Abstract
Quantified integer (linear) programs (QIP) are integer linear programs with variables being either existentially or universally quantified. They can be interpreted as two-person zero-sum games between an existential and a universal player on the one side, or multistage optimization problems under uncertainty on the other side. Solutions are so called winning strategies for the existential player that specify how to react on moves—certain fixations of universally quantified variables—of the universal player to certainly win the game. In this setting the existential player must ensure the fulfillment of a system of linear constraints, while the universal variables can range within given intervals, trying to make the fulfillment impossible. Recently, this approach was extended by adding a linear constraint system the universal player must obey. Consequently, existential and universal variable assignments in early decision stages now can restrain possible universal variable assignments later on and vice versa resulting in a multistage optimization problem with decision-dependent uncertainty. We present an attenuated variant, which instead of an NP-complete decision problem allows a polynomial-time decision on the legality of a move. Its usability is motivated by several examples.
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References
Apap, R., Grossmann, I.: Models and computational strategies for multistage stochastic programming under endogenous and exogenous uncertainties. Comput. Chem. Eng. 103, 233–274 (2017)
Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)
Birge, J., Louveaux, F.: Introduction to Stochastic Programming, 2nd edn. Springer, New York (2011)
Ederer, T., Hartisch, M., Lorenz, U., Opfer, T., Wolf, J.: Yasol: an open source solver for quantified mixed integer programs. In: Advances in Computer Games - 15th International Conferences, ACG 2017, pp. 224–233 (2017)
Gupta, V., Grossmann, I.: A new decomposition algorithm for multistage stochastic programs with endogenous uncertainties. Comput. Chem. Eng. 62, 62–79 (2014)
Hartisch, M., Lorenz, U.: Mastering uncertainty: towards robust multistage optimization with decision dependent uncertainty. In: PRICAI 2019: Trends in Artificial Intelligence, pp. 446–458. Springer, Berlin (2019)
Hartisch, M., Ederer, T., Lorenz, U., Wolf, J.: Quantified integer programs with polyhedral uncertainty set. In: Computers and Games - 9th International Conference, CG 2016, pp. 156–166 (2016)
Hellemo, L., Barton, P.I., Tomasgard, A.: Decision-dependent probabilities in stochastic programs with recourse. Comput. Manag. Sci. 15(3-4), 369–395 (2018)
Jonsbråten, T., Wets, R.J.B., Woodruff, D.: A class of stochastic programs with decision dependent random elements. Ann. Oper. Res. 82(0), 83–106 (1998)
Lappas, N., Gounaris, C.: Multi-stage adjustable robust optimization for process scheduling under uncertainty. AIChE J. 62(5), 1646–1667 (2016)
Lappas, N.H., Gounaris, C.E.: Robust optimization for decision-making under endogenous uncertainty. Comput. Chem. Eng. 111, 252–266 (2018)
Lorenz, U., Wolf, J.: Solving multistage quantified linear optimization problems with the alpha–beta nested benders decomposition. EURO J. Comput. Optim. 3(4), 349–370 (2015)
Nohadani, O., Sharma, K.: Optimization under decision-dependent uncertainty. SIAM J. Optim. 28(2), 1773–1795 (2018)
Papadimitriou, C.: Games against nature. J. Comput. Syst. Sci. 31(2), 288–301 (1985)
Poss, M.: Robust combinatorial optimization with variable cost uncertainty. Eur. J. Oper. Res. 237(3), 836–845 (2014)
Vujanic, R., Goulart, P., Morari, M.: Robust optimization of schedules affected by uncertain events. J. Optim. Theory Appl. 171(3), 1033–1054 (2016)
Acknowledgements
This research is partially supported by the German Research Foundation (DFG) project “Advanced algorithms and heuristics for solving quantified mixed - integer linear programs”.
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Hartisch, M., Lorenz, U. (2020). Robust Multistage Optimization with Decision-Dependent Uncertainty. In: Neufeld, J.S., Buscher, U., Lasch, R., Möst, D., Schönberger, J. (eds) Operations Research Proceedings 2019. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-48439-2_53
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