Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming

Multistage stochastic programs can be approximated by restricting policies to follow decision rules. Directly applying this idea to problems with integer decisions is difficult because of the need for decision rules that lead to integral decisions. In this work, we introduce Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed integer programming (MSMIP) which overcome this difficulty by applying decision rules in a Lagrangian dual of the MSMIP. We propose two new bounding techniques based on stagewise (SW) and nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies are restricted by LDDRs. We demonstrate how the solutions from these duals can be used to drive primal policies. Our proposal requires fewer assumptions than most existing MSMIP methods. We compare the theoretical strength of the restricted duals and show that the restricted NA dual can provide relaxation bounds at least as good as the ones obtained by the restricted SW dual. In our numerical study, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems.

[1]  T. Morton,et al.  Capacity Expansion and Replacement in Growing Markets with Uncertain Technological Breakthroughs , 1998 .

[2]  A. Wood,et al.  Transmission investment planning using SDDP , 2007, 2007 Australasian Universities Power Engineering Conference.

[3]  Werner Römisch,et al.  Duality gaps in nonconvex stochastic optimization , 2004, Math. Program..

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

[5]  Andy Philpott,et al.  MIDAS: A mixed integer dynamic approximation scheme , 2020, Math. Program..

[6]  Cosmin G. Petra,et al.  On Parallelizing Dual Decomposition in Stochastic Integer Programming , 2012, Oper. Res. Lett..

[7]  Andrzej Ruszczynski,et al.  On augmented Lagrangian decomposition methods for multistage stochastic programs , 1996, Ann. Oper. Res..

[8]  Andrzej Ruszczynski,et al.  A regularized decomposition method for minimizing a sum of polyhedral functions , 1986, Math. Program..

[9]  Laureano F. Escudero,et al.  BFC, A branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs , 2003, Eur. J. Oper. Res..

[10]  Werner Römisch,et al.  Stochastic Lagrangian Relaxation Applied to Power Scheduling in a Hydro-Thermal System under Uncertainty , 2000, Ann. Oper. Res..

[11]  Alexander Shapiro,et al.  On Complexity of Stochastic Programming Problems , 2005 .

[12]  Arild Helseth,et al.  Co-optimizing sales of energy and capacity in a hydropower scheduling model , 2015, 2015 IEEE Eindhoven PowerTech.

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

[14]  Mario V. F. Pereira,et al.  Long-term optimal allocation of hydro generation for a price-maker company in a competitive market: latest developments and a stochastic dual dynamic programming approach , 2010 .

[15]  Rommert Dekker,et al.  A Scenario Aggregation - Based Approach for Determining a Robust Airline Fleet Composition for Dynamic Capacity Allocation , 2002, Transp. Sci..

[16]  Kengy Barty,et al.  Decomposition of large-scale stochastic optimal control problems , 2009, RAIRO Oper. Res..

[17]  Georg Ch. Pflug,et al.  A branch and bound method for stochastic global optimization , 1998, Math. Program..

[18]  David L. Woodruff,et al.  Progressive hedging and tabu search applied to mixed integer (0,1) multistage stochastic programming , 1996, J. Heuristics.

[19]  Julia L. Higle,et al.  Stochastic Decomposition: A Statistical Method for Large Scale Stochastic Linear Programming , 1996 .

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

[21]  Daniel Kuhn,et al.  Generalized decision rule approximations for stochastic programming via liftings , 2014, Mathematical Programming.

[22]  Zhi-Long Chen,et al.  A scenario-based stochastic programming approach for technology and capacity planning , 2002, Comput. Oper. Res..

[23]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[24]  Bernardo Freitas Paulo da Costa,et al.  Stochastic Lipschitz dynamic programming , 2019, Math. Program..

[25]  Peng Sun,et al.  Information Relaxations and Duality in Stochastic Dynamic Programs , 2010, Oper. Res..

[26]  A. Gjelsvik,et al.  Optimisation of hydropower operation in a liberalised market with focus on price modelling , 2001, 2001 IEEE Porto Power Tech Proceedings (Cat. No.01EX502).

[27]  John R. Birge,et al.  A stochastic model for the unit commitment problem , 1996 .

[28]  R. Tyrrell Rockafellar,et al.  Scenarios and Policy Aggregation in Optimization Under Uncertainty , 1991, Math. Oper. Res..

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

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

[31]  James R. Luedtke,et al.  Two-stage linear decision rules for multi-stage stochastic programming , 2017, Math. Program..

[32]  R. Kevin Wood,et al.  Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems , 2009, Oper. Res..

[33]  Peng Sun,et al.  A Linear Decision-Based Approximation Approach to Stochastic Programming , 2008, Oper. Res..

[34]  Alexander Shapiro,et al.  Stochastic Approximation approach to Stochastic Programming , 2013 .

[35]  Jikai Zou.  Large scale multistage stochastic integer programming with applications in electric power systems , 2017 .

[36]  Brian T. Denton,et al.  A Progressive Hedging Approach for Surgery Planning Under Uncertainty , 2015, INFORMS J. Comput..

[37]  Rüdiger Schultz,et al.  Dual decomposition in stochastic integer programming , 1999, Oper. Res. Lett..

[38]  K. Kiwiel.  Approximations in proximal bundle methods and decomposition of convex programs , 1995 .

[39]  Changzheng Liu,et al.  Solving Stochastic Transportation Network Protection Problems Using the Progressive Hedging-based Method , 2010 .

[40]  Yongpei Guan,et al.  Cutting Planes for Multistage Stochastic Integer Programs , 2009, Oper. Res..

[41]  Stefan Minner,et al.  Optimizing Trading Decisions for Hydro Storage Systems Using Approximate Dual Dynamic Programming , 2013, Oper. Res..

[42]  Zhihao Cen.  Solving multi-stage stochastic mixed integer linear programs by the dual dynamic programming approach , 2012 .

[43]  Daniel Kuhn,et al.  Primal and dual linear decision rules in stochastic and robust optimization , 2011, Math. Program..

[44]  Santiago Cerisola,et al.  Stochastic dual dynamic programming applied to nonconvex hydrothermal models , 2012, Eur. J. Oper. Res..

[45]  David L. Woodruff,et al.  Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems , 2011, Comput. Manag. Sci..

[46]  Stefan Helber,et al.  Dynamic capacitated lot sizing with random demand and dynamic safety stocks , 2013, OR Spectr..

[47]  Daniel Kuhn,et al.  Scenario-free stochastic programming with polynomial decision rules , 2011, IEEE Conference on Decision and Control and European Control Conference.

[48]  David L. Woodruff,et al.  Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs , 2016, Math. Program..