Abstract
A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem.
Similar content being viewed by others
References
Yu.M. Ermoliev, Stochastic quasi-gradient methods and their application to systems optimization, Stochastics 4 (1983) 1–37.
J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research 33 (1985) 989–1007.
J.M. Mulvey, A. Ruszczyński, A new scenario decomposition method for large-scale stochastic optimization. Operations Research 43 (1995) 477–490.
R.T. Rockafellar, R.J.-B. Wets, Scenarios and policy aggregation in optimization under uncertainty, Mathematics of Operations Research 16 (1991) 1–23.
R.J.-B. Wets, Large scale linear programming techniques, in: Yu. Ermoliev, R.J.-B. Wets (Eds.), Numerical Methods in Stochastic Programming, Springer, Berlin, 1988, pp. 65–94.
V.S. Mikhalevich, A.M. Gupal, V.I. Norkin, Methods of Non-Convex Optimization, Nauka, Moscow, 1987 (in Russian).
V.I. Norkin, Yu.M. Ermoliev, A. Ruszczyński, On optimal allocation of indivisibles under uncertainty, Operations Research to appear in 1998.
R. Horst, P.M. Pardalos (Eds.), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, 1994.
R.Y. Rubinstein, A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, Wiley, New York, 1993.
V.I. Norkin, The analysis and optimization of probability functions, Working Paper WP-93-6, International Institute of Applied System Analysis, Laxenburg, Austria, 1993.
A. Prekopa, Logarithmic concave measures and related topics, in: M.A.H. Dempster (Ed.), Stochastic Programming, Academic Press, London, pp. 63–82.
P. Hansen, B. Jaumard, H. Tuy, Global optimization in location, in: Z. Drezner (Ed.), Facility Location—A Survey of Applications and Methods, Springer Series in Operations Research, Springer, Berlin, pp. 43–68.
M. Labbè, D. Peeters, J.-F. Thisse, Location on networks, in: M.O. Ball (Ed.), Handbooks in OR & MS, vol. 8, ch. 7, Elsevier, Amsterdam, pp. 551–624.
B.J. Lence, A. Ruszczyński, Managing water quality under uncertainty: application of a new stochastic branch and bound method, Working Paper WP-96-66, International Institute for Applied System Analysis, Laxenburg, Austria, accepted for publication in: Risk, Reliability, Uncertainty and Robustness of Water Resources Systems, Cambridge University Press, Cambridge, UK.
K. Hägglöf, The implementation of the stochastic branch and bound method for applications in river basin water quality management, Working paper WP-96-89, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1996.
R. Horst, H. Tuy, Global Optimization, Springer, Berlin, 1990.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Norkin, V.I., Pflug, G.C. & Ruszczyński, A. A branch and bound method for stochastic global optimization. Mathematical Programming 83, 425–450 (1998). https://doi.org/10.1007/BF02680569
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02680569