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On a mixture of the fix-and-relax coordination and Lagrangian substitution schemes for multistage stochastic mixed integer programming

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Abstract

We present a framework for solving large-scale multistage mixed 0–1 optimization problems under uncertainty in the coefficients of the objective function, the right-hand side vector, and the constraint matrix. A scenario tree-based scheme is used to represent the Deterministic Equivalent Model of the stochastic mixed 0–1 program with complete recourse. The constraints are modeled by a splitting variable representation via scenarios. So, a mixed 0–1 model for each scenario cluster is considered, plus the nonanticipativity constraints that equate the 0–1 and continuous so-called common variables from the same group of scenarios in each stage. Given the high dimensions of the stochastic instances in the real world, it is not realistic to obtain the optimal solution for the problem. Instead we use the so-called Fix-and-Relax Coordination (FRC) algorithm to exploit the characteristics of the nonanticipativity constraints of the stochastic model. A mixture of the FRC approach and the Lagrangian Substitution and Decomposition schemes is proposed for satisfying, both, the integrality constraints for the 0–1 variables and the nonanticipativity constraints.

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References

  • Ahmed S (2004) Mean-risk objectives in stochastic programming. Stochastic programming e-print series, http://dochost.rz.hu-berlin.de/speps

  • Ahmed S, King AJ, Parija G (2003) A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. J Glob Optim 26:3–24

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Garín A, Ortuño MT, Pérez G (2003a) An approach for strategic supply chain planning based on stochastic 0–1 programming. J Glob Optim 26:97–124

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Ortuño MT (2003b) BFC, a branch-and-fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0–1 programs. Eur J Oper Res 151:503–519

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Garín A, Ortuño MT, Pérez G (2005a) On the product selection and plant dimensioning problem under uncertainty. Omega 33:307–318

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Ortuño MT (2005b) Modeling production planning and scheduling under uncertainty. In: Wallace SW, Ziemba WT (eds) Applications of stochastic programming. MPS-SIAM-series in optimization, pp 217–252

  • Alonso-Ayuso A, Escudero LF, Pizarrro C, Romeijn HE, Romero Morales D (2006) On solving the multi-period single-sourcing problem under uncertainty. Comput Manag Sci 3:29–53

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Ortuño MT, Pizarro C (2007) On a stochastic sequencing and scheduling problem. Comput Oper Res 34:2604–2624

    Article  Google Scholar 

  • Alonso-Ayuso A, Escudero LF, Guignard M, Quinteros M, Weintraub A (2009) Forestry management under uncertainty. Ann Oper Res (accepted for publication)

  • Andrade R, Lisser A, Maculan N, Plateau G (2006) Enhancing a branch-and-bound algorithm for two-stage stochastic integer network design-based models. Manag Sci 52:1450–1455

    Article  Google Scholar 

  • Barahona F, Anbil R (2000) The volume algorithm: producing primal solutions with a subgradient method. Math Program 87:385–399

    Article  Google Scholar 

  • Benders JF (1962) Partitioning procedures for solving mixed variables programming problems. Numer Math 4:238–252

    Article  Google Scholar 

  • Bertsekas DP (1982) Constrained optimization and Lagrange multiplier methods. Academic Press, San Diego

    Google Scholar 

  • Bertsekas DP (1995) Nonlinear programming. Athena Scientific, pp 487–532

  • Birge JR, Louveaux FV (1997) Introduction to stochastic programming. Springer, Berlin

    Google Scholar 

  • Carøe CC, Tind J (1998) L-shaped decomposition of two-stage stochastic programs with integer recourse. Math Program 83:451–464

    Article  Google Scholar 

  • Carøe CC, Schultz R (1999) Dual decomposition in stochastic integer programming. Oper Res Lett 24:37–45

    Article  Google Scholar 

  • Escudero LF, de la Fuente JL, Garcia C, Prieto FJ (1999) A parallel computation approach for solving multistage stochastic network problems. Ann Oper Res 90:131–160

    Article  Google Scholar 

  • Escudero LF, Garín A, Merino M, Pérez G (2006) A multistage stochastic integer programming model and algorithm for incorporating logical constraints in assets and liabilities management under uncertainty. Comput Manag Sci. doi:10.1007/s10287-006-0035-7

    Google Scholar 

  • Escudero LF, Garín A, Merino M, Pérez G (2007) A two-stage stochastic integer programming approach as a mixture of branch-and-fix coordination and benders decomposition schemes. Ann Oper Res 152:395–420

    Article  Google Scholar 

  • Escudero LF, Garín A, Merino M, Pérez G (2009a) A general algorithm for solving two-stage stochastic mixed 0–1 first stage problems. Comput Oper Res 36:2590–2600

    Article  Google Scholar 

  • Escudero LF, Garín A, Merino M, Pérez G (2009b) BFC-MSMIP: an exact branch-and-fix coordination approach for solving multistage stochastic mixed 0–1 problems. TOP (this volume)

  • Gendreau M, Laporte G, Séguin R (1996) Stochastic vehicle routing. Eur J Oper Res 88:3–12

    Article  Google Scholar 

  • Groewe-Kuska N, Kiwiel K, Nowak MP, Römisch W, Wegner I (2001) Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation. In: Greengard C, Ruszczynski A (eds) Decision making under uncertainty: energy and power, pp 39–70

  • Guignard M (2003a) Lagrangean decomposition and Lagrangean substitution for stochastic integer programming. Technical note. OPIM Dept., Wharton School, University of Pennsylvania, Philadelphia, USA

  • Guignard M (2003b) Lagrangean relaxation. TOP 11:151–228

    Article  Google Scholar 

  • Held M, Karp RM (1970) The traveling salesman problem and the minimum spanning trees. Oper Res 18:1138–1162

    Article  Google Scholar 

  • Hemmecke R, Schultz R (2001) Decomposition methods for two-stage stochastic integer programs. In: Grötschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems. Springer, Berlin, pp 601–622

    Google Scholar 

  • Hernández P, Alonso-Ayuso A, Escudero LF, Guignard M, Marianov V, Weintraub A (2009) Prison facility site selection under uncertainty. Eur J Oper Res (referring process, in 2nd revision)

  • Jimenez Redondo N, Conejo AJ (1997) Short-term hydro-thermal coordination by Lagrangian relaxation: solution of the dual problem. IEEE Trans Power Syst 14:89–95

    Article  Google Scholar 

  • Klein Haneveld WK, van der Vlerk MH (2001) Optimizing electricity distribution using integer recourse models. In: Uryasev S, Pardalos PM (eds) Stochastic optimization: algorithms and applications. Kluwer Academic, Berlin, pp 137–154

    Google Scholar 

  • Laporte G, Louveaux FV (1993) The integer L-shaped method for stochastic integer programs with complete recourse. Oper Res Lett 13:133–142

    Article  Google Scholar 

  • Laporte G, Louveaux FV (2002) An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Oper Res 50:415–423

    Article  Google Scholar 

  • Lulli G, Sen S (2002) Stochastic batch-sizing problems: models and algorithms. In: Woodruff DF (ed) Stochastic integer programming and network interdiction models. Kluwer Academic, Dordrecht

    Google Scholar 

  • Lulli G, Sen S (2004) A branch-and-price algorithm for multi-stage stochastic integer programming with application to stochastic batch-sizing problems. Manag Sci 50:786–796

    Article  Google Scholar 

  • Lulli G, Sen S (2006) A heuristic procedure for stochastic integer programs with complete recourse. Eur J Oper Res 171:879–890

    Article  Google Scholar 

  • MirHassani SA, Lucas C, Mitra G, Poojari CA (2000) Computational solution of a capacity planning model under uncertainty. Parallel Comput J 26:511–538

    Article  Google Scholar 

  • Mulvey JM, Ruszczynski A (1992) A diagonal quadratic approximation method for large-scale linear programs. Oper Res Lett 12:205–221

    Article  Google Scholar 

  • Ntaimo L, Sen S (2005) The million variable “march” for stochastic combinatorial optimization. J Glob Optim 32:385–400

    Article  Google Scholar 

  • Nürnberg R, Römisch W (2002) A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty. Optim Eng 3:355–378

    Article  Google Scholar 

  • Ogryczak W, Ruszczynski A (1999) From stochastic dominance to mean-risk models: semi-deviations as risk measures. Eur J Oper Res 116:33–50

    Article  Google Scholar 

  • Polyak BT (1987) Introduction to optimization software. Inc. Publications Division, NY

  • Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–41

    Google Scholar 

  • Rockafellar RT, Wets RJ-B (1991) Scenario and policy aggregation in optimisation under uncertainty. Math Oper Res 16:119–147

    Article  Google Scholar 

  • Römisch W, Schultz R (2001) Multi-stage stochastic integer programs: an introduction. In: Grötschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems. Springer, Berlin, pp 581–600

    Google Scholar 

  • Ruszczynski A (1989) An augmented Lagrangian decomposition for block diagonal programming problems. Oper Res Lett 8:287–294

    Article  Google Scholar 

  • Santoso T, Ahmed S, Goetschalckx M, Shapiro A (2005) A stochastic programming approach for supply network design under uncertainty. Eur J Oper Res 167:96–115

    Article  Google Scholar 

  • Schultz R (2003) Stochastic programming with integer variables. Math Program, Ser B 97:285–309

    Google Scholar 

  • Schultz R, Tiedemann S (2004) Risk aversion via excess probabilities in stochastic programs with mixed-integer recourse. SIAM J Optim 14:115–138

    Article  Google Scholar 

  • Schultz R, Tiedemann S (2006) Conditional value-at-risk in stochastic programs with mixed integer recourse. Math Program, Ser B 105:365–386

    Article  Google Scholar 

  • Schultz R, Nowak M, Westphalen M (2005) A stochastic integer programming model for incorporating day-ahead trading of electricity into hydro-thermal unit commitment. Optim Eng 6:163–176

    Article  Google Scholar 

  • Sen S, Higle JL (2005) The C3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: set convexification. Math Program, Ser A 104:1–20

    Article  Google Scholar 

  • Sen S, Sherali HD (2006) Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math Program, Ser A 106:203–223

    Article  Google Scholar 

  • Sherali HD, Zhu X (2006) On solving discrete two stage stochastic programs having mixed-integer first and second stage variables. Math Program, Ser A 108:597–611

    Article  Google Scholar 

  • Takriti S, Birge JR (2000) Lagrangean solution techniques and bounds for loosely coupled mixed-integer stochastic programs. Oper Res 48:91–98

    Article  Google Scholar 

  • Triki Ch, Beraldi P, Gross G (2005) Optimal capacity allocation in multi-auction electricity markets under uncertainty. Comput Oper Res 32:201–217

    Google Scholar 

  • Tsiakis P, Shah N, Pantelides CC (2001) Design of a multiechelon supply chain network under demand uncertainty. Ind Eng Chem Res 40:3585–3604

    Article  Google Scholar 

  • Uryasev S, Pardalos PM (eds) (2001) Stochastic optimization: algorithms and applications. Kluwer Academic, Dordrecht

    Google Scholar 

  • van der Vlerk MH (2003) Integrated chance constraints in an ALM model for pension funds. Stochastic programming e-print series, http://dochost.rz.hu-berlin.de/speps

  • Wallace SW, Ziemba WT (eds) (2005) Applications of stochastic programming. MPS-SIAM series in optimization

  • Wolsey LA (1998) Integer programming. Wiley, New York

    Google Scholar 

  • Ziemba WT, Mulvey JM (eds) (1998) Worldwide asset and liability modeling. Cambridge University Press, Cambridge

    Google Scholar 

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Authors and Affiliations

  1. Dpto. de Estadística e Investigación Operativa, Universidad Rey Juan Carlos, Calle Tulipán, s/n. 28933, Móstoles, Madrid, Spain

    Laureano F. Escudero

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  1. Laureano F. Escudero
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Correspondence to Laureano F. Escudero.

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This invited paper is discussed in the comments available at: doi:10.1007/s11750-009-0091-6, doi:10.1007/s11750-009-0092-5, doi:10.1007/s11750-009-0093-4, doi:10.1007/s11750-009-0094-3.

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Escudero, L.F. On a mixture of the fix-and-relax coordination and Lagrangian substitution schemes for multistage stochastic mixed integer programming. TOP 17, 5–29 (2009). https://doi.org/10.1007/s11750-009-0090-7

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