Abstract
We consider stochastic programming problems with probabilistic and quantile criteria. We describe a method for approximating these problems with a sample of realizations for random parameters. When we use this method, criterial functions of the problems are replaced with their sample estimates. We show the hypoconvergence of sample probability functions to its exact value that guarantees the convergence of approximations for the probability function maximization problem on a compact set with respect to both the value of the criterial function and the optimization strategy. We prove a theorem on the convergence of approximation for the quantile function minimization problem with respect to the value of the criterial function and the optimization strategy.
Similar content being viewed by others
References
Kibzun, A.I. and Kan, Y.S., Stochastic Programming Problems with Probability and Quantile Functions, Chichester: Wiley, 1996.
Kibzun, A.I., Naumov, A.V., and Norkin, V.I., On Reducing a Quantile Optimization Problem with Discrete Distribution to a Mixed Integer Programming Problem, Autom. Remote Control, 2013, vol. 74, no. 6, pp. 951–967.
Kibzun, A.I., Comparison of Two Algorithms for Solving a Two-Stage Bilinear Stochastic Programming Problem with Quantile Criterion, Appl. Stoch. Model. Business Ind., 2015, vol. 31, no. 6, pp. 862–874.
Ivanov, S.V., Bilevel Stochastic Linear Programming Problems with Quantile Criterion, Autom. Remote Control, 2014, vol. 75, no. 1, pp. 107–118.
Sen, S., Relaxation for Probabilistically Constrained Programs with Discrete Random Variables, Oper. Res. Lett., 1992, vol. 11, pp. 81–86.
Ruszczyński, A., Probabilistic Programming with Discrete Distributions and Precedence Constrained Knapsack Polyhedra, Math. Program., 2002, vol. 93, pp. 195–215.
Benati, S. and Rizzi, R., A Mixed Integer Linear Programming Formulation of the Optimal Mean/Valueat-Risk Portfolio Problem, Eur. J. Oper. Res., 2007, vol. 176, pp. 423–434.
Artstein, Z. and Wets, R.J.-B., Consistency of Minimizers and the SLLN for Stochastic Programs, J. Convex Anal., 1996, vol. 2, pp. 1–17.
Rockafellar, R.T. and Wets, R.J.-B., Variational Analysis, Berlin: Springer, 1998.
Shapiro, A., Dentcheva, D., and Ruszczyński, A., Lectures on Stochastic Programming. Modeling and Theory, in MPS/SIAM Series on Optimization, 2009, vol. 9.
Pagnoncelli, B.K., Ahmed, S., and Shapiro, A., Sample Average ApproximationMethod for Chance Constrained Programming: Theory and Applications, J. Optim. Theory Appl., 2009, vol. 142, pp. 399–416.
Luedtke, J. and Ahmed, S., A Sample Approximation Approach for Optimization with Probabilistic Constraints, SIAM J. Optim., 2008, vol. 19, no. 2, pp. 674–699.
Lepp, R., Approximate Solution of Stochastic Programming Problems with Recourse, Kybernetika, 1987, vol. 23, no. 6, pp. 476–482.
Lepp, R., Projection and Discretization Methods in Stochastic Programming, J. Comput. Appl. Math., 1994, vol. 56, pp. 55–64.
Choirat, C., Hess, C., and Seri, R., Approximation of Stochastic Programming Problems, in Monte Carlo and Quasi-Monte Carlo Methods 2004, Niederreiter, H. and Talay, D., Eds., Berlin: Springer-Verlag, 2006, pp. 45–59.
Pennanen, T. and Koivu, M., Epi-Convergent Discretizations of Stochastic Programs via Integration Quadratures, Numer. Math., 2005, vol. 100, pp. 141–163.
Pennanen, T., Epi-Convergent Discretizations of Multistage Stochastic Programs via Integration Quadratures, Math. Program., Ser. B, 2009, vol. 116, pp. 461–479.
Kibzun, A.I. and Ivanov, S.V., Convergence of Discrete Approximations of Stochastic Programming Problems with Probabilistic Criteria, in DOOR-2016, Lecture Notes Comput. Sci., Kochetov, Yu. et al., Eds., Heidelberg: Springer, 2016, vol. 9869, pp. 525–537.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.V. Ivanov, A.I. Kibzun, 2018, published in Avtomatika i Telemekhanika, 2018, No. 2, pp. 19–35.
Rights and permissions
About this article
Cite this article
Ivanov, S.V., Kibzun, A.I. On the Convergence of Sample Approximations for Stochastic Programming Problems with Probabilistic Criteria. Autom Remote Control 79, 216–228 (2018). https://doi.org/10.1134/S0005117918020029
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117918020029