Adaptive Partition-based SDDP Algorithms for Multistage Stochastic Linear Programming

In this paper, we extend the adaptive partition-based approach for solving two-stage stochastic programs with fixed recourse to the multistage stochastic programming setting. The proposed algorithms integrate the adaptive partition-based strategy with a popular approach for solving multistage stochastic programs, the stochastic dual dynamic programming, via different tree-traversal strategies in order to enhance its computational efficiency. Our numerical experiments on a hydro-thermal power generation planning problem show the effectiveness of the proposed algorithms.

[1]  Paul H. Zipkin,et al.  Bounds for Row-Aggregation in Linear Programming , 1980, Oper. Res..

[2]  Yongjia Song,et al.  Adaptive Partition-Based Level Decomposition Methods for Solving Two-Stage Stochastic Programs with Fixed Recourse , 2018, INFORMS J. Comput..

[3]  Warrren B Powell,et al.  Convergent Cutting-Plane and Partial-Sampling Algorithm for Multistage Stochastic Linear Programs with Recourse , 1999 .

[4]  Ignacio E. Grossmann,et al.  A stochastic programming approach to planning of offshore gas field developments under uncertainty in reserves , 2004, Comput. Chem. Eng..

[5]  M. Tzur,et al.  The multilocation transshipment problem , 2006 .

[6]  Wittrock.  Advances in a nested decomposition algorithm for solving staircase linear programs. Technical report SOL 83-2 , 1983 .

[7]  Andy Philpott,et al.  On the Convergence of Sampling-Based Decomposition Algorithms for Multistage Stochastic Programs , 2005 .

[8]  R. Wets,et al.  Stochastic programming , 1989 .

[9]  J. Jeffry Howbert,et al.  The Maximum Clique Problem , 2007 .

[10]  Golbon Zakeri,et al.  A deterministic algorithm for solving multistage stochastic programming problems , 2017 .

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

[12]  Laureano F. Escudero,et al.  A stochastic 0-1 program based approach for the air traffic flow management problem , 2000, Eur. J. Oper. Res..

[13]  Pantuso Giovanni.  The Football Team Composition Problem: a Stochastic Programming approach , 2017 .

[14]  S. Granville,et al.  Strategic bidding under uncertainty: a binary expansion approach , 2005, IEEE Transactions on Power Systems.

[15]  Warren B. Powell,et al.  Regularized Decomposition of High-Dimensional Multistage Stochastic Programs with Markov Uncertainty , 2015, SIAM J. Optim..

[16]  James R. Luedtke,et al.  An Adaptive Partition-Based Approach for Solving Two-Stage Stochastic Programs with Fixed Recourse , 2015, SIAM J. Optim..

[17]  Andrew B. Philpott,et al.  On the convergence of stochastic dual dynamic programming and related methods , 2008, Oper. Res. Lett..

[18]  S. E. Wright,et al.  Primal-Dual Aggregation and Disaggregation for Stochastic Linear Programs , 1994, Math. Oper. Res..

[19]  Vincent Guigues.  Inexact Cuts in Stochastic Dual Dynamic Programming , 2020, SIAM J. Optim..

[20]  Zhihong Zhou,et al.  Multistage Stochastic Decomposition: A Bridge between Stochastic Programming and Approximate Dynamic Programming , 2014, SIAM J. Optim..

[21]  Jitka Dupačová,et al.  Portfolio Optimization and risk management via stochastic programming , 2009 .

[22]  Claudia A. Sagastizábal,et al.  Level bundle methods for oracles with on-demand accuracy , 2014, Optim. Methods Softw..

[23]  Vincent Guigues.  Inexact cuts for Deterministic and Stochastic Dual Dynamic Programming applied to convex nonlinear optimization problems , 2017 .

[24]  Lewis Ntaimo,et al.  Adaptive multicut aggregation for two-stage stochastic linear programs with recourse , 2010, Eur. J. Oper. Res..

[25]  R. Wets,et al.  L-SHAPED LINEAR PROGRAMS WITH APPLICATIONS TO OPTIMAL CONTROL AND STOCHASTIC PROGRAMMING. , 1969 .

[26]  Shabbir Ahmed,et al.  A Multi-Stage Stochastic Integer Programming Approach for Capacity Expansion under Uncertainty , 2003, J. Glob. Optim..

[27]  David P. Morton,et al.  Assessing policy quality in a multistage stochastic program for long-term hydrothermal scheduling , 2017, Ann. Oper. Res..

[28]  Paul H. Zipkin,et al.  Bounds on the Effect of Aggregating Variables in Linear Programs , 1980, Oper. Res..

[29]  M. V. F. Pereira,et al.  Multi-stage stochastic optimization applied to energy planning , 1991, Math. Program..

[30]  Steffen Rebennack,et al.  Combining sampling-based and scenario-based nested Benders decomposition methods: application to stochastic dual dynamic programming , 2015, Mathematical Programming.

[31]  Giovanni Pantuso,et al.  The Football Team Composition Problem: a Stochastic Programming approach , 2017 .

[32]  Alexander Shapiro,et al.  Risk neutral and risk averse Stochastic Dual Dynamic Programming method , 2013, Eur. J. Oper. Res..

[33]  James R. Evans,et al.  Aggregation and Disaggregation Techniques and Methodology in Optimization , 1991, Oper. Res..

[34]  G. Pflug,et al.  Modeling, Measuring and Managing Risk , 2008 .

[35]  Monia Rekik,et al.  Stochastic dual dynamic programming for transportation planning under demand uncertainty , 2013, 2013 International Conference on Advanced Logistics and Transport.

[36]  Alexander Shapiro,et al.  Lectures on Stochastic Programming - Modeling and Theory, Second Edition , 2014, MOS-SIAM Series on Optimization.

[37]  Jitka Dupacová,et al.  Asset-liability management for Czech pension funds using stochastic programming , 2009, Ann. Oper. Res..

[38]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

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

[40]  Iain Dunning,et al.  JuMP: A Modeling Language for Mathematical Optimization , 2015, SIAM Rev..

[41]  David P. Morton,et al.  An enhanced decomposition algorithm for multistage stochastic hydroelectric scheduling , 1996, Ann. Oper. Res..

[42]  Jean-Philippe Chancelier,et al.  Exact Converging Bounds for Stochastic Dual Dynamic Programming via Fenchel Duality , 2020, SIAM J. Optim..

[43]  Tito Homem-de-Mello,et al.  Sampling strategies and stopping criteria for stochastic dual dynamic programming: a case study in long-term hydrothermal scheduling , 2011 .

[44]  Alexander Shapiro,et al.  Analysis of stochastic dual dynamic programming method , 2011, Eur. J. Oper. Res..

[45]  Yongjia Song,et al.  On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems , 2019, Comput. Optim. Appl..

[46]  Robert L. Smith,et al.  Aggregation in Dynamic Programming , 1987, Oper. Res..

[47]  Sverre Storøy,et al.  Aggregation and Disaggregation in Integer Programming Problems , 1990, Oper. Res..