# Stochastic programming

**DOI:**https://doi.org/10.1007/1-4020-0611-X_1005

## INTRODUCTION

Stochastic programming (SP) deals with a class of optimization models and algorithms in which some of the data may be subject to significant uncertainty. Such models are appropriate when data evolve over time and decisions need to be made prior to observing the entire data stream. For instance, investment decisions in portfolio planning problems must be implemented before stock performance can be observed. Similarly, utilities must plan power generation before the demand for electricity is realized. Such inherent uncertainty is amplified by technological innovation and market forces.

As a special example, consider the electric power industry. Deregulation of the electric power market and the possibility of personal electricity generators (e.g. gas turbines) are some of the causes of uncertainty in the industry. Under these circumstances it pays to develop models in which plans are evaluated against a variety of future scenarios that represent alternative outcomes of data....

## References

- [1]Beale, E.M.L (1955). “On minimizing a convex function subject to linear inequalities,” Jl. Royal Statistical Society, Series B, 17, 173–184.Google Scholar
- [2]Benders, J.F. (1962). “Partitioning procedures for solving mixed variables programming problems,” Numerische Mathematik, 4, 238–252.Google Scholar
- [3]Bertsekas, D. and Tsitsiklis, J. (1996). Neuro-Dynamic Programming, Athena Scientific, Belmont, Massachusetts.Google Scholar
- [4]Birge, J.R. (1985). “Decomposition and partitioning methods for multi-stage stochastic linear programs,” Operations Research, 33, 989–1007.Google Scholar
- [5]Birge, J.R. (1997). “Stochastic Programming Computation and Applications,” INFORMS Jl. Computing, 9, 111–133.Google Scholar
- [6]Birge, J.R. and Louveaux, F.V. (1997). Introduction to Stochastic Programming, Springer, New York.Google Scholar
- [7]Cariño, D.R., Kent, T., Meyers, D.H., Stacy, C., Sylvanus, M., Turner, A.L., Watanabe, K., and Ziemba, W.T. (1994). “The Russell-Yasuda Kasai Model: An Asset/Liability Model for a Japanese Insurance Company Using Multistage Stochastic Programming,” Interfaces, 24(1), 29–49.Google Scholar
- [8]Charnes, A. and Cooper, W.W. (1959). “Chance-constrained Programming,” Management Science,5,73–79.Google Scholar
- [9]Dantzig, G.B. (1955). “Linear Programming Under Uncertainty,” Management Science,1, 197–206.Google Scholar
- [10]Edirisinghe, N.C.P. and Ziemba, W.T. (1996). “Implementing bounds-based approximations in convex-concave two stage programming,” Mathematical Programming, 19, 314–340.Google Scholar
- [11]Fisher, M., Hammond, J., Obermeyer, W., and Raman, A. (1997). “Configuring a Supply Chain to Reduce the Cost of Demand Uncertainty,” Production and Operations Management, 6, 211–225.Google Scholar
- [12]Frauendorfer, K. (1992). “Stochastic Two-Stage Programming,” Lecture Notes in Economics and Mathematical Systems, 392, Springer-Verlag, Berlin.Google Scholar
- [13]Frauendorfer, K. (1994). “Multistage stochastic programming: error analysis for the convex case,” Zeitschrift für Operations Research, 39, 93–122.Google Scholar
- [14]Gassmann, H.I. and Ireland, A.M. (1996). “On the formulation of stochastic linear programs using algebraic modelling languages,” Annals Operations Research, 64, 83–112.Google Scholar
- [15]Higle, J.L. and Sen, S. (1991). “Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse,” Math. of Operations Research, 16, 650–669.Google Scholar
- [16]Higle, J.L. and Sen, S. (1992). “On the Convergence of Algorithms with Implications for Stochastic and Nondifferentiable Optimization,” Math. of Operations Research, 17, 112–131.Google Scholar
- [17]Higle, J.L. and Sen, S. (1995). “Epigraphical Nesting: A Unifying Theory for the Convergence of Algorithms,” Jl. Optimization Theory and Applications, 84, 339–360.Google Scholar
- [18]Higle, J.L. and Sen, S. (1996). Stochastic Decomposition: A Statistical Method for Large Scale Stochastic Linear Programming, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
- [19]Higle, J.L. and Sen, S. (1999). “Statistical approximations for stochastic linear programming problems,” Annals Operations Research, 85, 173–192.Google Scholar
- [20]Kall, P. (1976). Stochastic Linear Programming, Springer-Verlag, Berlin.Google Scholar
- [21]Kall, P. and Wallace, S.W. (1994). Stochastic Programming, John Wiley, Chichester, England.Google Scholar
- [22]Kall, P. and Mayer, J. (1996). “SLP-IOR: an interactive model management system for stochastic linear programs,” Mathematical Programming, Series B, 75, 221–240.Google Scholar
- [23]King, A.J. and Wets, R.J-B. (1991). “Epi-consistency of convex stochastic programs,” Stochastics, 34, 83–92.Google Scholar
- [24]Laporte, G. and Louveaux, F.V. (1993). “The Integer L-shaped Method for Stochastic Integer Programs with Complete Recourse,” Operations Research Letters, 13, 133–142.Google Scholar
- [25]Medova, E. (1998). “Chance constrained stochastic programming for integrated services network management,” Annals Operations Research, 81, 213–229.Google Scholar
- [26]Mulvey, J.M. and Ruszczyński, A. (1995). “A New Scenario Decomposition Method for Large Scale Stochastic Optimization,” Operations Research, 43, 477–490.Google Scholar
- [27]Murphy, F.H., Sen, S., and Soyster, A.L. (1982). “Electric Utility Capacity Expansion Planning with Uncertain Load Forecasts,” AIIE Transactions, 14, 52–59.Google Scholar
- [28]Nielsen, S.S. and Zenios, S.A. (1993). “A Massively Parallel Algorithm for Nonlinear Stochastic Network Problems,” Operations Research, 41, 319–337.Google Scholar
- [29]Norkin, V.I., Ermoliev, Y.M., and Ruszczyński, A. (1998). On Optimal Allocation of Indivisibles under Uncertainty,” Operations Research, 46, 381–395.Google Scholar
- [30]Prékopa, A. (1971). “Logarithmic concave measures with application to stochastic programming,”Acta Scientiarium Mathematiarum(
*Szeged*), 32, 301–316.*Strategic choice*789Google Scholar - [31]Prékopa, A. (1995). Stochastic Programming, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
- [32]Robinson, S.M. (1996). “Analysis of Sample Path Optimization,” Math. of Operations Research, 21, 513–528.Google Scholar
- [33]Rockafellar, R.T. and Wets, R.J-B. (1991). “Scenarios and Policy Aggregation in Optimization Under Uncertainty,” Math. of Operations Research, 16, 119–147.Google Scholar
- [34]Rockafellar, R.T and Wets, R.J-B. (1998). Variational Analysis, Springer-Verlag, Berlin.Google Scholar
- [35]Romisch, W. and Schultz, R. (1991). “Distribution sensitivity in stochastic programming,” Mathematical Programming, 50, 197–226.Google Scholar
- [36]Schultz, R. (1993). “Continuity Properties of Expectation Functions in Stochastic Integer Programming,” Math. of Operations Research, 18, 578–589.Google Scholar
- [37]Sen, S. (1992). “Relaxations for Probabilistically Constrained Programs with Discrete Random Variables,” Operations Research Letters, 11, 81–86.Google Scholar
- [38]Sen, S.R., D. Doverspike, and Cosares, S. (1994). “Network Planning with Random Demand,” Telecommunication Systems, 3, 11–30.Google Scholar
- [39]Sen, S. and Higle, J.L. (1999). “An Introductory Tutorial on Stochastic Linear Programming Models,” Interfaces, 29(2), 33–61.Google Scholar
- [40]Shapiro, A. (1991). “Asymptotic analysis of stochastic programs,” Annals Operations Research, 30, 169–186.Google Scholar
- [41]Shapiro, A. and Homem-deMello, T. (1998). “A simulation-based approach to stochastic programming with recourse,” Mathematical Programming, 81, 301–325.Google Scholar
- [42]Van Slyke, R. and Wets, R.J-B. (1969). “L-Shaped Linear Programs with Application to Optimal Control and Stochastic Programming,” SIAM Jl. Appl. Math., 17, 638–663.Google Scholar