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A survey of decision making and optimization under uncertainty

  • S.I.: Integrated Uncertainty in Knowledge Modelling & Decision Making 2018
  • Published:
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Abstract

Recent advances in decision making have incorporated both risk and ambiguity in decision theory and optimization methods. These methods implement a variety of uncertainty representations from probabilistic and non-probabilistic foundations, including traditional probability theory, sets of probability measures, uncertainty sets, ambiguity sets, possibility theory, evidence theory, fuzzy measures, and imprecise probability. The choice of uncertainty representation impacts the expressiveness and tractability of the decision models. We survey recent approaches for representing uncertainty in both decision making and optimization to clarify the trade-offs among the alternative representations. Robust and distributionally robust optimization are surveyed, with particular attention to standard form ambiguity sets. Applications of uncertainty and decision models are also reviewed, with a focus on recent optimization applications. These applications highlight common practices and potential research gaps. The intersection of behavioral decision making and robust optimization is a promising area for future research and there is also opportunity for further advances in distributionally robust optimization in sequential and multi-agent settings.

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References

  • Aberdeen, D. (2003). A (revised) survey of approximate methods for solving partially observable Markov decision processes. Technical report. National ICT Australia, Canberra.

  • Aghassi, M., & Bertsimas, D. (2006). Robust game theory. Mathematical Programming, 107(1), 231–273.

    Google Scholar 

  • Agra, A., Santos, M. C., Nace, D., & Poss, M. (2016). A dynamic programming approach for a class of robust optimization problems. SIAM Journal on Optimization, 26(3), 1799–1823.

    Google Scholar 

  • Ahipaşaoğlu, S. D., Meskarian, R., Magnanti, T. L., & Natarajan, K. (2015). Beyond normality: A cross moment-stochastic user equilibrium model. Transportation Research Part B: Methodological, 81, 333–354. https://doi.org/10.1016/j.trb.2015.01.005.

    Article  Google Scholar 

  • Ahmed, A., Varakantham, P., Lowalekar, M., Adulyasak, Y., & Jaillet, P. (2017). Sampling based approaches for minimizing regret in uncertain Markov decision processes (MDPs). Journal of Artificial Intelligence Research, 59, 229–264.

    Google Scholar 

  • Alismail, F., Xiong, P., & Singh, C. (2018). Optimal wind farm allocation in multi-area power systems using distributionally robust optimization approach. IEEE Transactions on Power Systems, 33(1), 536.

    Google Scholar 

  • Aoki, M. (1965). Optimal control of partially observable Markovian systems. Journal of The Franklin Institute, 280(5), 367–386.

    Google Scholar 

  • Arad, A., & Gayer, G. (2012). Imprecise data sets as a source of ambiguity: A model and experimental evidence. Management Science, 58(1), 188–202. https://doi.org/10.1287/mnsc.1110.1463.

    Article  Google Scholar 

  • Arrow, K. J., & Hurwicz, L. (1972). An optimality criterion for decision making under ignorance. In C. Carter & J. Ford (Eds.), Uncertainty and expectations in economics, B (pp. 1–11). Oxford: Blackwell.

    Google Scholar 

  • Artzner, P., Delbaen, F., Eber, J. M., Heath, D., & Ku, H. (2007). Coherent multiperiod risk adjusted values and Bellman’s principle. Annals of Operations Research, 152(1), 5–22. https://doi.org/10.1007/s10479-006-0132-6.

    Article  Google Scholar 

  • Astrom, K. J. (1965). Optimal control of Markov processes with incomplete state information. Journal of Mathematical Analysis and Applications, 10(1), 174–205. https://doi.org/10.1016/0022-247X(65)90154-X.

    Article  Google Scholar 

  • Augustin, T., Coolen, F., de Cooman, G., & Troffaes, M. (Eds.). (2014). Introduction to imprecise probability. Hoboken: Wiley.

    Google Scholar 

  • Baillon, A., Huang, Z., Selim, A., & Wakker, P. P. (2018). Measuring ambiguity attitudes for all (natural) events. Econometrica, 86(5), 1839–1858. https://doi.org/10.3982/ecta14370.

    Article  Google Scholar 

  • Bai, M., & Yang, Z. (2014). Distributionally robust self-scheduling optimization with CO\(_2\) emissions constraints under uncertainty of prices. Journal of Applied Mathematics, 2014, 1–7.

    Google Scholar 

  • Barberis, N. C. (2013). Thirty years of prospect theory in economics: A review and assessment. Journal of Economic Perspectives, 27(1), 173–196. https://doi.org/10.1257/jep.27.1.173.

    Article  Google Scholar 

  • Bazovkin, P., & Mosler, K. (2015). A general solution for robust linear programs with distortion risk constraints. Annals of Operations Research, 229(1), 103–120. https://doi.org/10.1007/s10479-015-1786-8.

    Article  Google Scholar 

  • Beale, E. M. L. (1955). On minizing a convex function subject to linear inequalities. Journal of the Royal Statistical Society Series B, 17, 173–184.

    Google Scholar 

  • Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17(4), B141–B164.

    Google Scholar 

  • Ben-Haim, Y. (2006). Info-Gap decision theory: Decisions under severe uncertainty. Oxford: Academic Press.

    Google Scholar 

  • Ben-Tal, A., Bertsimas, D., & Brown, D. B. (2010). A soft robust model for optimization under ambiguity. Operations Research, 58(4–part–2), 1220–1234. https://doi.org/10.1287/opre.1100.0821.

    Article  Google Scholar 

  • Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., & Rennen, G. (2013). Robust solutions of optimization problems affected by uncertain probabilities. Management Science, 59(2), 341–357. https://doi.org/10.1287/mnsc.1120.1641.

    Article  Google Scholar 

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

    Google Scholar 

  • Berger, J. O., Moreno, E., Pericchi, L. R., Bayarri, M. J., Bernardo, J. M., Cano, J. A., et al. (1994). An overview of robust Bayesian analysis. Test, 3(1), 5–124.

    Google Scholar 

  • Bertsimas, D., & Brown, D. B. (2009). Constructing uncertainty sets for robust linear optimization. Operations Research, 57(6), 1483–1495. https://doi.org/10.1287/opre.1080.0646.

    Article  Google Scholar 

  • Bertsimas, D., Brown, D. B., & Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464–501.

    Google Scholar 

  • Bertsimas, D., & Goyal, V. (2010). On the power of robust solutions in two-stage stochastic and adaptive optimization problems. Mathematics of Operations Research, 35(2), 284–305. https://doi.org/10.1287/moor.1090.0440.

    Article  Google Scholar 

  • Bertsimas, D., Gupta, V., & Kallus, N. (2018a). Data-driven robust optimization. Mathematical Programming, 167(2), 235–292. https://doi.org/10.1007/s10107-017-1125-8.

    Article  Google Scholar 

  • Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53. https://doi.org/10.1287/opre.1030.0065.

    Article  Google Scholar 

  • Bertsimas, D., Sim, M., & Zhang, M. (2018b). Adaptive distributionally robust optimization. Management Science, 65(2), 604–618.

    Google Scholar 

  • Bertuccelli, L. F., Bethke, B., & How, J. P. (2009). Robust adaptive Markov decision processes in multi-vehicle applications. In 2009 American control conference (pp. 1304–1309). https://doi.org/10.1109/ACC.2009.5160511.

  • Binmore, K., Stewart, L., & Voorhoeve, A. (2012). How much ambiguity aversion? Finding indifferences between Ellsberg’s risky and ambiguous bets. Journal of Risk and Uncertainty, 45(3), 215–238.

    Google Scholar 

  • Boloori, A., & Cook, C. B. (2017). Data-driven management of post-transplant medications: An APOMDP approach. HKS Working Paper RWP17-036.

  • Camerer, C., & Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. Journal of Risk and Uncertainty, 5(4), 325–370.

    Google Scholar 

  • Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., & Montrucchio, L. (2011). Uncertainty averse preferences. Journal of Economic Theory, 146(4), 1275–1330. https://doi.org/10.1016/j.jet.2011.05.006.

    Article  Google Scholar 

  • Chateauneuf, A., Eichberger, J., & Grant, S. (2007). Choice under uncertainty with the best and worst in mind: Neo-additive capacities. Journal of Economic Theory, 137(1), 538–567.

    Google Scholar 

  • Chen, Z., Yu, P., & Haskell, W. B. (2018). Distributionally robust optimization for sequential decision making. arXiv Preprint.

  • Choquet, G. (1954). Theory of capacities. Annales de l’Institut Fourier, 5, 131–295.

    Google Scholar 

  • Claßen, G., Koster, A. M. C., & Schmeink, A. (2015). The multi-band robust knapsack problem—A dynamic programming approach. Discrete Optimization, 18, 123–149.

    Google Scholar 

  • Conlisk, J. (1996). Why bounded rationality? Journal of Economic Literature, 34(2), 669–700.

    Google Scholar 

  • Cox, L. A. T. (2012). Confronting deep uncertainties in risk analysis. Risk Analysis, 32(10), 1607–1629. https://doi.org/10.1111/j.1539-6924.2012.01792.x.

    Article  Google Scholar 

  • Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science, 1(3–4), 197–206.

    Google Scholar 

  • de Finetti, B. (1974). Theory of probability: A critical introductory treatment. New York: Wiley.

    Google Scholar 

  • Delage, E., & Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research, 58(3), 595–612. https://doi.org/10.1287/opre.1090.0741.

    Article  Google Scholar 

  • Delgado, K. V., de Barros, L. N., Cozman, F. G., & Sanner, S. (2011a). Using mathematical programming to solve factored Markov decision processes with imprecise probabilities. International Journal of Approximate Reasoning, 52(7), 1000–1017. https://doi.org/10.1016/j.ijar.2011.04.002.

    Article  Google Scholar 

  • Delgado, K. V., de Barros, L. N., Dias, D. B., & Sanner, S. (2016). Real-time dynamic programming for Markov decision processes with imprecise probabilities. Artificial Intelligence, 230, 192–223. https://doi.org/10.1016/j.artint.2015.09.005.

    Article  Google Scholar 

  • Delgado, K. V., Sanner, S., & de Barros, L. N. (2011b). Efficient solutions to factored MDPs with imprecise transition probabilities. Artificial Intelligence, 175(9–10), 1498–1527. https://doi.org/10.1016/j.artint.2011.01.001.

    Article  Google Scholar 

  • Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics, 458(2), 325–339. https://doi.org/10.1016/j.jmaa.2017.10.006.

    Article  Google Scholar 

  • Destercke, S., Dubois, D., & Chojnacki, E. (2008). Unifying practical uncertainty representations—I: Generalized p-boxes. International Journal of Approximate Reasoning, 49(3), 649–663. https://doi.org/10.1016/j.ijar.2008.07.003.

    Article  Google Scholar 

  • Diecidue, E., & Wakker, P. (2001). On the intuition of rank-dependent utility. Journal of Risk and Uncertainty, 23(3), 281–298.

    Google Scholar 

  • Dimitrova, R., Fu, J., & Topcu, U. (2016). Robust optimal policies for Markov decision processes with safety-threshold constraints. In: IEEE 55th conference on decision and control (pp. 7081–7086). https://doi.org/10.1109/CDC.2016.7799360.

  • Dimitrov, N. B., Dimitrov, S., & Chukova, S. (2014). Robust decomposable Markov decision processes motivated by allocating school budgets. European Journal of Operational Research, 239(1), 199–213. https://doi.org/10.1016/j.ejor.2014.05.003.

    Article  Google Scholar 

  • Dimmock, S. G., Kouwenberg, R., Mitchell, O. S., & Peijnenburg, K. (2015a). Estimating ambiguity preferences and perceptions in multiple prior models: Evidence from the field. Journal of Risk and Uncertainty, 51(3), 219–244. https://doi.org/10.1007/s11166-015-9227-2.

    Article  Google Scholar 

  • Dimmock, S. G., Kouwenberg, R., & Wakker, P. P. (2015b). Ambiguity attitudes in a large representative sample. Management Science, 62(5), 1363–1380. https://doi.org/10.2139/ssrn.1876580.

    Article  Google Scholar 

  • Doria, S. (2017). On the disintegration property of coherent upper conditional prevision defined by the Choquet integral with respect to its associated Hausdorff outer measure. Annals of Operations Research, 256(2), 253–269. https://doi.org/10.1007/s10479-016-2270-9.

    Article  Google Scholar 

  • Dubois, D., & Prade, H., (2003). Possibility theory and its applications: A retrospective and prospective view. In The 12th IEEE international conference on fuzzy systems (pp. 5–11). https://doi.org/10.1109/FUZZ.2003.1209314.

  • Dubois, D., & Prade, H. (1987). Properties of measures of information in evidence and possibility theories. Fuzzy Sets and Systems, 24(2), 161–182.

    Google Scholar 

  • Dubois, D., & Prade, H. (2001). Possibility theory, probability theory and multiple-valued logics: A clarification. Annals of Mathematics and Artificial Intelligence, 32(1–4), 35–66. https://doi.org/10.1023/A:1016740830286.

    Article  Google Scholar 

  • Dubois, D., Prade, H., & Sabbadin, R. (2001). Decision-theoretic foundations of qualitative possibility theory. European Journal of Operational Research, 128(3), 459–478. https://doi.org/10.1016/S0377-2217(99)00473-7.

    Article  Google Scholar 

  • Dubra, J., Maccheroni, F., & Ok, E. A. (2004). Expected utility theory without the completeness axiom. Journal of Economic Theory, 115(1), 118–133. https://doi.org/10.1016/S0022-0531(03)00166-2.

    Article  Google Scholar 

  • Dupacova, J., Growe-Kuska, N., & Romisch, W. (2003). Scenario reduction in stochastic programming. Mathematical Programming, Series A, 95, 493–511. https://doi.org/10.1007/s10107-002-0331-0.

    Article  Google Scholar 

  • Dynkin, E. (1965). Markov processes. Berlin: Springer. https://doi.org/10.1007/978-3-662-00031-1.

    Book  Google Scholar 

  • Eichberger, J., & Kelsey, D. (1999). E-capacities and the Ellsberg paradox. Theory and Decision, 46(2), 107–140. https://doi.org/10.1023/A:1004994630014.

    Article  Google Scholar 

  • Eliaz, K., & Ok, E. A. (2006). Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences. Games and Economic Behavior, 56(1), 61–86. https://doi.org/10.1016/j.geb.2005.06.007.

    Article  Google Scholar 

  • Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4), 643–669.

    Google Scholar 

  • Epstein, L., & Schneider, M. (2007). Learning under ambiguity. The Review of Economic Studies, 74(4), 1275–1303.

    Google Scholar 

  • Ergin, H., & Gul, F. (2009). A theory of subjective compound lotteries. Journal of Economic Theory, 144(3), 899–929. https://doi.org/10.1016/j.jet.2008.08.003.

    Article  Google Scholar 

  • Etner, J., Jeleva, M., & Tallon, J. M. (2012). Decision theory under ambiguity. Journal of Economic Surveys, 26(2), 234–270. https://doi.org/10.1111/j.1467-6419.2010.00641.x.

    Article  Google Scholar 

  • Fehr, E., & Rangel, A. (2011). Neuroeconomic foundations of economic choice—Recent advances. Journal of Economic Perspectives, 25(4), 3–30. https://doi.org/10.1257/jep.25.4.3.

    Article  Google Scholar 

  • Ferson, S., Joslyn, C. A., Helton, J. C., Oberkampf, W. L., & Sentz, K. (2004). Summary from the epistemic uncertainty workshop: Consensus amid diversity. Reliability Engineering and System Safety, 85(1–3), 355–369. https://doi.org/10.1016/j.ress.2004.03.023.

    Article  Google Scholar 

  • Fine, T. L. (1977). Review: Glenn Shafer, A mathematical theory of evidence. Bulletin of the American Mathematical Society, 83(4), 667–672. https://doi.org/10.1093/ije/6.1.83.

    Article  Google Scholar 

  • Fishburn, P. C. (1986). The axioms of subjective probability. Statistical Science, 1(3), 335–345. https://doi.org/10.1214/ss/1177013611.

    Article  Google Scholar 

  • Fussuma, F. L., Delgado, K. V., & de Barros, L. N. (2014). B2RTDP: An efficient solution for bounded-parameter Markov decision process. In 2014 Brazilian conference on intelligent systems (pp. 128–133).

  • Gabrel, V., Murat, C., & Thiele, A. (2014). Recent advances in robust optimization: An overview. European Journal of Operational Research, 235(3), 471–483. https://doi.org/10.1016/j.ejor.2013.09.036.

    Article  Google Scholar 

  • Gajdos, T., Hayashi, T., Tallon, J. M., & Vergnaud, J. C. (2008). Attitude toward imprecise information. Journal of Economic Theory, 140(1), 27–65. https://doi.org/10.1016/j.jet.2007.09.002.

    Article  Google Scholar 

  • Ganzfried, S., & Sandholm, T. (2015). Safe opponent exploitation. ACM Transactions on Economics and Computation, 3(2), 8:1–8:28. https://doi.org/10.1016/j.apcatb.2006.06.014.

    Article  Google Scholar 

  • Ghirardato, P., Maccheroni, F., & Marinacci, M. (2004). Differentiating ambiguity and ambiguity attitude. Journal of Economic Theory, 118(2), 133–173.

    Google Scholar 

  • Giang, P. H., & Shenoy, P. P. (2005). Two axiomatic approaches to decision making using possibility theory. European Journal of Operational Research, 162(2), 450–467. https://doi.org/10.1016/j.ejor.2003.05.004.

    Article  Google Scholar 

  • Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153. https://doi.org/10.1016/0304-4068(89)90018-9.

    Article  Google Scholar 

  • Givan, R., Leach, S., & Dean, T. (2000). Bounded-Parameter Markov decision processes. Artificial Intelligence, 122(1–2), 71–109.

    Google Scholar 

  • Goerigk, M., & Schöbel, A. (2016). Algorithm engineering in robust optimization. In Algorithm Engineering (pp. 245–279). Cham: Springer.

  • Good, I. J. (1977). Bruno de Finetti, theory of probability. Bulletin of the American Mathematical Society, 83(1), 94–97.

    Google Scholar 

  • Gotoh, J., & Uryasev, S. (2017). Support vector machines based on convex risk functions and general norms. Annals of Operations Research, 249(1–2), 301–328. https://doi.org/10.1007/s10479-016-2326-x.

    Article  Google Scholar 

  • Gotoh, J. Y., Kim, M. J., & Lim, A. E. B. (2017). Calibration of distributionally robust empirical optimization models. arXiv Preprint.

  • Gul, F., & Pesendorfer, W. (2006). Random expected utility. Econometrica, 74(1), 121–146.

    Google Scholar 

  • Guo, P. (2011). One-shot decision theory. IEEE Transactions on Systems, Man, and Cybernetics Part A: Systems and Humans, 41(5), 917–926. https://doi.org/10.1109/TSMCA.2010.2093891.

    Article  Google Scholar 

  • Guo, P. (2019). Focus theory of choice and its application to resolving the St. Petersburg, Allais, and Ellsberg paradoxes and other anomalies. European Journal of Operational Research, 276(3), 1034–1043. https://doi.org/10.1016/j.ejor.2019.01.019.

    Article  Google Scholar 

  • Guo, P., & Li, Y. (2014). Approaches to multistage one-shot decision making. European Journal of Operational Research, 236(2), 612–623. https://doi.org/10.1016/j.ejor.2013.12.038.

    Article  Google Scholar 

  • Guo, P., & Tanaka, H. (2010). Decision making with interval probabilities. European Journal of Operational Research, 203(2), 444–454. https://doi.org/10.1016/j.ejor.2009.07.020.

    Article  Google Scholar 

  • Gutin, E., Kuhn, D., & Wiesemann, W. (2015). Interdiction games on Markovian PERT networks. Management Science, 61(5), 999–1017. https://doi.org/10.1287/mnsc.2014.1973.

    Article  Google Scholar 

  • Hahn, E. M., Hashemi, V., Hermanns, H., Lahijanian, M., & Turrini, A. (2017). Multi-objective robust strategy synthesis for interval Markov decision processes. In N. Bertrand & L. Bortolussi (Eds.), Quantitative evaluation of systems (pp. 207–223). Cham: Springer. https://doi.org/10.1007/978-3-319-66335-7_13.

    Chapter  Google Scholar 

  • Hansen, L. P., & Sargent, T. J. (2001). Robust control and model uncertainty. American Economic Review, 91(2), 60–66.

    Google Scholar 

  • Harless, B. Y. D. W., & Camerer, C. F. (1994). The predictive utility of generalized expected utility theories. Econometrica, 62(6), 1251–1289.

    Google Scholar 

  • Harmanec, D. (2002). Generalizing Markov decision processes to imprecise probabilities. Journal of Statistical Planning and Inference, 105(1), 199–213.

    Google Scholar 

  • Haskell, W. B., Kar, D., Fang, F., Tambe, M., Cheung, S., & Denicola, E. (2014). Robust protection of fisheries with COmPASS. AAAI, 4, 2978–2983.

    Google Scholar 

  • Hazen, G. (1989). Ambiguity aversion and ambiguity content in decision making under uncertainty. Annals of Operations Research, 19(1), 415–433.

    Google Scholar 

  • Hu, Z., & Hong, L. (2012). Kullback–Leibler divergence constrained distributionally robust optimization. Optimization Online Preprint.

  • Huang, J., Zhou, K., & Guan, Y. (2017). A study of distributionally robust multistage stochastic optimization. arXiv Preprint.

  • Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1), 73–101.

    Google Scholar 

  • Iancu, D. A., & Trichakis, N. (2014). Pareto efficiency in robust optimization. Management Science, 60(1), 130–147. https://doi.org/10.1287/mnsc.2013.1753.

    Article  Google Scholar 

  • Itoh, H., & Nakamura, K. (2007). Partially observable Markov decision processes with imprecise parameters. Artificial Intelligence, 171(8–9), 453–490.

    Google Scholar 

  • Iyengar, G. N. (2005). Robust dynamic programming. Mathematics of Operations Research, 30(2), 257–280.

    Google Scholar 

  • Jaffray, J. Y. (1989). Linear utility theory for belief functions. Operations Research Letters, 8(2), 107–112.

    Google Scholar 

  • Jakubovskis, A. (2017). Strategic facility location, capacity acquisition, and technology choice decisions under demand uncertainty: Robust vs. non-robust optimization approaches. European Journal of Operational Research, 260(3), 1095–1104.

    Google Scholar 

  • Jech, T. (1992). The logarithmic distribution of leading digits and finitely additive measures. Discrete Mathematics, 108(1–3), 53–57. https://doi.org/10.1016/0012-365X(92)90659-4.

    Article  Google Scholar 

  • Jensen, F. V., & Nielsen, T. D. (2013). Probabilistic decision graphs for optimization under uncertainty. Annals of Operations Research, 204(1), 223–248. https://doi.org/10.1007/s10479-012-1263-6.

    Article  Google Scholar 

  • Jiang, H., Netessine, S., & Savin, S. (2011). Robust newsvendor competition under asymmetric information. Operations Research, 59(1), 254–261. https://doi.org/10.1287/opre.1100.0858.

    Article  Google Scholar 

  • Johanson, M., & Bowling, M. (2009). Data biased robust counter strategies. In Proceedings of the 12th international conference on artificial intelligence and statistics (AISTATS) (vol. 5, pp. 264–271).

  • Johanson, M. B. (2016). Robust strategies and counter-strategies: From superhuman to optimal play. Ph.D. thesis, University of Alberta.

  • Julien, B. (1994). An extension to possibilistic linear programming. Fuzzy Sets and Systems, 64(2), 195–206.

    Google Scholar 

  • Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263.

    Google Scholar 

  • Kardeş, E. (2005). Robust stochastic games and applications to counter-terrorism strategies. Technical report. CREATE.

  • Kardeş, E. (2014). On discounted stochastic games with incomplete information on payoffs and a security application. Operations Research Letters, 42(1), 7–11. https://doi.org/10.1016/j.orl.2013.10.005.

    Article  Google Scholar 

  • Kardeş, E., Ordóñez, F., & Hall, R. W. (2011). Discounted robust stochastic games and an application to queueing control. Operations Research, 59(2), 365–382. https://doi.org/10.1287/opre.1110.0931.

    Article  Google Scholar 

  • Kataoka, S. (2016). A stochastic programming model. Econometrica, 31(1), 181–196.

    Google Scholar 

  • Killian, T. W., Daulton, S., Konidaris, G., Doshi-Velez, F., & Konidaris, G. (2017). Robust and efficient transfer learning with hidden parameter Markov decision processes. In Advances in neural information processing systems (pp. 6251–6262). https://doi.org/10.1038/nature14236.

  • Klibanoff, P., Marinacci, M., Alberto, C. C., & Torino, U. (2009). Recursive smooth ambiguity preferences. Journal of Economic Theory, 144(3), 930–976.

    Google Scholar 

  • Klibanoff, P., Marinacci, M., Applicata, M., & Torino, U. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892.

    Google Scholar 

  • Knight, F. H. (1921). Risk, uncertainty and profit. New York: Hart Schaffner and Marx.

    Google Scholar 

  • Kolmogorov, A. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer.

    Google Scholar 

  • Kolmogorov, A. (2018). Foundations of the theory of probability (2nd ed.). New York: Dover.

    Google Scholar 

  • Kothiyal, A., Spinu, V., & Wakker, P. P. (2011). Prospect theory for continuous distributions: A preference foundation. Journal of Risk and Uncertainty, 42(3), 195–210. https://doi.org/10.1007/s11166-011-9118-0.

    Article  Google Scholar 

  • Krajbich, I., Oud, B., & Fehr, E. (2014). Benefits of neuroeconomic modeling: new policy interventions and predictors of preference. American Economic Review, 104(5), 501–506.

    Google Scholar 

  • Lang, M. (2017). First-order and second-order ambiguity aversion. Management Science, 63(4), 1254–1269.

    Google Scholar 

  • Lei, C., Lin, W. H., & Miao, L. (2016). A two-stage robust optimization approach for the mobile facility fleet sizing and routing problem under uncertainty. Computers & Operations Research, 67, 75–89.

    Google Scholar 

  • Lempert, R. J. (2003). Shaping the next one hundred years: New methods for quantitative, long-term policy analysis. Santa Monica: Rand Corporation.

    Google Scholar 

  • Levin, J. (2006). Choice under uncertainty. Lecture Notes, 2.

  • Li, B., & Si, J. (2010). Approximate robust policy iteration using multilayer perceptron neural networks for discounted infinite-horizon Markov decision processes with uncertain correlated transition matrices. IEEE Transactions on Neural Networks, 21(8), 1270–1280.

    Google Scholar 

  • Li, Z., Müller, J., Wakker, P. P., & Wang, T. V. (2017). The rich domain of ambiguity explored. Management Science, 64(7), 3227–3240. https://doi.org/10.1287/mnsc.2017.2777.

    Article  Google Scholar 

  • Lim, A. E. B., Shanthikumar, J. G., & Shen, Z. J. (2006). Model uncertainty, robust optimization, and learning. In Tutorials in operations research: Models, methods, and applications for innovative decision making (pp. 66–94). https://doi.org/10.1287/educ.1063.0023.

  • Lim, S. H., Xu, H., & Mannor, S. (2016). Reinforcement learning in robust Markov decision processes. Mathematics of Operations Research, 41(4), 1325–1353.

    Google Scholar 

  • Liu, J., Jin, X., Wang, T., & Yuan, Y. (2015). Robust multi-period portfolio model based on prospect theory and ALMV-PSO algorithm. Expert Systems with Applications, 42(20), 7252–7262.

    Google Scholar 

  • Liu, Y., Xu, H., Yang, S. J. S., & Zhang, J. (2018). Distributionally robust equilibrium for continuous games: Nash and Stackelberg models. European Journal of Operational Research, 265(2), 631–643. https://doi.org/10.1016/j.ejor.2017.07.050.

    Article  Google Scholar 

  • Loizou, N. (2016). Distributionally robust games with risk-averse players. arXiv Preprint. https://doi.org/10.5220/0005753301860196.

  • Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The Economic Journal, 92(368), 805–824.

    Google Scholar 

  • Loomes, G., & Sugden, R. (1995). Incorporating a stochastic element into decision theories. European Economic Review, 39(3–4), 641–648. https://doi.org/10.1016/0014-2921(94)00071-7.

    Article  Google Scholar 

  • Lovejoy, W. S. (1991). A survey of algorithmic methods for partially observed Markov decision processes. Annals of Operations Research, 28(1), 47–66.

    Google Scholar 

  • Maccheroni, F., Marinacci, M., & Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica, 74(6), 1447–1498. https://doi.org/10.1111/j.1468-0262.2006.00716.x.

    Article  Google Scholar 

  • Machina, M. J. (1982). “Expected utility” analysis without the independence axiom. Econometrica, 50(2), 277–323.

    Google Scholar 

  • Maiers, J., & Sherif, Y. S. (1985). Applications of fuzzy set theory. IEEE Transactions on Systems, Man and Cybernetics, 15(1), 175–189. https://doi.org/10.1109/TSMC.1985.6313408.

    Article  Google Scholar 

  • Mannor, S., Mebel, O., & Xu, H. (2016). Robust MDPs with k-rectangular uncertainty. Mathematics of Operations Research, 41(4), 1484–1509.

    Google Scholar 

  • Marinacci, M. (2015). Model uncertainty. Journal of the European Economic Association, 13(6), 1022–1100. https://doi.org/10.1111/jeea.12164.

    Article  Google Scholar 

  • Miranda, E. (2008). A survey of the theory of coherent lower previsions. International Journal of Approximate Reasoning, 48(2), 628–658. https://doi.org/10.1016/j.ijar.2007.12.001.

    Article  Google Scholar 

  • Mohajerin Esfahani, P., & Kuhn, D. (2017). Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1–2), 115–166. https://doi.org/10.1007/s10107-017-1172-1.

    Article  Google Scholar 

  • Monahan, G. E. (1982). State of the art—A survey of partially observable Markov decision processes: Theory, models, and algorithms. Management Science, 28(1), 1–16.

    Google Scholar 

  • Murofushi, T., & Sugeno, M. (1991). A theory of fuzzy measures: Representations, the Choquet integral, and null sets. Journal of Mathematical Analysis and Applications, 159(2), 532–549. https://doi.org/10.1016/0022-247X(91)90213-J.

    Article  Google Scholar 

  • Nakao, H., Shen, S., & Chen, Z. (2017). Network design in scarce data environment using moment-based distributionally robust optimization. Computers and Operations Research, 88, 44–57.

    Google Scholar 

  • Nascimento, L., & Riella, G. (2013). Second-order ambiguous beliefs. Economic Theory, 52(3), 1005–1037. https://doi.org/10.1007/s00199-011-0675-x.

    Article  Google Scholar 

  • Natarajan, K., Pachamanova, D., & Sim, M. (2009). Constructing risk measures from uncertainty sets. Operations Research, 57(5), 1129–1141. https://doi.org/10.1287/opre.1080.0683.

    Article  Google Scholar 

  • Natenzon, P. (2019). Random choice and learning. Journal of Political Economy, 127(1), 419–457. https://doi.org/10.1086/700762.

    Article  Google Scholar 

  • Nau, R. F. (2006). Uncertainty aversion with second-order utilities and probabilities. Management Science, 52(1), 136–145. https://doi.org/10.1287/mnsc.1050.0469.

    Article  Google Scholar 

  • Neumaier, A. (2004). Clouds, fuzzy sets, and probability intervals. Reliable Computing, 10(4), 249–272.

    Google Scholar 

  • Neyshabouri, S., & Berg, B. P. (2017). Discrete optimization: Two-stage robust optimization approach to elective surgery and downstream capacity planning. European Journal of Operational Research, 260(1), 21–40.

    Google Scholar 

  • Nguyen, T. H., Jiang, A. X., & Tambe, M. (2014). Stop the compartmentalization: Unified robust algorithms for handling uncertainties in security games. In Proceedings of the 13th international conference on autonomous agents and multiagent systems (AAMAS 2014) (pp. 317–324).

  • Nguyen, T. H., Sinha, A., & Tambe, M. (2016). Addressing behavioral uncertainty in security games: An efficient robust strategic solution for defender patrols. In Proceedings—2016 IEEE 30th international parallel and distributed processing symposium. IPDPS 2016 (pp. 1831–1838). https://doi.org/10.1109/IPDPSW.2016.195.

  • Nguyen, H., & Sriboonchitta, S. (2010). On Choquet integral risk measures. In V. Huynh, Y. Nakamori, J. Lawry, & M. Inuiguchi (Eds.), Integrated uncertainty management and applications (pp. 15–22). Berlin: Springer.

    Google Scholar 

  • Ni, Y., & Liu, Z. Q. (2013). Bounded-parameter partially observable Markov decision processes: Framework and algorithm. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21(6), 821–863.

    Google Scholar 

  • Nilim, A., & El Ghaoui, L. (2005). Robust control of Markov decision processes with uncertain transition matrices. Operations Research, 53(5), 780–798.

    Google Scholar 

  • Oh, E., & Kim, K. E. (2011). A geometric traversal algorithm for reward-uncertain MDPs. In Proceedings of the 27th conference on uncertainty in artificial intelligence. AUAI Press.

  • Osogami, T. (2015). Robust partially observable Markov decision process. In Proceedings of the 32nd international conference on machine learning (pp. 106–115).

  • Paç, A. B., & Pınar, M. (2018). On robust portfolio and naïve diversification: mixing ambiguous and unambiguous assets. Annals of Operations Research, 266(1–2), 223–253. https://doi.org/10.1007/s10479-017-2619-8.

    Article  Google Scholar 

  • Perny, P., Spanjaard, O., & Storme, L. X. (2006). A decision-theoretic approach to robust optimization in multivalued graphs. Annals of Operations Research, 147(1), 317–341. https://doi.org/10.1007/s10479-006-0073-0.

    Article  Google Scholar 

  • Ponsen, M. J. V., de Jong, S., & Lanctot, M. (2011). Computing approximate Nash equilibria and robust best-responses using sampling. Journal of Artificial Intelligence Research, 42, 575–605.

    Google Scholar 

  • Popescu, I. (2005). A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Mathematics of Operations Research, 30(3), 632–657. https://doi.org/10.1287/moor.1040.0137.

    Article  Google Scholar 

  • Puterman, M. L. (2014). Markov decision processes: Discrete stochastic dynamic programming. New York: Wiley.

    Google Scholar 

  • Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior & Organization, 3(4), 323–343.

    Google Scholar 

  • Qu, S., Meng, D., Zhou, Y., & Dai, Y. (2017). Distributionally robust games with an application to supply chain. Journal of Intelligent & Fuzzy Systems, 33(5), 2749–2762.

    Google Scholar 

  • Romanko, O., & Mausser, H. (2016). Robust scenario-based value-at-risk optimization. Annals of Operations Research, 237(1–2), 203–218.

    Google Scholar 

  • Saghafian, S. (2018). Ambiguous partially observable Markov decision processes: Structural results and applications. Journal of Economic Theory, 178, 1–35. https://doi.org/10.1016/j.jet.2018.08.006.

    Article  Google Scholar 

  • Samuelson, S., Yang, I. (2017). Data-Driven distributionally robust control of energy storage to manage wind power fluctuations. In 2017 IEEE conference on control technology and applications (pp. 199–204).

  • Santos, M. C., Poss, M., & Nace, D. (2018). A perfect information lower bound for robust lot-sizing problems. Annals of Operations Research, 271(2), 887–913. https://doi.org/10.1007/s10479-018-2908-x.

    Article  Google Scholar 

  • Sariddichainunta, P., & Inuiguchi, M. (2017). Global optimality test for maximin solution of bilevel linear programming with ambiguous lower-level objective function. Annals of Operations Research, 256(2), 285–304. https://doi.org/10.1007/s10479-016-2293-2.

    Article  Google Scholar 

  • Sasaki, Y. (2017). Generalized Nash equilibrium with stable belief hierarchies in static games with unawareness. Annals of Operations Research, 256(2), 271–284. https://doi.org/10.1007/s10479-016-2266-5.

    Article  Google Scholar 

  • Satia, J. K., & Lave, R. E. (1973). Markovian decision processes with uncertain transition probabilities. Operations Research, 21(3), 728–740.

    Google Scholar 

  • Savage, L. J. (1954). The foundations of statistics. New York: Wiley.

    Google Scholar 

  • Scheftelowitsch, D., Buchholz, P., Hashemi, V., & Hermanns, H. (2017). Multi-Objective approaches to Markov decision processes with uncertain transition parameters. arXiv Preprint.

  • Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.

    Google Scholar 

  • Seo, K. (2009). Ambiguity and second-order belief. Econometrica, 77(5), 1575–1605. https://doi.org/10.3982/ecta6727.

    Article  Google Scholar 

  • Shafer, G. (1976). A mathematical theory of evidence. Princeton: Princeton University Press.

    Google Scholar 

  • Shapiro, A. (2011). A dynamic programming approach to adjustable robust optimization. Operations Research Letters, 39(2), 83–87. https://doi.org/10.1016/j.orl.2011.01.001.

    Article  Google Scholar 

  • Shapiro, A. (2016). Rectangular sets of probability measures. Operations Research, 64(2), 528–541. https://doi.org/10.1287/opre.2015.1466.

    Article  Google Scholar 

  • Shashua, S. D. C, & Mannor, S. (2017). Deep robust Kalman filter. arXiv Preprint.

  • Singh, V. V., Jouini, O., & Lisser, A. (2017). Distributionally robust chance-constrained games: Existence and characterization of Nash equilibrium. Optimization Letters, 11(7), 1385–1405. https://doi.org/10.1007/s11590-016-1077-6.

    Article  Google Scholar 

  • Sinha, S., & Ghate, A. (2016). Policy iteration for robust nonstationary Markov decision processes. Optimization Letters, 10(8), 1613–1628. https://doi.org/10.1007/s11590-016-1040-6.

    Article  Google Scholar 

  • Sinha, S., Kotas, J., & Ghate, A. (2016). Robust response-guided dosing. Operations Research Letters, 44(3), 394–399.

    Google Scholar 

  • Siniscalchi, M. (2009). Vector expected utility and attitudes toward variation. Econometrica, 77(3), 801–855. https://doi.org/10.2139/ssrn.1030407.

    Article  Google Scholar 

  • Siniscalchi, M. (2011). Dynamic choice under ambiguity. Theoretical Economics, 6(3), 379–421. https://doi.org/10.3982/TE571.

    Article  Google Scholar 

  • Smets, P. (1999). Practical uses of belief functions. In Proceedings of the 15th conference on Uncertainty in artificial intelligence (pp. 612–621).

  • Smets, P., & Kennes, R. (1994). The transferable belief model. Artificial Intelligence, 66(2), 191–234. https://doi.org/10.1016/0004-3702(94)90026-4.

    Article  Google Scholar 

  • Snow, A. (2010). Ambiguity and the value of information. Journal of Risk and Uncertainty, 40(2), 133–145.

    Google Scholar 

  • Soyster, A. L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154–1157. https://doi.org/10.1287/opre.21.5.1154.

    Article  Google Scholar 

  • Starmer, C. (2010). Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature, 38(2), 332–382. https://doi.org/10.1257/jel.38.2.332.

    Article  Google Scholar 

  • Sugeno, M. (1974). Theory of fuzzy integrals and its applications. Ph.D. thesis, Tokyo Institute of Technology.

  • Sun, H., & Xu, H. (2016). Convergence analysis for distributionally robust optimization and equilibrium problems. Mathematics of Operations Research, 41(2), 377–401. https://doi.org/10.1287/moor.2015.0732.

    Article  Google Scholar 

  • Tan, C. H., & Hartman, J. C. (2011). Sensitivity analysis in Markov decision processes with uncertain reward parameters. Journal of Applied Probability, 48(4), 954–967.

    Google Scholar 

  • Trevizan, F. W., de Barros, L. N., & Cozman, F. G. (2007). Planning under risk and Knightian uncertainty. In International joint conference on artificial intelligence (pp. 2023–2028).

  • Troffaes, M. C. (2007). Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning, 45(1), 17–29. https://doi.org/10.1016/j.ijar.2006.06.001.

    Article  Google Scholar 

  • Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323. https://doi.org/10.1007/BF00122574.

    Article  Google Scholar 

  • Tversky, A., & Koehler, D. J. (1994). Support theory: A nonextensional representation of subjective probability. Psychological Review, 101(4), 547–567. https://doi.org/10.1037/0033-295X.101.4.547.

    Article  Google Scholar 

  • Ure, N. K, Geramifard, A., Chowdhary, G., & How, J. P. (2012). Adaptive planning for Markov decision processes with uncertain transition models via incremental feature dependency discovery. In Joint European conference on machine learning and knowledge discovery in databases (pp. 99–115).

  • Van Parys, B. P. G., Esfahani, P. M., & Kuhn, D. (2017). From data to decisions: Distributionally robust optimization is optimal. arXiv Preprint.

  • von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior. Princeton: Princeton University Press.

    Google Scholar 

  • Wakker, P. P. (2010). Prospect theory: For risk and ambiguity. Cambridge: Cambridge University Press.

    Google Scholar 

  • Wald, A. (1945). Statistical decision functions which minimize the maximum risk. Annals of Mathematics, 46(2), 265–280.

    Google Scholar 

  • Walker, W. E., Harremoës, P., Rotmans, J., van der Sluijs, J. P., van Asselt, M. B. A., Janssen, P., et al. (2003). Defining uncertainty: A conceptual basis for uncertainty management in model-based decision support. Integrated Assessment, 4(1), 5–17.

    Google Scholar 

  • Wallenius, J., Dyer, J. S., Fishburn, P. C., Steuer, R. E., Zionts, S., & Deb, K. (2008). Multiple criteria decision making, multiattribute utility theory: Recent accomplishments and what lies ahead. Management Science, 54(7), 1336–1349. https://doi.org/10.1287/mnsc.1070.0838.

    Article  Google Scholar 

  • Walley, P. (1987). Belief function representations of statistical evidence. Annals of Statistics, 14(2), 590–606.

    Google Scholar 

  • Walley, P. (1991). Statistical reasoning with imprecise probabilities. New York: Chapman & Hall.

    Google Scholar 

  • Walley, P. (2000). Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning, 24(2–3), 125–148.

    Google Scholar 

  • Wang, C., & Guo, P. (2017). Behavioral models for first-price sealed-bid auctions with the one-shot decision theory. European Journal of Operational Research, 261(3), 994–1000. https://doi.org/10.1016/j.ejor.2017.03.024.

    Article  Google Scholar 

  • Wang, W., Sun, H. J., & Wu, J. J. (2015a). Robust user equilibrium model based on cumulative prospect theory under distribution-free travel time. Journal of Central South University, 22(2), 761–770.

    Google Scholar 

  • Wang, X., Chen, J., Dutta, A., & Chiang, M. (2015b). Adaptive video streaming over whitespace: SVC for 3-tiered spectrum sharing. In 2015 IEEE conference on computer communications (pp. 28–36). IEEE.

  • Wang, X., Fan, N., & Pardalos, P. M. (2018). Robust chance-constrained support vector machines with second-order moment information. Annals of Operations Research, 263(1–2), 45–68. https://doi.org/10.1007/s10479-015-2039-6.

    Article  Google Scholar 

  • Wang, Z., Boularias, A., Mülling, K., & Peters, J. (2011). Balancing safety and exploitability in opponent modeling. In 25th AAAI conference on artificial intelligence (AAAI 2011) (pp. 1515–1520).

  • Weber, M. (1987). Decision making with incomplete information. European Journal of Operational Research, 28(1), 44–57. https://doi.org/10.1016/0377-2217(87)90168-8.

    Article  Google Scholar 

  • Weng, P., Qiu, Z., Costanzo, J., Yin, X., & Sinopoli, B. (2017). Optimal threshold policies for robust data center control. Lecture Notes in Electrical Engineering, 465, 104–114.

    Google Scholar 

  • Whalen, T. (1984). Decisionmaking under uncertainty with various assumptions about available information. IEEE Transactions on Systems, Man and Cybernetics, SMC–14(6), 888–900. https://doi.org/10.1109/TSMC.1984.6313316.

    Article  Google Scholar 

  • White, C. C., & Eldeib, H. K. (1986). Parameter imprecision in finite state, finite action dynamic programs. Operations Research, 34(1), 120–129.

    Google Scholar 

  • White, C. C., & Eldeib, H. K. (1994). Markov decision processes with imprecise transition probabilities. Operations Research, 42(4), 739–749.

    Google Scholar 

  • Wiesemann, W., Kuhn, D., & Rustem, B. (2013). Robust Markov decision processes. Mathematics of Operations Research, 38(1), 152–183.

    Google Scholar 

  • Wiesemann, W., Kuhn, D., & Sim, M. (2014). Distributionally robust convex optimization. Operations Research, 62(6), 1358–1376. https://doi.org/10.1287/opre.2014.1314.

    Article  Google Scholar 

  • Wolff, E. M., Topcu, U., & Murray, R. M. (2012). Robust control of uncertain Markov decision processes with temporal logic specifications. In IEEE 51st annual conference on decision and control (pp. 3372–3379). IEEE.

  • Wong, K. Y., Lalmazloumian, M., Wong, K. Y., Govindan, K., & Kannan, D. (2016). A robust optimization model for agile and build-to-order supply chain planning under uncertainties. Annals of Operations Research, 240(2), 435–470.

    Google Scholar 

  • Woodford, M. (2014). Stochastic choice: An optimizing neuroeconomic model. American Economic Review, 104(5), 495–500.

    Google Scholar 

  • Wozabal, D. (2012). A framework for optimization under ambiguity. Annals of Operations Research, 193(1), 21–47. https://doi.org/10.1007/s10479-010-0812-0.

    Article  Google Scholar 

  • Xiao, H., Yang, K., Wang, X., & Shao, H. (2012). A robust MDP approach to secure power control in cognitive radio networks. In 2012 IEEE international conference on communications (pp. 4642–4647). IEEE.

  • Xin, L., & Goldberg, D. A. (2015). Distributionally robust inventory control when demand is a martingale. arXiv Preprint.

  • Xu, H., & Mannor, S. (2012). Distributionally robust Markov decision processes. Mathematics of Operations Research, 37(2), 288–300. https://doi.org/10.1287/moor.1120.0540.

    Article  Google Scholar 

  • Yager, R. R. (1979). Possibilistic decision making. IEEE Transactions on Systems Man and Cybernetics, 9(7), 388–392.

    Google Scholar 

  • Yang, I. (2017a). A convex optimization approach to distributionally robust Markov decision processes with Wasserstein distance. IEEE Control Systems Letters, 1(1), 164–169.

    Google Scholar 

  • Yang, I. (2017b). Distributionally robust stochastic control with conic confidence sets. In 56th IEEE conference on decision and control (pp. 4291–4296).

  • Yang, L., Li, Y., Chen, K., & Zhou, Z. (2014). Distributionally robust return-risk optimization models and their applications. Journal of Applied Mathematics, 2014, 1–9.

    Google Scholar 

  • Yu, P., & Xu, H. (2016). Distributionally robust counterpart in Markov decision processes. IEEE Transactions on Automatic Control, 61(9), 2538–2543. https://doi.org/10.1109/TAC.2015.2495174.

    Article  Google Scholar 

  • Zadeh, L. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X.

    Article  Google Scholar 

  • Zadeh, L. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1), 3–28. https://doi.org/10.1016/0165-0114(78)90029-5.

    Article  Google Scholar 

  • Zadeh, L. (1984). Review of a mathematical theory of evidence. AI Magazine, 5(3), 81. https://doi.org/10.1609/aimag.v5i3.452.

    Article  Google Scholar 

  • Zadeh, L., Bellman, R., & Robbins, H. (2005). Toward a generalized theory of uncertainty (GTU)—An outline. Information Sciences, 172(1–2), 1–40. https://doi.org/10.1016/j.ins.2005.01.017.

    Article  Google Scholar 

  • Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1(1), 45–55. https://doi.org/10.1016/0165-0114(78)90031-3.

    Article  Google Scholar 

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    Andrew J. Keith & Darryl K. Ahner

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Keith, A.J., Ahner, D.K. A survey of decision making and optimization under uncertainty. Ann Oper Res 300, 319–353 (2021). https://doi.org/10.1007/s10479-019-03431-8

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