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Parallel processors for planning under uncertainty

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Abstract

Our goal is to demonstrate for an important class of multistage stochastic models that three techniques — namely nested decomposition, Monte Carlo importance sampling, and parallel computing — can be effectively combined to solve this fundamental problem of large-scale linear programming.

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References

  1. P.G. Abrahamson, A nested decomposition approach for solving staircase linear programs, Technical Report SOL 83-4, Department of Operations Research, Stanford University, CA (1983).

    Google Scholar 

  2. E.M.L. Beale, On minimizing a convex function subject to linear inequalities, J. Roy. Stat. Soc. 17b(1955)173–184.

    Google Scholar 

  3. E.M.L. Beale, G.B. Dantzig and R.D. Watson, A first-order approach to a class of multi-time-period stochastic programming problems, Math. Progr. Study 27(1986)103–117.

    Google Scholar 

  4. J.F. Benders, Partitioning procedures for solving mixed-variable programming problems, Numerische Mathematik 4(1962)238–252.

    Article  Google Scholar 

  5. J.R. Birge, Solution methods for stochastic dynamic linear programs, Technical Report SOL 80-29, Department of Operations Research, Stanford University, CA (1980).

    Google Scholar 

  6. J.R. Birge, Aggregation in stochastic linear programming, Math. Progr. 31(1984)25–41.

    Google Scholar 

  7. J.R. Birge, Decomposition and partitioning methods for multi-stage stochastic linear programming, Oper. Res. 33(1985)989–1007.

    Google Scholar 

  8. J.R. Birge and S.W. Wallace, A separable piecewise linear upper bound for stochastic linear programs, SIAM J. Control and Optimization 26, 3(1988).

  9. J.R. Birge and R.J.-B. Wets, Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse, Math. Progr. Study 27(1986)54–102.

    Google Scholar 

  10. J.R. Birge and R.J.-B. Wets, Computing bounds for stochastic programming problems by means of a generalized moment problem, Math. Oper. Res. 12(1987)149–162.

    Google Scholar 

  11. J.R. Birge and R.J.-B. Wets, Sublinear upper bounds for stochastic programs with recourse, Math. Progr. 3, 43(1989)131–149.

    Article  Google Scholar 

  12. Bisschop and Meeraus, Towards successful modeling applications in a strategic planning environment,Large-Scale Linear Programming, Vol. 2, ed. G.B. Dantzig, M.A.H. Dempster and M.J. Kallio, CP-81-51, IIASA Collaborative Proceedings Series, Laxenburg, Austria (1981) pp. 712–745.

  13. P. Bratley, B. Fox and L. Schrage,A Guide to Simulation (Springer-Verlag, New York, 1983).

    Google Scholar 

  14. W.G. Cochran,Sampling Techniques (Wiley, New York, 1977).

    Google Scholar 

  15. K.J. Cohen and S. Thore, Programming bank portfolios under uncertainty, J. Bank Research 1(1970)42–61.

    Google Scholar 

  16. G.B. Dantzig, Linear programming under uncertainty, Mang. Sci. 1(1955)197–206.

    Google Scholar 

  17. G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963) Chs. 22–25 and 28.

    Google Scholar 

  18. G.B. Dantzig, Time-staged methods in linear programs, in:Studies in Management Science and Systems, Vol. 7:Large-Scale Systems, ed. Y.Y. Haims (North-Holland, Amsterdam, 1982) pp. 19–30.

    Google Scholar 

  19. G.B. Dantzig, Planning under uncertainty using parallel computing, Ann. Oper. Res. 14(1988)1–16.

    Google Scholar 

  20. G.B. Dantzig, M.A.H. Dempster and M.J. Kallio, eds.,Large-Scale Linear Programming, Vols. 1, 2, IIASA Collaborative Proceedings Series, CP-81-51, IIASA, Laxenburg, Austria, 1981.

    Google Scholar 

  21. G.B. Dantzig, P.W. Glynn, M. Avriel, J.C. Stone, R. Entriken and M. Nakayama, Decomposition techniques for multi-area transmission planning under uncertainty. Final report of EPRI Project RP 2940-1, prepared by Systems Optimization Laboratory, Operations Research Department, Stanford University, CA (1989).

    Google Scholar 

  22. G.B. Dantzig and M. Madansky, On the solution of two-staged linear programs under uncertainty, in:Proc. 4th Berekely Symp. on Mathematical Statistics and Probability I, ed. J. Neyman (1961) pp. 165–176.

  23. G.B. Dantzig, M.V.F. Pereira et al., Mathematical decomposition techniques for power system expansion planning, EPRI EL-5299, Vols. 1–5, Electric Power Research Institute, Palo Alto, CA (1988).

    Google Scholar 

  24. G.B. Dantzig and A.F. Perold, A basic factorization method for block triangular linear programs, Technical Report SOL 78-7, Department of Operations Research, Stanford University, CA (1979);Sparse Matrix Proceedings, ed. I.S. Duff and G.W. Steward (SIAM, 1978) pp. 283–312.

    Google Scholar 

  25. G.B. Dantzig and P. Wolfe, The decomposition principle for linear programs, Oper. Res. 8(1960)110–111.

    Google Scholar 

  26. P.J. Davis and P. Rabinowitz,Methods of Numerical Integration (Academic Press, London, 1984).

    Google Scholar 

  27. I. Deák, Multidimensional integration and stochastic programming, in:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 187–200.

    Google Scholar 

  28. M.A.H. Dempster and E. Sole, Stochastic scheduling via stochastic control, Bernoulli 1(1987)783–788.

    Google Scholar 

  29. J. Dupačová, Minimax approach to stochastic linear programming and the moment problem: Selected results, Zeitschrift für Angewandte Mathematik und Mechanik 58T(1978)466–467.

    Google Scholar 

  30. J. Dupačová and R.J.-B. Wets, Asymptotic behavior of statistical estimators of optimal solutions of stochastic optimization problems, Ann. Math. Statis. 16(1988)1517–1549.

    Google Scholar 

  31. R. Entriken, The parallel decomposition of linear programs, Ph.D. Dissertation, Department of Operations Research, Stanford University, CA (1989).

    Google Scholar 

  32. R. Entriken, A parallel decomposition algorithm for staircase linear programs, Oak Ridge National Laboratory Report ORNL/TM 11011 (1988).

  33. Y. Ermoliev, Stochastic quasi-gradient methods and their applications to systems optimization, Stochastics 9(1983)1–36.

    Google Scholar 

  34. Y. Ermoliev, Stochastic quasi-gradient methods, In:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 141–186.

    Google Scholar 

  35. Y. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization (Springer-Verlag, Berlin, 1988).

    Google Scholar 

  36. R.H. Fourer, Solving staircase linear programs by the simplex method, 1: Inversion, Math. Progr. 23(1982)274–313.

    Article  Google Scholar 

  37. R.H. Fourer, Solving staircase linear programs by the simplex method, 2: Pricing, Math. Progr. 25(1983)251–292.

    Google Scholar 

  38. R.H. Fourer, Staircase matrices and systems, SIAM Rev. 26(1984)1–70.

    Article  Google Scholar 

  39. K. Frauendorfer and P. Kall, A solution method for SLP recourse problems with arbitrary multivariate distributions — the independent case, Problems of Control and Information Theory, 17, 4(1988)177–205.

    Google Scholar 

  40. A. Gaivoronski, Implementation of stochastic quasi-gradient methods, in:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 313–352.

    Google Scholar 

  41. A. Gaivoronski and L. Nazereth, Combining generalized programming and sampling techniques for stochastic program with recourse, in:Planning Under Uncertainty for Electric Power Systems, Proc. Workshop (in preparation), Department of Operations Research, Stanford University, CA (1989).

    Google Scholar 

  42. H. Gassmann and W.T. Ziemba, A tight upper bound for the expectation of a convex function of a multivariate random variable, Math. Progr. Study 27(1986)39–52.

    Google Scholar 

  43. P.E. Gill, W. Murray, M.A. Saunders, J.A. Tomlin and M.H. Wright, On Newton-barrier methods for linear programming and on equivalence to Karmarkar's projective method, Technical Report SOL 85-9, Department of Operations Research, Stanford University, CA (1985).

    Google Scholar 

  44. R. Glassey, Nested decomposition and multi-stage linear programs, Manag. Sci. 20(1973)282–292.

    Google Scholar 

  45. P.W. Glynn and W. Whitt, Efficiency of simulation estimates (1988), submitted for publication.

  46. P.W. Glynn and D.L. Iglehart, Importance sampling for stochastic simulation (1988), submitted for publication.

  47. J.M. Hammersley and D.C. Handscomb,Monte Carlo Methods (Methuen, London, 1964).

    Google Scholar 

  48. J.K. Ho, T.C. Lee and R.P. Sundarraj, Decomposition of linear programs using parallel computation, Math. Progr. 42(1988)391–405.

    Article  Google Scholar 

  49. J.K. Ho and E. Loute, An advanced implementation of the Dantzig—Wolfe decomposition algorithm for linear programming, Discussion Paper 8014, Center for Operations Research and Econometrics (CORE), Belgium (1980).

    Google Scholar 

  50. J.K. Ho and A.S. Manne, Nested decomposition for dynamic models, Math. Progr. 6(1974)121–140.

    Article  Google Scholar 

  51. C.C. Huang, W.T. Ziemba and A. Ben-Tal, Bounds on the expectation of a convex function with a random variable with applications to stochastic programming, Oper. Res. 25(1977)315–325.

    Google Scholar 

  52. P.L. Jackson and D.F. Lynch, Revised Dantzig-Wolfe decomposition for staircase-structured linear programs, Technical Report 558, School of Operations Research and Industrial Engineering, Cornell University (1982), revised 1985.

  53. P. Kall, A. Ruszczyński and K. Frauendorfer, Approximation techniques in stochastic programming, in:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 33–64.

    Google Scholar 

  54. P. Kall and D. Stoyan, Solving stochastic programming problems with recourse including error bounds, Mathematische Operationsforschung und Statistik, Series Optimization 13(1982)431–447.

    Google Scholar 

  55. N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4(1984)373–395.

    Google Scholar 

  56. A.J. King and R.J.-B. Wets, Epi-consistency of convex stochastic programs, Technical Report (Alan King, IBM Research Division, P.O. Box 218, Yorktown Heights, NY 10598, USA, 1989).

  57. S.S. Lavenberg and P.D. Welch, A perspective on the use of control variables to increase the efficiency of Monte Carlo simulation, Manag. Sci. 27(1981)322–335.

    Google Scholar 

  58. F.V. Louveaux, Multistage stochastic programs with block-separable recourse, Math. Progr. Study 28(1986)48–62.

    Google Scholar 

  59. F.V. Louveaux and Y. Smeers, Optimal investment for electricity generation: A stochastic model and a test problem, in:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 445–454.

    Google Scholar 

  60. A. Madansky, Bounds on the expectation of a convex function of a multivariante random variable, Ann. Math. Statis. 30(1959)743–746.

    Google Scholar 

  61. J. Mulvey and H. Vladimirou, Solving multistage investment problems: An application of scenario aggregation, Technical Report SOR 88-1, Princeton University, Princeton, NJ (1988).

    Google Scholar 

  62. B.A. Murtagh and M.A. Saunders, MINOS User's Guide, Technical Report SOL 82-20, Department of Operations Research, Stanford University, CA (1983).

    Google Scholar 

  63. M. Nakayama, Section 6 in Dantzig, Glynn, Avriel et al. (1989).

  64. L. Nazereth and R.J.-B. Wets, Algorithms for stochastic programs: The case of nonstochastic tenders, Math. Progr. Study 28(1986)1–28.

    Google Scholar 

  65. H. Niederreiter, Multidimensional numerical integration using pseudo random numbers, Math. Progr. Study 27(1986)17–38.

    Google Scholar 

  66. T. Nishiya, A basis factorization method for multi-stage linear programming with an application to optimal operation of an energy plant, Draft Report (1983).

  67. J. Parida and A. Sen, A variational-like inequality for multifunctions with applications, J. Math. Analysis and Applications 124(1987)73–81.

    Article  Google Scholar 

  68. M.V.F. Pereira and L.M.V.G. Pinto, Stochastic optimization of a multi-reservoir hydroelectric system — a decomposition approach, CEPAL (Centro del Pesquisas de Energia Electrica), Rio de Janeiro, Brazil (1983).

    Google Scholar 

  69. M.V.F. Pereira, L.M.V.G. Pinto, G.C. Oliveira and S.H.F. Cunha, A technique for solving LP problems with stochastic right-hand sides, CEPAL (Centro del Pesquisas de Energia Electria), Rio de Janeiro, Brazil (1989), 13 pages.

    Google Scholar 

  70. A. Prékopa, Dynamic type stochastic programming models, in:Studies in Applied Stochastic Programming, ed. A. Prékopa (Hungarian Academy of Science, Budapest, 1978) pp. 179–209.

    Google Scholar 

  71. S.M. Robinson and R.J.-B. Wets, Stability in two-stage stochastic programming, SIAM J. on Control and Optimization 25(1987)1409–1416.

    Article  Google Scholar 

  72. R.T. Rockafellar and R.J.-B. Wets, Scenario and policy aggregation in optimization under uncertainty, Math. Oper. Res., to appear.

  73. R.Y. Rubinstein and R. Marcus, Efficiency of multivariate control variates in Monte Carlo simulation, Oper. Res. 33(1985)661–677.

    Article  Google Scholar 

  74. G. Salinetti and R.J.-B. Wets, On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infinima, Math. Oper. Res. (1986)385–419.

  75. D.M. Scott, A dynamic programming approach to time-staged convex programs, Technical Report SOL 85-3, Department of Operations Research, Stanford University, University, CA (1985).

    Google Scholar 

  76. B. Strazicky, Computational experience with an algorithm for discrete recourse problems, in:Stochastic Programming, ed. M. Dempster (Academic Press, London, 1980) pp. 263–274.

    Google Scholar 

  77. M.J. Todd, Probabilistic models for linear programming, Technical Report No. 836, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY (1989).

    Google Scholar 

  78. R. Van Slyke and R.J.-B. Wets, Programming under uncertainty and stochastic optimal control, SIAM J. on Control and Optimization 4(1966)179–193.

    Article  Google Scholar 

  79. R. Van Slyke and R.J.-B. Wets, L-shaped linear programs with application to optimal control and stochastic programming, SIAM J. on Appl. Math. 17(1969)638–663.

    Article  Google Scholar 

  80. R.J.-B. Wets, Programming under uncertainty: The equivalent convex program, SIAM J. on Appl. Math. 14(1984)89–105.

    Article  Google Scholar 

  81. R.J.-B. Wets, On parallel processor design for solving stochastic programs, in:Proc. 6th Mathematical Programming Symposium (Japanese Mathematical Programming Society, Tokyo, 1985) pp. 13–36.

    Google Scholar 

  82. R.J.-B. Wets, Large-scale linear programming techniques, in:Numerical Techniques for Stochastic Optimization, ed. Y. Ermoliev and R.J.-B. Wets (Springer-Verlag, Berlin, 1988) pp. 65–94.

    Google Scholar 

  83. R.J. Wittrock, Advances in a nested decomposition algorithm for solving staircase linear programs, Technical Report SOL 83-2, Department of Operations Research, Stanford University, CA (1983).

    Google Scholar 

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  1. Department of Operations Research, Stanford University, 94305, Stanford, CA, USA

    George B. Dantzig & Peter W. Glynn

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  1. George B. Dantzig
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Dantzig, G.B., Glynn, P.W. Parallel processors for planning under uncertainty. Ann Oper Res 22, 1–21 (1990). https://doi.org/10.1007/BF02023045

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