Relaxations and approximations of chance constraints under finite distributions
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Optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications. Such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity. In the setting of a finite distribution of the stochastic parameters, an optimization problem with linear chance constraints can be formulated as a mixed integer linear program (MILP). The natural MILP formulation has a weak relaxation bound and is quite difficult to solve. In this paper, we review some recent results on improving the relaxation bounds and constructing approximate solutions for MILP formulations of chance constraints. We also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems. This algorithm uses a relaxation to construct a solution whose (constraint violation) risk level may be larger than the pre-specified threshold, but is within a constant factor of it, and whose objective value is also within a constant factor of the true optimal value. Finally, we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios.
Mathematics Subject Classification90C11 90C15 90C27
This research has been supported in part by the National Science Foundation Award 1633196 and the Office of Naval Research Grant N00014-18-1-2075.
- 3.Ahmed, S., Shapiro, A.: Solving chance-constrained stochastic programs via sampling and integer programming. Tutor. Oper. Res. (INFORMS) 10, 261–269 (2008)Google Scholar
- 14.Goyal, V., Ravi, R.: Approximation algorithms for robust covering problems with chance constraints. http://repository.cmu.edu/cgi/viewcontent.cgi?article=1365&context=tepper (2008)
- 20.Liu, X., Kılınç-Karzan, F., Küçükyavuz, S.: On intersection of two mixing sets with applications to joint chance-constrained programs. Math. Program. (2018). https://doi.org/10.1007/s10107-018-1231-2
- 27.Nemirovski, A., Shapiro, A.: Scenario approximation of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–48. Springer, London (2005)Google Scholar
- 39.Snyder, L.V., Daskin, M.S: Models for reliable supply chain network design. In: Murray, A.T., Grubesic, T.H. (eds.) Critical Infrastructure, pp. 257–289. Springer, Berlin (2007)Google Scholar
- 41.Swamy, C.: Risk-averse stochastic optimization: probabilistically-constrained models and algorithms for black-box distributions. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 1627–1646. SIAM (2011)Google Scholar
- 44.van Ackooij, W., Zorgati, R., Henrion, R., Möller, A.: Chance constrained programming and its applications to energy management. In: Dritsas I. (ed.) Stochastic Optimization—Seeing the Optimal for the Uncertain. InTech (2011)Google Scholar
- 45.Xie, W., Ahmed, S.: Bicriteria approximation of chance constrained covering problems. Submitted for publication, Preprint available at Optimization Online (2018)Google Scholar
- 47.Xie, W., Ahmed, S.: On quantile cuts and their closure for chance constrained optimization problems. Math. Program. (2017). https://doi.org/10.1007/s10107-017-1190-z