Ahmed and X. A. Sun, Stochastic dual dynamic integer programming, Mathematical Programming, 175 (2019), 461-502.  doi: 10.1007/s10107-018-1249-5.  Google Scholar

[205]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

show all references

References:
[1]

F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, 95 (2003), 3-51.  doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory, Springer Science & Business Media, 2006.  Google Scholar

[3]

I. E. Bardakci and C. M. Lagoa, Distributionally robust portfolio optimization, in 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE, (2019), 1526–1531. doi: 10.1007/s11579-019-00241-1.  Google Scholar

[4]

G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, in The Operations Research Revolution, INFORMS, (2015), 1–19. Google Scholar

[5]

A. Ben-TalD. den HertogA. De WaegenaereB. Melenberg and G. Rennen, Robust solutions of optimization problems affected by uncertain probabilities, Management Science, 59 (2013), 341-357.   Google Scholar

[6] A. Ben-TalL. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, 2009.  doi: 10.1515/9781400831050.  Google Scholar
[7]

A. Ben-TalE. HazanT. Koren and S. Mannor, Oracle-based robust optimization via online learning, Operations Research, 63 (2015), 628-638.  doi: 10.1287/opre.2015.1374.  Google Scholar

[8]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[9]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations Research Letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[10]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424.  doi: 10.1007/PL00011380.  Google Scholar

[11]

D. BertsimasD. B. Brown and C. Caramanis, Theory and applications of robust optimization, SIAM Review, 53 (2011), 464-501.  doi: 10.1137/080734510.  Google Scholar

[12]

D. Bertsimas and M. S. Copenhaver, Characterization of the equivalence of robustification and regularization in linear and matrix regression, European Journal of Operational Research, 270 (2018), 931-942.  doi: 10.1016/j.ejor.2017.03.051.  Google Scholar

[13]

D. BertsimasX. V. DoanK. Natarajan and C.-P. Teo, Models for minimax stochastic linear optimization problems with risk aversion, Mathematics of Operations Research, 35 (2010), 580-602.  doi: 10.1287/moor.1100.0445.  Google Scholar

[14]

D. BertsimasV. Gupta and N. Kallus, Data-driven robust optimization, Mathematical Programming, 167 (2018), 235-292.  doi: 10.1007/s10107-017-1125-8.  Google Scholar

[15]

D. BertsimasV. Gupta and N. Kallus, Robust sample average approximation, Mathematical Programming, 171 (2018), 217-282.  doi: 10.1007/s10107-017-1174-z.  Google Scholar

[16]

D. Bertsimas and I. Popescu, Optimal inequalities in probability theory: A convex optimization approach, SIAM Journal on Optimization, 15 (2005), 780-804.  doi: 10.1137/S1052623401399903.  Google Scholar

[17]

D. Bertsimas and M. Sim, The price of robustness, Operations Research, 52 (2004), 35-53.  doi: 10.1287/opre.1030.0065.  Google Scholar

[18]

D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Mathematical Programming, 107 (2006), 5-36.  doi: 10.1007/s10107-005-0677-1.  Google Scholar

[19]

D. BertsimasM. Sim and M. Zhang, Adaptive distributionally robust optimization, Management Science, 65 (2019), 604-618.   Google Scholar

[20]

H.-G. Beyer and B. Sendhoff, Robust optimization–a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3190-3218.  doi: 10.1016/j.cma.2007.03.003.  Google Scholar

[21]

C. Bhattacharyya, Second order cone programming formulations for feature selection, Journal of Machine Learning Research, 5 (2004), 1417-1433.   Google Scholar

[22]

J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2nd edition, Springer Publishing Company, Incorporated, 2011. doi: 10.1007/978-1-4614-0237-4.  Google Scholar

[23]

J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis and L. Rey-Bellet, $(f, \gamma) $-divergences: Interpolating between $ f $-divergences and integral probability metrics, arXiv: 2011.05953. Google Scholar

[24]

J. Blanchet, L. Chen and X. Y. Zhou, Distributionally robust mean-variance portfolio selection with wasserstein distances, arXiv: 1802.04885. Google Scholar

[25]

J. Blanchet, P. W. Glynn, J. Yan and Z. Zhou, Multivariate distributionally robust convex regression under absolute error loss, arXiv: 1905.12231. Google Scholar

[26]

J. Blanchet and Y. Kang, Sample out-of-sample inference based on wasserstein distance, arXiv: 1605.01340. doi: 10.1287/opre.2020.2028.  Google Scholar

[27]

J. Blanchet and Y. Kang, Semi-supervised learning based on distributionally robust optimization, Data Analysis and Applications 3: Computational, Classification, Financial, Statistical and Stochastic Methods, 5 (2020), 1-33.   Google Scholar

[28]

J. BlanchetY. Kang and K. Murthy, Robust wasserstein profile inference and applications to machine learning, Journal of Applied Probability, 56 (2019), 830-857.  doi: 10.1017/jpr.2019.49.  Google Scholar

[29]

J. Blanchet, K. Murthy and N. Si, Confidence regions in wasserstein distributionally robust estimation, reprint arXiv: 1906.01614. Google Scholar

[30]

V. I. Bogachev and A. V. Kolesnikov, The monge-kantorovich problem: achievements, connections, and perspectives, Russian Mathematical Surveys, 67 (2012), 785-890.  doi: 10.1070/rm2012v067n05abeh004808.  Google Scholar

[31]

F. BolleyA. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probability Theory and Related Fields, 137 (2007), 541-593.  doi: 10.1007/s00440-006-0004-7.  Google Scholar

[32]

M. Breton and S. El Hachem, Algorithms for the solution of stochastic dynamic minimax problems, Computational Optimization and Applications, 4 (1995), 317-345.  doi: 10.1007/BF01300861.  Google Scholar

[33]

M. Breton and S. El Hachem, A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems, Stochastics and Stochastic Reports, 53 (1995), 305-322.  doi: 10.1080/17442509508833994.  Google Scholar

[34]

G. C. Calafiore, Ambiguous risk measures and optimal robust portfolios, SIAM Journal on Optimization, 18 (2007), 853-877.  doi: 10.1137/060654803.  Google Scholar

[35]

G. C. Calafiore and L. El Ghaoui, On distributionally robust chance-constrained linear programs, Journal of Optimization Theory and Applications, 130 (2006), 1-22.  doi: 10.1007/s10957-006-9084-x.  Google Scholar

[36]

R. Chen and I. C. Paschalidis, A robust learning approach for regression models based on distributionally robust optimization, Journal of Machine Learning Research, 19 (2018), 1-48.   Google Scholar

[37]

Y. ChenQ. GuoH. SunZ. LiW. Wu and Z. Li, A distributionally robust optimization model for unit commitment based on kullback–leibler divergence, IEEE Transactions on Power Systems, 33 (2018), 5147-5160.   Google Scholar

[38]

J. ChengE. Delage and A. Lisser, Distributionally robust stochastic knapsack problem, SIAM Journal on Optimization, 24 (2014), 1485-1506.  doi: 10.1137/130915315.  Google Scholar

[39]

J. ChengR. Li-Yang ChenH. N. NajmA. PinarC. Safta and J.-P. Watson, Distributionally robust optimization with principal component analysis, SIAM Journal on Optimization, 28 (2018), 1817-1841.  doi: 10.1137/16M1075910.  Google Scholar

[40]

A. Cherukuri and J. Cortés, Cooperative data-driven distributionally robust optimization, IEEE Transactions on Automatic Control, 65 (2019), 4400-4407.   Google Scholar

[41]

V. K. Chopra and W. T. Ziemba, The effect of errors in means, variances, and covariances on optimal portfolio choice, in Handbook of the Fundamentals of Financial Decision Making: Part I, World Scientific, (2013), 365–373. Google Scholar

[42]

K. L. ClarksonE. Hazan and D. P. Woodruff, Sublinear optimization for machine learning, Journal of the ACM (JACM), 59 (2012), 1-49.  doi: 10.1145/2371656.2371658.  Google Scholar

[43]

A. R. Conn and N. I. Gould, An exact penalty function for semi-infinite programming, Mathematical Programming, 37 (1987), 19-40.  doi: 10.1007/BF02591681.  Google Scholar

[44]

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd edition, Wiley Online Library, Hoboken, 2012.  Google Scholar

[45]

G. B. Dantzig, Linear programming under uncertainty, Management Science, 50 (2004), 1764-1769.  doi: 10.1287/opre.50.1.42.17798.  Google Scholar

[46]

E. del BarrioE. Giné and C. Matrán, Central limit theorems for the wasserstein distance between the empirical and the true distributions, Annals of Probability, 27 (1999), 1009-1071.  doi: 10.1214/aop/1022677394.  Google Scholar

[47]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Operations Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[48]

S. Dharmadhikari and K. Joag-Dev, Unimodality, Convexity, and Applications, Elsevier, 1988.  Google Scholar

[49]

T. DinhR. Fukasawa and J. Luedtke, Exact algorithms for the chance-constrained vehicle routing problem, Mathematical Programming, 172 (2018), 105-138.  doi: 10.1007/s10107-017-1151-6.  Google Scholar

[50]

X. V. DoanX. Li and K. Natarajan, Robustness to dependency in portfolio optimization using overlapping marginals, Operations Research, 63 (2015), 1468-1488.  doi: 10.1287/opre.2015.1424.  Google Scholar

[51]

N. DuY. Liu and Y. Liu, A new data-driven distributionally robust portfolio optimization method based on wasserstein ambiguity set, IEEE Access, 9 (2021), 3174-3194.   Google Scholar

[52]

C. DuanW. FangL. JiangL. Yao and J. Liu, Distributionally robust chance-constrained approximate ac-opf with wasserstein metric, IEEE Transactions on Power Systems, 33 (2018), 4924-4936.   Google Scholar

[53]

J. C. Duchi and H. Namkoong, Variance-based regularization with convex objectives, Journal of Machine Learning Research, 20 (2019), 1-68.   Google Scholar

[54]

J. C. Duchi and H. Namkoong, Learning models with uniform performance via distributionally robust optimization, arXiv: 1810.08750. doi: 10.1214/20-aos2004.  Google Scholar

[55]

J. Dupačová, Stochastic programming: Minimax approach, Encyclopedia Optimization, 5 (2001), 327-330.   Google Scholar

[56]

J. Dupačová, On minimax solutions of stochastic linear programming problems, Časopis Pro Pěstování Matematiky, 091 (1966), 423–430.  Google Scholar

[57]

J. Dupačová, The minimax approach to stochastic programming and an illustrative application, Stochastics, 20 (1987), 73-88.  doi: 10.1080/17442508708833436.  Google Scholar

[58]

J. Dupačová, Uncertainties in minimax stochastic programs, Optimization, 60 (2011), 1235-1250.  doi: 10.1080/02331934.2010.532214.  Google Scholar

[59]

L. El Ghaoui and H. Lebret, Robust solutions to least-squares problems with uncertain data, SIAM Journal on Matrix Analysis and Applications, 18 (1997), 1035-1064.  doi: 10.1137/S0895479896298130.  Google Scholar

[60]

E. Erdoğan and G. Iyengar, Ambiguous chance constrained problems and robust optimization, Mathematical Programming, 107 (2006), 37-61.  doi: 10.1007/s10107-005-0678-0.  Google Scholar

[61]

P. M. Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations, Mathematical Programming, 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1.  Google Scholar

[62]

F. Farnia and D. Tse, A minimax approach to supervised learning, Advances in Neural Information Processing Systems, 29 (2016), 4240-4248.   Google Scholar

[63]

L. FauryU. TanielianE. DohmatobE. Smirnova and F. Vasile, Distributionally robust counterfactual risk minimization, Proceedings of the AAAI Conference on Artificial Intelligence, 34 (2020), 3850-3857.   Google Scholar

[64]

N. Fournier and A. Guillin, On the rate of convergence in wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.  Google Scholar

[65]

D. Fouskakis and D. Draper, Stochastic optimization: a review, International Statistical Review, 70 (2002), 315-349.   Google Scholar

[66]

C. FrognerS. ClaiciE. Chien and J. Solomon, Incorporating unlabeled data into distributionally robust learning, Journal of Machine Learning Research, 22 (2021), 1-46.   Google Scholar

[67]

V. GabrelC. Murat and A. Thiele, Recent advances in robust optimization: An overview, European Journal of Operational Research, 235 (2014), 471-483.  doi: 10.1016/j.ejor.2013.09.036.  Google Scholar

[68]

G. Gallego and I. Moon, The distribution free newsboy problem: review and extensions, Journal of the Operational Research Society, 44 (1993), 825-834.   Google Scholar

[69]

R. Gao, Finite-sample guarantees for wasserstein distributionally robust optimization: Breaking the curse of dimensionality, arXiv: 2009.04382. Google Scholar

[70]

R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with wasserstein distance, arXiv: 1604.02199. Google Scholar

[71]

R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with dependence structure, arXiv: 1701.04200. Google Scholar

[72]

R. Gao, L. Xie, Y. Xie and H. Xu, Robust hypothesis testing using wasserstein uncertainty sets, in NeurIPS, (2018), 7913–7923. Google Scholar

[73]

L. E. GhaouiM. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.  Google Scholar

[74]

S. Ghosal and W. Wiesemann, The distributionally robust chance-constrained vehicle routing problem, Operations Research, 68 (2020), 716-732.  doi: 10.1287/opre.2019.1924.  Google Scholar

[75]

A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, International Statistical Review, 70 (2002), 419-435.   Google Scholar

[76]

M. Á. Goberna and M. A. López, Semi-Infinite Programming: Recent Advances, Springer Science & Business Media, 2013. doi: 10.1007/0-387-26771-9_1.  Google Scholar

[77]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operations Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.  Google Scholar

[78]

I. J. Goodfellow, J. Shlens and C. Szegedy, Explaining and harnessing adversarial examples, arXiv: 1412.6572. Google Scholar

[79]

S. GuoH. Xu and L. Zhang, Probability approximation schemes for stochastic programs with distributionally robust second-order dominance constraints, Optimization Methods and Software, 32 (2017), 770-789.  doi: 10.1080/10556788.2016.1175003.  Google Scholar

[80]

V. Gupta, Near-optimal bayesian ambiguity sets for distributionally robust optimization, Management Science, 65 (2019), 4242-4260.   Google Scholar

[81]

S.-Å. Gustafson, Semi-infinite programming: Methods for linear problems, in Encyclopedia of Optimization, Springer, (2009), 3424–3429. Google Scholar

[82]

G. A. Hanasusanto and D. Kuhn, Conic programming reformulations of two-stage distributionally robust linear programs over wasserstein balls, Operations Research, 66 (2018), 849-869.  doi: 10.1287/opre.2017.1698.  Google Scholar

[83]

G. A. HanasusantoD. KuhnS. W. Wallace and S. Zymler, Distributionally robust multi-item newsvendor problems with multimodal demand distributions, Mathematical Programming, 152 (2015), 1-32.  doi: 10.1007/s10107-014-0776-y.  Google Scholar

[84]

G. A. HanasusantoV. RoitchD. Kuhn and W. Wiesemann, A distributionally robust perspective on uncertainty quantification and chance constrained programming, Mathematical Programming, 151 (2015), 35-62.  doi: 10.1007/s10107-015-0896-z.  Google Scholar

[85]

R. Hettich, A. Kaplan and R. Tichatschke, Semi-infinite programming: Numerical methods, in Encyclopedia of Optimization, Springer, (2009), 3429–3434. Google Scholar

[86]

R. Hettich and K. O. Kortanek, Semi-infinite programming: theory, methods, and applications, SIAM Review, 35 (1993), 380-429.  doi: 10.1137/1035089.  Google Scholar

[87]

Z. Hu and L. J. Hong, Kullback-leibler divergence constrained distributionally robust optimization, Available at Optimization Online. Google Scholar

[88]

K. HuangH. YangI. KingM. R. Lyu and L. Chan, The minimum error minimax probability machine, Journal of Machine Learning Research, 5 (2004), 1253-1286.   Google Scholar

[89]

G. Infanger, Planning under uncertainty solving large-scale stochastic linear programs, Technical report, Stanford University, 1992. Google Scholar

[90]

K. Isii, On sharpness of tchebycheff-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962), 185-197.  doi: 10.1007/BF02868641.  Google Scholar

[91]

R. Ji and M. A. Lejeune, Data-driven distributionally robust chance-constrained optimization with wasserstein metric, Journal of Global Optimization, (2020), 1–33. doi: 10.1007/s10898-020-00966-0.  Google Scholar

[92]

R. Jiang and Y. Guan, Data-driven chance constrained stochastic program, Mathematical Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7.  Google Scholar

[93]

R. Jiang and Y. Guan, Risk-averse two-stage stochastic program with distributional ambiguity, Operations Research, 66 (2018), 1390-1405.  doi: 10.1287/opre.2018.1729.  Google Scholar

[94]

R. Jiang, M. Ryu and G. Xu, Data-driven distributionally robust appointment scheduling over wasserstein balls, arXiv: 1907.03219. Google Scholar

[95]

P. Kall, S. W. Wallace and P. Kall, Stochastic Programming, Springer, 1994.  Google Scholar

[96]

Z. KangX. LiZ. Li and S. Zhu, Data-driven robust mean-cvar portfolio selection under distribution ambiguity, Quantitative Finance, 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057.  Google Scholar

[97]

L. V. Kantorovich, On the translocation of masses, Journal of Mathematical Sciences, 133 (2006), 1381-1382.  doi: 10.1007/s10958-006-0049-2.  Google Scholar

[98]

D. KlabjanD. Simchi-Levi and M. Song, Robust stochastic lot-sizing by means of histograms, Production and Operations Management, 22 (2013), 691-710.   Google Scholar

[99]

Ç. KoçyiğitG. IyengarD. Kuhn and W. Wiesemann, Distributionally robust mechanism design, Management Science, 66 (2020), 159-189.   Google Scholar

[100]

D. Kuhn, P. M. Esfahani, V. A. Nguyen and S. Shafieezadeh-Abadeh, Wasserstein distributionally robust optimization: Theory and applications in machine learning, in Operations Research & Management Science in the Age of Analytics, INFORMS, (2019), 130–166. doi: 10.3770/j.issn:2095-2651.2021.01.010.  Google Scholar

[101]

S. Kullback, Information Theory and Statistics, Courier Corporation, Mineola, 1997.  Google Scholar

[102]

G. Lanckriet, L. E. Ghaoui, C. Bhattacharyya and M. I. Jordan, Minimax probability machine, in Advances in neural information processing systems, (2001), 801–807. Google Scholar

[103]

G. R. LanckrietL. E. GhaouiC. Bhattacharyya and M. I. Jordan, A robust minimax approach to classification, Journal of Machine Learning Research, 3 (2002), 555-582.  doi: 10.1162/153244303321897726.  Google Scholar

[104]

H. J. Landau, Maximum entropy and the moment problem, Bulletin of the American Mathematical Society, 16 (1987), 47-77.  doi: 10.1090/S0273-0979-1987-15464-4.  Google Scholar

[105]

C. Lee and S. Mehrotra, A distributionally-robust approach for finding support vector machines, 2015. Google Scholar

[106]

S. Lee, H. Kim and I. Moon, A data-driven distributionally robust newsvendor model with a wasserstein ambiguity set, Journal of the Operational Research Society, (2020), 1–19. Google Scholar

[107]

D. Levy, Y. Carmon, J. C. Duchi and A. Sidford, Large-scale methods for distributionally robust optimization, Advances in Neural Information Processing Systems, 33. Google Scholar

[108]

A. S. Lewis and C. J. Pang, Lipschitz behavior of the robust regularization, SIAM Journal on Control and Optimization, 48 (2010), 3080-3104.  doi: 10.1137/08073682X.  Google Scholar

[109]

J. Y. Li and R. H. Kwon, Portfolio selection under model uncertainty: a penalized moment-based optimization approach, Journal of Global Optimization, 56 (2013), 131-164.  doi: 10.1007/s10898-012-9969-1.  Google Scholar

[110]

G. D. Lin, Recent developments on the moment problem, Journal of Statistical Distributions and Applications, 4 (2017), 1-17.   Google Scholar

[111]

Q. LinR. LoxtonK. L. TeoY. H. Wu and C. Yu, A new exact penalty method for semi-infinite programming problems, Journal of Computational and Applied Mathematics, 261 (2014), 271-286.  doi: 10.1016/j.cam.2013.11.010.  Google Scholar

[112]

J. LiuZ. ChenA. Lisser and Z. Xu, Closed-form optimal portfolios of distributionally robust mean-cvar problems with unknown mean and variance, Applied Mathematics & Optimization, 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y.  Google Scholar

[113]

Y. LiuR. Meskarian and H. Xu, Distributionally robust reward-risk ratio optimization with moment constraints, SIAM Journal on Optimization, 27 (2017), 957-985.  doi: 10.1137/16M106114X.  Google Scholar

[114]

F. Luo and S. Mehrotra, Decomposition algorithm for distributionally robust optimization using wasserstein metric with an application to a class of regression models, European Journal of Operational Research, 278 (2019), 20-35.  doi: 10.1016/j.ejor.2019.03.008.  Google Scholar

[115]

C. Lyu, K. Huang and H.-N. Liang, A unified gradient regularization family for adversarial examples, in 2015 IEEE International Conference on Data Mining, IEEE, (2015), 301–309.  Google Scholar

[116]

A. MajumdarG. Hall and A. A. Ahmadi, Recent scalability improvements for semidefinite programming with applications in machine learning, control, and robotics, Annual Review of Control, Robotics, and Autonomous Systems, 3 (2020), 331-360.   Google Scholar

[117]

S. Mehrotra and D. Papp, A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization, SIAM Journal on Optimization, 24 (2014), 1670-1697.  doi: 10.1137/130925013.  Google Scholar

[118]

S. Mehrotra and H. Zhang, Models and algorithms for distributionally robust least squares problems, Mathematical Programming, 146 (2014), 123-141.  doi: 10.1007/s10107-013-0681-9.  Google Scholar

[119]

R. O. Michaud, The markowitz optimization enigma: Is 'optimized' optimal?, Financial Analysts Journal, 45 (1989), 31-42.   Google Scholar

[120]

H. Mostafaei and S. Kordnourie, Probability metrics and their applications, Applied Mathematical Sciences, 5 (2011), 181-192.   Google Scholar

[121]

H. Namkoong and J. C. Duchi, Stochastic gradient methods for distributionally robust optimization with f-divergences, NIPS, 29 (2016), 2208-2216.   Google Scholar

[122]

K. NatarajanM. Sim and J. Uichanco, Tractable robust expected utility and risk models for portfolio optimization, Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 20 (2010), 695-731.  doi: 10.1111/j.1467-9965.2010.00417.x.  Google Scholar

[123]

K. Natarajan and C.-P. Teo, On reduced semidefinite programs for second order moment bounds with applications, Mathematical Programming, 161 (2017), 487-518.  doi: 10.1007/s10107-016-1019-1.  Google Scholar

[124]

A. NemirovskiA. JuditskyG. Lan and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization, 19 (2009), 1574-1609.  doi: 10.1137/070704277.  Google Scholar

[125]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM Journal on Optimization, 17 (2007), 969-996.  doi: 10.1137/050622328.  Google Scholar

[126]

V. A. Nguyen, D. Kuhn and P. M. Esfahani, Distributionally robust inverse covariance estimation: The wasserstein shrinkage estimator, arXiv: 1805.07194. Google Scholar

[127]

V. A. Nguyen, S. Shafieezadeh-Abadeh, D. Kuhn and P. M. Esfahani, Bridging bayesian and minimax mean square error estimation via wasserstein distributionally robust optimization, arXiv: 1911.03539. Google Scholar

[128] A. B. Owen, Empirical Likelihood, CRC Press, 2001.   Google Scholar
[129] L. Pardo, Statistical Inference Based on Divergence Measures, CRC Press, 2018.   Google Scholar
[130]

G. C. Pflug and A. Pichler, Approximations for probability distributions and stochastic optimization problems, in Stochastic Optimization Methods in Finance and Energy, Springer, 2011,343–387. doi: 10.1007/978-1-4419-9586-5_15.  Google Scholar

[131]

R. R. Phelps, Lectures on Choquet's Theorem, Springer Science & Business Media, 2001. doi: 10.1007/b76887.  Google Scholar

[132]

A. B. PhilpottV. L. de Matos and L. Kapelevich, Distributionally robust sddp, Computational Management Science, 15 (2018), 431-454.  doi: 10.1007/s10287-018-0314-0.  Google Scholar

[133]

I. Popescu, A semidefinite programming approach to optimal-moment bounds for convex classes of distributions, Mathematics of Operations Research, 30 (2005), 632-657.  doi: 10.1287/moor.1040.0137.  Google Scholar

[134]

I. Popescu, Robust mean-covariance solutions for stochastic optimization, Operations Research, 55 (2007), 98-112.  doi: 10.1287/opre.1060.0353.  Google Scholar

[135]

K. PostekD. den Hertog and B. Melenberg, Computationally tractable counterparts of distributionally robust constraints on risk measures, SIAM Review, 58 (2016), 603-650.  doi: 10.1137/151005221.  Google Scholar

[136] M. A. Proschan and P. A. Shaw, Essentials of Probability Theory for Statisticians, CRC Press, 2018.   Google Scholar
[137]

H. RahimianG. Bayraksan and T. Homem-de Mello, Identifying effective scenarios in distributionally robust stochastic programs with total variation distance, Mathematical Programming, 173 (2019), 393-430.  doi: 10.1007/s10107-017-1224-6.  Google Scholar

[138]

H. Rahimian and S. Mehrotra, Distributionally robust optimization: A review, arXiv: 1908.05659. Google Scholar

[139]

M. Riis and K. A. Andersen, Applying the minimax criterion in stochastic recourse programs, European Journal of Operational Research, 165 (2005), 569-584.  doi: 10.1016/j.ejor.2003.09.033.  Google Scholar

[140] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.   Google Scholar
[141]

Y. RubnerC. Tomasi and L. J. Guibas, The earth mover's distance as a metric for image retrieval, International Journal of Computer Vision, 40 (2000), 99-121.  doi: 10.1007/3-540-46238-4_2.  Google Scholar

[142]

N. RujeerapaiboonD. Kuhn and W. Wiesemann, Robust growth-optimal portfolios, Management Science, 62 (2016), 2090-2109.   Google Scholar

[143]

N. RujeerapaiboonD. Kuhn and W. Wiesemann, Chebyshev inequalities for products of random variables, Mathematics of Operations Research, 43 (2018), 887-918.  doi: 10.1287/moor.2017.0888.  Google Scholar

[144]

L. Rüschendorf, Bounds for distributions with multivariate marginals, Lecture Notes-Monograph Series, (1991), 285–310. doi: 10.1214/lnms/1215459862.  Google Scholar

[145]

H. Scarf, A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production. Google Scholar

[146]

S. Shafieezadeh-AbadehD. Kuhn and P. M. Esfahani, Regularization via mass transportation, Journal of Machine Learning Research, 20 (2019), 1-68.   Google Scholar

[147]

S. Shafieezadeh AbadehP. M. Mohajerin Esfahani and D. Kuhn, Distributionally robust logistic regression, Advances in Neural Information Processing Systems, 28 (2015), 1576-1584.   Google Scholar

[148]

S. Shafieezadeh-Abadeh, V. A. Nguyen, D. Kuhn and P. M. Esfahani, Wasserstein distributionally robust kalman filtering, in Advances in Neural Information Processing Systems, vol. 31, Curran Associates, Inc., 2018, 8474–8483. Google Scholar

[149]

U. ShahamY. Yamada and S. Negahban, Understanding adversarial training: Increasing local stability of supervised models through robust optimization, Neurocomputing, 307 (2018), 195-204.   Google Scholar

[150]

S. Shalev-Shwartz and Y. Wexler, Minimizing the maximal loss: How and why, in International Conference on Machine Learning, PMLR, 2016,793–801. Google Scholar

[151]

C. Shang and F. You, Distributionally robust optimization for planning and scheduling under uncertainty, Computers & Chemical Engineering, 110 (2018), 53-68.   Google Scholar

[152]

A. Shapiro, On duality theory of conic linear problems, in Semi-infinite Programming, Springer, Boston, MA, 2001,135–165. doi: 10.1007/978-1-4757-3403-4_7.  Google Scholar

[153]

A. Shapiro, Worst-case distribution analysis of stochastic programs, Mathematical Programming, 107 (2006), 91-96.  doi: 10.1007/s10107-005-0680-6.  Google Scholar

[154]

A. Shapiro, Semi-infinite programming, duality, discretization and optimality conditions, Optimization, 58 (2009), 133-161.  doi: 10.1080/02331930902730070.  Google Scholar

[155]

A. Shapiro, Distributionally robust stochastic programming, SIAM Journal on Optimization, 27 (2017), 2258-2275.  doi: 10.1137/16M1058297.  Google Scholar

[156]

A. Shapiro, Tutorial on risk neutral, distributionally robust and risk averse multistage stochastic programming, European Journal of Operational Research, 288 (2020), 1-13.  doi: 10.1016/j.ejor.2020.03.065.  Google Scholar

[157]

A. Shapiro and S. Ahmed, On a class of minimax stochastic programs, SIAM Journal on Optimization, 14 (2004), 1237-1249.  doi: 10.1137/S1052623403434012.  Google Scholar

[158]

A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, 2nd edition, SIAM, Philadelphia, PA, 2014. Google Scholar

[159]

A. Shapiro and A. Kleywegt, Minimax analysis of stochastic problems, Optimization Methods and Software, 17 (2002), 523-542.  doi: 10.1080/1055678021000034008.  Google Scholar

[160]

K. S. ShehadehA. E. Cohn and R. Jiang, A distributionally robust optimization approach for outpatient colonoscopy scheduling, European Journal of Operational Research, 283 (2020), 549-561.  doi: 10.1016/j.ejor.2019.11.039.  Google Scholar

[161]

A. Sinha, H. Namkoong and J. Duchi, Certifying some distributional robustness with principled adversarial training, in International Conference on Learning Representations, 2018. Google Scholar

[162]

J. E. Smith, Generalized chebychev inequalities: theory and applications in decision analysis, Operations Research, 43 (1995), 807-825.  doi: 10.1287/opre.43.5.807.  Google Scholar

[163]

J. E. Smith and R. L. Winkler, The optimizer's curse: Skepticism and postdecision surprise in decision analysis, Management Science, 52 (2006), 311-322.   Google Scholar

[164]

A. M.-C. So, Moment inequalities for sums of random matrices and their applications in optimization, Mathematical Programming, 130 (2011), 125-151.  doi: 10.1007/s10107-009-0330-5.  Google Scholar

[165]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 21 (1973), 1154-1157.  doi: 10.1287/opre.22.4.892.  Google Scholar

[166] A. Spanos, Probability Theory and Statistical Inference: Econometric Modeling with Observational Data, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511754081.  Google Scholar
[167]

G. Still, Optimization problems with infinitely many constraints, Buletinul tiinific al Universitatii Baia Mare, Seria B, Fascicola matematică-informatică, 18 (2002), 343–354.  Google Scholar

[168]

T. Strohmann and G. Z. Grudic, A formulation for minimax probability machine regression, in NIPS, Citeseer, 2002,769–776. Google Scholar

[169]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equilibrium problems, Mathematics of Operations Research, 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.  Google Scholar

[170]

S. Takriti and S. Ahmed, Managing short-term electricity contracts under uncertainty: A minimax approach, 2002. Google Scholar

[171]

B. P. Van Parys, P. M. Esfahani and D. Kuhn, From data to decisions: Distributionally robust optimization is optimal, Management Science, (2020), preprint. Google Scholar

[172]

B. P. Van ParysP. J. Goulart and D. Kuhn, Generalized gauss inequalities via semidefinite programming, Mathematical Programming, 156 (2016), 271-302.  doi: 10.1007/s10107-015-0878-1.  Google Scholar

[173]

B. P. Van ParysP. J. Goulart and M. Morari, Distributionally robust expectation inequalities for structured distributions, Mathematical Programming, 173 (2019), 251-280.  doi: 10.1007/s10107-017-1220-x.  Google Scholar

[174]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95.  doi: 10.1137/1038003.  Google Scholar

[175]

L. VandenbergheS. Boyd and K. Comanor, Generalized chebyshev bounds via semidefinite programming, SIAM Review, 49 (2007), 52-64.  doi: 10.1137/S0036144504440543.  Google Scholar

[176]

C. Villani, Optimal Transport: Old and New, vol. 338, Springer Science & Business Media, 2008. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[177]

M. R. Wagner, Stochastic 0–1 linear programming under limited distributional information, Operations Research Letters, 36 (2008), 150-156.  doi: 10.1016/j.orl.2007.07.003.  Google Scholar

[178]

A. Wald, Statistical decision functions which minimize the maximum risk, Annals of Mathematics, 46 (1945), 265-280.  doi: 10.2307/1969022.  Google Scholar

[179]

A. Wald, Statistical decision functions, in Breakthroughs in Statistics, Springer, 1992,342–357. Google Scholar

[180]

C. WangR. GaoF. QiuJ. Wang and L. Xin, Risk-based distributionally robust optimal power flow with dynamic line rating, IEEE Transactions on Power Systems, 33 (2018), 6074-6086.   Google Scholar

[181]

C. WangR. GaoW. WeiM. Shafie-khahT. Bi and J. P. Catalao, Risk-based distributionally robust optimal gas-power flow with wasserstein distance, IEEE Transactions on Power Systems, 34 (2018), 2190-2204.   Google Scholar

[182]

S. Wang and Y. Yuan, Feasible method for semi-infinite programs, SIAM Journal on Optimization, 25 (2015), 2537-2560.  doi: 10.1137/140982143.  Google Scholar

[183]

Z. WangK. YouS. Song and Y. Zhang, Wasserstein distributionally robust shortest path problem, European Journal of Operational Research, 284 (2020), 31-43.  doi: 10.1016/j.ejor.2020.01.009.  Google Scholar

[184]

Z. WangP. W. Glynn and Y. Ye, Likelihood robust optimization for data-driven problems, Computational Management Science, 13 (2016), 241-261.  doi: 10.1007/s10287-015-0240-3.  Google Scholar

[185]

W. WiesemannD. Kuhn and B. Rustem, Robust markov decision processes, Mathematics of Operations Research, 38 (2013), 153-183.  doi: 10.1287/moor.1120.0566.  Google Scholar

[186]

W. WiesemannD. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[187]

L. A. Wolsey, Integer Programming, Wiley Online Library, 1998.  Google Scholar

[188]

D. Wozabal, Robustifying convex risk measures for linear portfolios: A nonparametric approach, Operations Research, 62 (2014), 1302-1315.  doi: 10.1287/opre.2014.1323.  Google Scholar

[189]

H. XuC. Caramanis and S. Mannor, Robustness and regularization of support vector machines, Journal of Machine Learning Research, 10 (2009), 1485-1510.   Google Scholar

[190]

H. XuC. Caramanis and S. Mannor, Robust regression and lasso, IEEE Transactions on Information Theory, 56 (2010), 3561-3574.  doi: 10.1109/TIT.2010.2048503.  Google Scholar

[191]

H. XuC. Caramanis and S. Mannor, A distributional interpretation of robust optimization, Mathematics of Operations Research, 37 (2012), 95-110.  doi: 10.1287/moor.1110.0531.  Google Scholar

[192]

H. XuY. Liu and H. Sun, Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods, Mathematical Programming, 169 (2018), 489-529.  doi: 10.1007/s10107-017-1143-6.  Google Scholar

[193]

M. XuS.-Y. Wu and J. Y. Jane, Solving semi-infinite programs by smoothing projected gradient method, Computational Optimization and Applications, 59 (2014), 591-616.  doi: 10.1007/s10589-014-9654-z.  Google Scholar

[194]

I. Yang, A convex optimization approach to distributionally robust markov decision processes with wasserstein distance, IEEE Control Systems Letters, 1 (2017), 164-169.   Google Scholar

[195]

I. Yang, Wasserstein distributionally robust stochastic control: A data-driven approach, IEEE Transactions on Automatic Control, (2020), 1-8.   Google Scholar

[196]

X. YangZ. Chen and J. Zhou, Optimality conditions for semi-infinite and generalized semi-infinite programs via lower order exact penalty functions, Journal of Optimization Theory and Applications, 169 (2016), 984-1012.  doi: 10.1007/s10957-016-0914-1.  Google Scholar

[197]

Y. Ye, Interior Point Algorithms: Theory and Analysis, John Wiley & Sons, 2011. doi: 10.1002/9781118032701.  Google Scholar

[198]

M. YildirimX. A. Sun and N. Z. Gebraeel, Sensor-driven condition-based generator maintenance scheduling-part i: Maintenance problem, IEEE Transactions on Power Systems, 31 (2016), 4253-4262.   Google Scholar

[199]

J. YueB. Chen and M.-C. Wang, Expected value of distribution information for the newsvendor problem, Operations Research, 54 (2006), 1128-1136.  doi: 10.1287/opre.1060.0318.  Google Scholar

[200]

Y. ZhangR. Jiang and S. Shen, Ambiguous chance-constrained binary programs under mean-covariance information, SIAM Journal on Optimization, 28 (2018), 2922-2944.  doi: 10.1137/17M1158707.  Google Scholar

[201]

Y. ZhangS. SongZ.-J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE transactions on intelligent transportation systems, 19 (2017), 1080-1090.   Google Scholar

[202]

A. ZhouM. YangM. Wang and Y. Zhang, A linear programming approximation of distributionally robust chance-constrained dispatch with wasserstein distance, IEEE Transactions on Power Systems, 35 (2020), 3366-3377.   Google Scholar

[203]

Z. ZhuJ. Zhang and Y. Ye, Newsvendor optimization with limited distribution information, Optimization Methods and Software, 28 (2013), 640-667.  doi: 10.1080/10556788.2013.768994.  Google Scholar

[204]

J. ZouS. Ahmed and X. A. Sun, Stochastic dual dynamic integer programming, Mathematical Programming, 175 (2019), 461-502.  doi: 10.1007/s10107-018-1249-5.  Google Scholar

[205]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

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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