Abstract
The simple integer recourse (SIR) function of a decision variable is the expectation of the integer round-up of the shortage/surplus between a random variable with a known distribution and the decision variable. It is the integer analogue of the simple (continuous) recourse function in two-stage stochastic linear programming. Structural properties and approximations of SIR functions have been extensively studied in the seminal works of van der Vlerk and coauthors. We study a distributionally robust SIR function (DR-SIR) that considers the worst-case expectation over a given family of distributions. Under the assumption that the distribution family is specified by its mean and support, we derive a closed form analytical expression for the DR-SIR function. We also show that this nonconvex DR-SIR function can be represented using a mixed-integer second-order conic program.
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References
Bayraksan G, Love DK (2015) Data-driven stochastic programming using phi-divergences. In: Tutorials in operations research, INFORMS, pp 1–19
Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper Res 58(3):595–612
Mohajerin Esfahani, P, Kuhn D (2017) Data-driven distributionally robust optimization using the wasserstein metric: performance guarantees and tractable reformulations. Math Program. https://doi.org/10.1007/s10107-017-1172-1
Jiang R, Guan Y (2016) Data-driven chance constrained stochastic program. Math Program 158(1–2):291–327
Haneveld WKKlein, Stougie L, van der Vlerk M (1995) On the convex hull of the simple integer recourse objective function. Ann Oper Res 56:209–224
Haneveld WKKlein, Stougie L, van der Vlerk M (1996) An algorithm for the construction of convex hulls in simple integer recourse programming. Ann Oper Res 64:67–81
Haneveld WKKlein, Stougie L, van der Vlerk M (2006) Simple integer recourse models. Math Program Ser B 108:435–473
Louveaux FV, van der Vlerk MH (1993) Stochastic programming with simple integer recourse. Math Program 61(1–3):301–325
Popescu I (2007) Robust mean-covariance solutions for stochastic optimization. Oper Res 55:98–112
Postek K, Ben-Tal A, Den Hertog D, Melenberg B (2015) Exact robust counterparts of ambiguous stochastic constraints under mean and dispersion information. Available at Optimization Online
Romeijnders W, Morton DP, van der Vlerk MH (2017) Assessing the quality of convex approximations for two-stage totally unimodular integer recourse models. INFORMS J. Comput. 29(2):211–231
Romeijnders W, Schultz R, van der Vlerk MH, Haneveld WKK (2016a) A convex approximation for two-stage mixed-integer recourse models with a uniform error bound. SIAM J Optim 26:426–447
Romeijnders W, Stougie L, van der Vlerk MH (2014) Approximation in two-stage stochastic integer programming. Surv Oper Res Manag Sci 19(1):17–33
Romeijnders W, van der Vlerk MH, Haneveld WKK (2015) Convex approximations for totally unimodular integer recourse models: a uniform error bound. SIAM J Optim 25:130–158
Romeijnders W, van der Vlerk MH, Haneveld WKKlein (2016b) Total variation bounds on the expectation of periodic functions with applications to recourse approximations. Math Program 157:3–46
Scarf H (1958) A min-max solution of an inventory problem. In: Arrow KJ, Karlin S, Scarf HE (eds) Studies in the mathematical theory of inventory and production. Stanford University Press, Palo Alto, pp 201–209
Schultz R (2011) Risk aversion in two-stage stochastic integer programming. In: Infanger G (ed) Stochastic programming: the state of the art in honor of George B. Dantzig. Springer, New York, pp 165–187
Shapiro A (2017) Distributionally robust stochastic programming. SIAM J Optim 27(4):2258–2275
Shapiro A, Ahmed S (2004) On a class of minimax stochastic programs. SIAM J Optim 14:1237–1249
Shapiro A, Dentcheva D, Ruszczynski A (2014) Lectures on stochastic programming: modeling and theory, 2nd edn. SIAM Publishers, Philadelphia
Van der Vlerk MH (2010) Convex approximations for a class of mixed-integer recourse models. Ann Oper Res 177:139–150
Wiesemann W, Kuhn D, Sim M (2014) Distributionally robust convex optimization. Oper Res 62(6):1358–1376
Xie W, Ahmed S (2018) On deterministic reformulations of distributionally robust joint chance constrained optimization problems. SIAM J Optim 28(2):1151–1182
Zackova J (1966) On minimax solutions of stochastic linear programming problems. Casopis pro pestovani matematiky 091(4):423–430
Zymler S, Kuhn D, Rustem B (2013) Distributionally robust joint chance constraints with second-order moment information. Math Program 137(1–2):167–198
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This research has been supported in part by the National Science Foundation Grant #1633196. The authors thank the editor and two anonymous referees for constructive comments.
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Dedicated to the memory of Maarten van der Vlerk.
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Xie, W., Ahmed, S. Distributionally robust simple integer recourse. Comput Manag Sci 15, 351–367 (2018). https://doi.org/10.1007/s10287-018-0313-1
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DOI: https://doi.org/10.1007/s10287-018-0313-1