Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.

[1]  Maarten H. van der Vlerk,et al.  Stochastic integer programming:General models and algorithms , 1999, Ann. Oper. Res..

[2]  R. Schultz,et al.  Two-stage stochastic integer programming : a survey , 1996 .

[3]  R. Schultz,et al.  Multistage Stochastic Integer Programs: An Introduction , 2001 .

[4]  Daniel Kuhn,et al.  Aggregation and discretization in multistage stochastic programming , 2008, Math. Program..

[5]  Nikolaos V. Sahinidis,et al.  A finite branch-and-bound algorithm for two-stage stochastic integer programs , 2004, Math. Program..

[6]  Daniel Kuhn,et al.  An Information-Based Approximation Scheme for Stochastic Optimization Problems in Continuous Time , 2009, Math. Oper. Res..

[7]  A. Ruszczynski.  Stochastic Programming Models , 2003 .

[8]  Marida Bertocchi,et al.  Bounds in Multistage Linear Stochastic Programming , 2014, J. Optim. Theory Appl..

[9]  S. Sen.  Algorithms for Stochastic Mixed-Integer Programming Models , 2005 .

[10]  Alexander Shapiro,et al.  Stochastic programming approach to optimization under uncertainty , 2007, Math. Program..

[11]  Hanif D. Sherali,et al.  Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming , 2006, Math. Program..

[12]  Hanif D. Sherali,et al.  On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables , 2006, Math. Program..

[13]  Marida Bertocchi,et al.  A scenario-based framework for supply planning under uncertainty: stochastic programming versus robust optimization approaches , 2016, Computational Management Science.

[14]  Maarten H. van der Vlerk.  Convex approximations for a class of mixed-integer recourse models , 2010, Ann. Oper. Res..

[15]  A. Ben-Tal,et al.  Bounds on the Expectation of a Convex Function of a Random Variable: With Applications to Stochastic Programming , 1977, Oper. Res..

[16]  Georg Ch. Pflug,et al.  Bounds and Approximations for Multistage Stochastic Programs , 2016, SIAM J. Optim..

[17]  Stein W. Wallace,et al.  Analyzing the quality of the expected value solution in stochastic programming , 2012, Ann. Oper. Res..

[18]  C. HuangC.,et al.  Bounds on the Expectation of a Convex Function of a Random Variable , 1977 .

[19]  W. T. Ziemba,et al.  Bounds on the value of information in uncertain decision problems , 1975 .

[20]  L. Escudero,et al.  The value of the stochastic solution in multistage problems , 2007 .

[21]  John R. Birge,et al.  Aggregation bounds in stochastic linear programming , 1985, Math. Program..

[22]  Marida Bertocchi,et al.  Monotonic bounds in multistage mixed-integer stochastic programming , 2016, Computational Management Science.

[23]  John R. Birge,et al.  The value of the stochastic solution in stochastic linear programs with fixed recourse , 1982, Math. Program..

[24]  Kai Huang,et al.  The Value of Multistage Stochastic Programming in Capacity Planning Under Uncertainty , 2009, Oper. Res..

[25]  A. Madansky.  Inequalities for Stochastic Linear Programming Problems , 1960 .

[26]  Karl Frauendorfer,et al.  Solving SLP Recourse Problems with Arbitrary Multivariate Distributions - The Dependent Case , 1988, Math. Oper. Res..

[27]  Suvrajeet Sen,et al.  A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems , 2004, Manag. Sci..

[28]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[29]  Mordecai Avriel,et al.  The Value of Information and Stochastic Programming , 1970, Oper. Res..

[30]  Andrew J. Schaefer,et al.  A hierarchy of bounds for stochastic mixed-integer programs , 2013, Math. Program..

[31]  Samer Takriti,et al.  Improving aggregation bounds for two-stage stochastic programs , 1999, Oper. Res. Lett..

[32]  Ilan Vertinsky,et al.  Sharp Bounds on the Value of Perfect Information , 1977, Oper. Res..

[33]  H. Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[34]  Burhaneddin Sandıkçı,et al.  A Scalable Bounding Method for Multi-Stage Stochastic Integer Programs , 2014 .

[35]  J. Jensen.  Sur les fonctions convexes et les inégalités entre les valeurs moyennes , 1906 .

[36]  A. Madansky.  Bounds on the Expectation of a Convex Function of a Multivariate Random Variable , 1959 .

[37]  Kurt M. Anstreicher,et al.  Linear Programming in O([n3/ln n]L) Operations , 1999, SIAM J. Optim..