Distributionally robust simple integer recourse
- 34 Downloads
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.
KeywordsDistributionally robust Stochastic integer recourse Mixed integer conic program
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.
- Bayraksan G, Love DK (2015) Data-driven stochastic programming using phi-divergences. In: Tutorials in operations research, INFORMS, pp 1–19Google Scholar
- 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 OnlineGoogle Scholar
- Romeijnders W, Stougie L, van der Vlerk MH (2014) Approximation in two-stage stochastic integer programming. Surv Oper Res Manag Sci 19(1):17–33Google Scholar
- 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 201–209Google Scholar
- 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 165–187Google Scholar
- Shapiro A, Dentcheva D, Ruszczynski A (2014) Lectures on stochastic programming: modeling and theory, 2nd edn. SIAM Publishers, PhiladelphiaGoogle Scholar
- Zackova J (1966) On minimax solutions of stochastic linear programming problems. Casopis pro pestovani matematiky 091(4):423–430Google Scholar