Abstract
We consider stochastic programming problems with probabilistic and quantile objective functions. The original distribution of the random variable is replaced by a discrete one. We thus consider a sequence of problems with discrete distributions. We suggest conditions, which guarantee that the sequence of optimal strategies converges to an optimal strategy of the original problem. We consider the case of a symmetrical distribution, the case of the loss function increasing in the random variable, and the case of the loss function increasing in the optimization strategy.
Keywords
S.V. Ivanov—Supported by Russian Science Foundation (project 15-11-10009).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Kibzun, A.I., Kan, Yu.S: Stochastic Programming Problems with Probability and Quantile Functions. Wiley, New York (1996)
Kibzun, A.I., Naumov, A.V., Norkin, V.I.: On reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem. Autom. Remote Control. 74, 951–967 (2013)
Vishnaykov, B.V., Kibzun, A.I.: Deterministic equivalents for the problems of stochastic programming with probabilistic criteria. Autom. Remote Control. 67, 945–961 (2006)
Lepp, R.: Projection and discretization methods in stochastic programming. J. Comput. Appl. Math. 56, 55–64 (1994)
Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. Ser. B. 116, 461–479 (2009)
Choirat, C., Hess, C., Seri, R.: Approximation of stochastic programming problems. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 45–59. Springer, Heidelberg (2006)
Pflug, G.Ch.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. 89, 251–271 (2001)
Lepp, R.: Approximation of value-at-risk problems with decision rules. In: Uryasev, S.P. (ed.) Probabilistic Constrained Optimization. pp. 186–197. Kluwer Academic Publishers, Norwell (2000)
Kibzun, A., Lepp, R.: Discrete approximation in quantile problem of portfolio selection. In: Uryasev, S., Pardalos, P.M. (eds.) Stochastic Optimization: Algorithms and Applications, pp. 119–133. Kluwer Academic Publishers, Norwell (2000)
Chistyakov, V.P.: Probability Theory Course. Nauka, Moscow (1978). (in Russian)
Kibzun, A.I., Naumov, A.V., Ivanov, S.V.: Bilevel optimization problem for railway transport hub planning. Upravlenie bol’simi sistemami 38, 140–160 (2012). (in Russian)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Kibzun, A.I., Ivanov, S.V. (2016). Convergence of Discrete Approximations of Stochastic Programming Problems with Probabilistic Criteria. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds) Discrete Optimization and Operations Research. DOOR 2016. Lecture Notes in Computer Science(), vol 9869. Springer, Cham. https://doi.org/10.1007/978-3-319-44914-2_41
Download citation
DOI: https://doi.org/10.1007/978-3-319-44914-2_41
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44913-5
Online ISBN: 978-3-319-44914-2
eBook Packages: Computer ScienceComputer Science (R0)