Parallel Scenario Decomposition of Risk-Averse 0-1 Stochastic Programs

In this paper, we extend a recently proposed scenario decomposition algorithm for risk-neutral 0-1 stochastic programs to the risk-averse setting. Specifically, we consider two-stage risk-averse 0-1 stochastic programs with objective functions based on coherent risk measures. Using a dual representation of a coherent risk measure, we first derive an equivalent minimax reformulation of the considered problem. We then develop three variants of the scenario decomposition algorithm for this minimax formulation based on different relaxations of the nonanticipaticity constraints. The algorithms proceed by solving scenario subproblems to obtain candidate solutions and bounds and subsequently cutting off the candidate solutions from the search space to achieve convergence to an optimal solution. We design three parallelization schemes for implementing the algorithms with different tradeoffs between overhead time and computation time. Our computational results with risk-averse extensions of two standard stochastic...

[1]  Lewis Ntaimo,et al.  The Million-Variable “March” for Stochastic Combinatorial Optimization , 2005, J. Glob. Optim..

[2]  Peter S. Pacheco.  Parallel programming with MPI , 1996 .

[3]  Alexander Shapiro,et al.  Minimax and risk averse multistage stochastic programming , 2012, Eur. J. Oper. Res..

[4]  John R. Birge,et al.  A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs , 1996, Math. Program..

[5]  Shabbir Ahmed,et al.  Improving the Integer L-Shaped Method , 2016, INFORMS J. Comput..

[6]  Alexander Shapiro,et al.  On a Class of Minimax Stochastic Programs , 2004, SIAM J. Optim..

[7]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[8]  David L. Woodruff,et al.  Scalable Heuristics for a Class of Chance-Constrained Stochastic Programs , 2010, INFORMS J. Comput..

[9]  Michel Gendreau,et al.  Progressive hedging‐based metaheuristics for stochastic network design , 2011, Networks.

[10]  John R. Birge,et al.  Parallel decomposition of large-scale stochastic nonlinear programs , 1996, Ann. Oper. Res..

[11]  Naomi Miller,et al.  Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition , 2011, Oper. Res..

[12]  Anthony Skjellum,et al.  A High-Performance, Portable Implementation of the MPI Message Passing Interface Standard , 1996, Parallel Comput..

[13]  Shabbir Ahmed,et al.  Scenario Decomposition for 0-1 Stochastic Programs: Improvements and Asynchronous Implementation , 2016, 2016 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW).

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

[15]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[16]  Stephen J. Wright,et al.  Decomposition Algorithms for Stochastic Programming on a Computational Grid , 2001, Comput. Optim. Appl..

[17]  George Bosilca,et al.  Open MPI: Goals, Concept, and Design of a Next Generation MPI Implementation , 2004, PVM/MPI.

[18]  Stavros A. Zenios,et al.  Scalable Parallel Benders Decomposition for Stochastic Linear Programming , 1997, Parallel Comput..

[19]  R. Rockafellar,et al.  Nonanticipativity and L1-martingales in stochastic optimization problems , 1976 .

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

[21]  Jacek Gondzio,et al.  Building and Solving Large-Scale Stochastic Programs on an Affordable Distributed Computing System , 2000, Ann. Oper. Res..

[22]  Teodor Gabriel Crainic,et al.  Scenario grouping in a progressive hedging-based meta-heuristic for stochastic network design , 2014, Comput. Oper. Res..

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

[24]  Andrzej Ruszczynski,et al.  Scenario decomposition of risk-averse multistage stochastic programming problems , 2012, Ann. Oper. Res..

[25]  Andrzej Ruszczynski,et al.  Parallel decomposition of multistage stochastic programming problems , 1993, Math. Program..

[26]  James R. Luedtke,et al.  Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs , 2016, Mathematical Programming.

[27]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

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

[29]  Stephen J. Wright,et al.  5. Computational Grids for Stochastic Programming , 2005, Applications of Stochastic Programming.

[30]  John M. Mulvey,et al.  A New Scenario Decomposition Method for Large-Scale Stochastic Optimization , 1995, Oper. Res..

[31]  Stephen J. Wright.  Computational Grids for Stochastic Programming , 2005 .

[32]  Antonio Alonso Ayuso,et al.  Introduction to Stochastic Programming , 2009 .

[33]  Andrzej Ruszczyński,et al.  Advances in Risk-Averse Optimization , 2013 .

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

[35]  Shabbir Ahmed,et al.  A scenario decomposition algorithm for 0-1 stochastic programs , 2013, Oper. Res. Lett..