Bounds in multi-horizon stochastic programs

In this paper, we present bounds for multi-horizon stochastic optimization problems, a class of problems introduced in Kaut et al. (Comput Manag Sci 11:179–193, 2014) relevant in many industry-life applications typically involving strategic and operational decisions on two different time scales. After providing three general mathematical formulations of a multi-horizon stochastic program, we extend the definition of the traditional Expected Value problem and Wait-and-See problem from stochastic programming in a multi-horizon framework. New measures are introduced allowing to quantify the importance of the uncertainty at both strategic and operational levels. Relations among the solution approaches are then determined and chain of inequalities provided. Numerical experiments based on an energy planning application are finally presented.

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

[2]  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..

[3]  Asgeir Tomasgard,et al.  CenSES working paper 2 / 201 5 ISBN : 978-82-93198-15-4 A Multi-Stage Multi-Horizon Stochastic Equilibrium Model of Multi-Fuel Energy Markets , 2015 .

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

[5]  Karl Frauendorfer,et al.  Multistage Stochastic Programming: Barycentric Approximation , 2009, Encyclopedia of Optimization.

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

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

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

[9]  Siqian Shen,et al.  Scenario Grouping and Decomposition Algorithms for Chance-Constrained Programs , 2021, INFORMS J. Comput..

[10]  Asgeir Tomasgard,et al.  Risk measures in multi-horizon scenario trees , 2013 .

[11]  Alan Scheller-Wolf,et al.  Strategic analysis of technology and capacity investments in the liquefied natural gas industry , 2013, Eur. J. Oper. Res..

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

[13]  Asgeir Tomasgard,et al.  Natural Gas Infrastructure Design with an Operational Perspective , 2012 .

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

[15]  Georg Ch. Pflug,et al.  Guaranteed Bounds for General Nondiscrete Multistage Risk-Averse Stochastic Optimization Programs , 2019, SIAM J. Optim..

[16]  Michal Kaut,et al.  Multi-horizon stochastic programming , 2014, Comput. Manag. Sci..

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

[18]  Goran Andersson,et al.  Multi-horizon Modeling in Hydro Power Planning , 2016 .

[19]  Marida Bertocchi,et al.  Monotonic bounds in multistage mixed-integer stochastic programming , 2016, Comput. Manag. Sci..

[20]  D. Kuhn,et al.  Barycentric Bounds in Stochastic Programming: Theory and Application , 2010 .

[21]  Gerardo A. Pérez-Valdés,et al.  A multi-horizon stochastic programming model for the European power system , 2016 .

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

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

[24]  A. Tomasgard,et al.  The impact of policy actions and future energy prices on the cost-optimal development of the energy system in Norway and Sweden , 2017 .

[25]  A. Tomasgard,et al.  Short-term uncertainty in long-term energy system models — A case study of wind power in Denmark , 2015 .

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

[27]  D. Kuhn.  Generalized Bounds for Convex Multistage Stochastic Programs , 2004 .

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

[29]  John M. Wilson,et al.  Introduction to Stochastic Programming , 1998, J. Oper. Res. Soc..

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

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

[32]  Asgeir Tomasgard,et al.  Multi-Stage Stochastic Programming for Natural Gas Infrastructure Design with a Production Perspective , 2013 .

[33]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[34]  Shabbir Ahmed,et al.  Supply chain design under uncertainty using sample average approximation and dual decomposition , 2009, Eur. J. Oper. Res..

[35]  Hubert Abgottspon.  Hydro power planning: Multi-horizon modeling and its applications , 2015 .

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