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Primal—Dual Constraint Aggregation with Application to Stochastic Programming

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Abstract

The special constraint structure and large dimension are characteristic for multistage stochastic optimization. This results from modeling future uncertainty via branching process or scenario tree. Most efficient algorithms for solving this type of problems use certain decomposition schemes, and often only a part of the whole set of scenarios is taken into account in order to make the problem tractable.

We propose a primal–dual method based on constraint aggregation, which constructs a sequence of iterates converging to a solution of the initial problem. At each iteration, however, only a reduced sub-problem with smaller number of aggregate constraints has to be solved. Number of aggregates and their composition are determined by the user, and the procedure for calculating aggregates can be parallelized. The method provides a posteriori estimates of the quality of the current solution approximation in terms of the objective function value and the residual.

Results of numerical tests for a portfolio allocation problem with quadratic utility function are presented.

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References

  1. A. Antipin, Gradient and proximal control processes, in: Problems of Cybernetics, Russian Academy of Sciences (1992) pp. 32–66 (in Russian).

  2. J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res. 33 (1985) 989–1007.

    Google Scholar 

  3. J.R. Birge, The relationship between the L-shaped method and dual basis factorization for stochastic linear programming, in: Numerical Techniques for Stochastic Optimization, Springer Ser. Comput. Math., Vol. 10 (1988) pp. 267–272.

    Google Scholar 

  4. J.R. Birge, Current trends in stochastic programming computation and applications, Technical report, Dept. of Industrial and Operational Engineering, University of Michigan, Ann Arbor, MI (1995).

  5. J.R. Birge, C.J. Donohue, D.F. Holmes and O.G. Svintsitski, A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs, Math. Program. 75 (1996) 327–352.

    Google Scholar 

  6. J.R. Birge and A.J. King, Unbiased methods for sampled cuts in stochastic programming, Technical report, Dept. of Industrial and Operational Engineering, University of Michigan, Ann Arbor, MI (1995).

  7. J.R. Birge and F. Louveaux, Introduction to Stochastic Programming (Springer, New York, 1997).

    Google Scholar 

  8. J.R. Birge and J.-B.Wets, Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse, Math. Program. Study 2 (1986) 754–102.

    Google Scholar 

  9. B.J. Chun and S.M. Robinson, Scenario analysis via bundle decomposition, Ann. Oper. Res. 56 (1995 39–63.

    Google Scholar 

  10. G.B. Dantzig and P.W. Glynn, Parallel processors for planning under uncertainty, Ann. Oper. Res. 22 (1990) 1–21.

    Google Scholar 

  11. G.B. Dantzig and G. Infanger, Large-scale stochastic linear programs-importance sampling and Benders decomposition, in: Computational and Applied Mathematics, I. Algorithms and Theory, Sel. Rev. Pap. IMACS 13th World Congr., Dublin/Ireland, 1991 (1992) pp. 111–120.

  12. M. Davidson, Proximal point mappings and constraint aggregation principle, Technical report, WP–96–102, IIASA, Laxenburg, Austria (1996).

  13. M. Davidson, Primal-dual constraint aggregation method in multistage stochastic programming, Technical report, WP 97–26, University of Trier, Germany (1997).

  14. M.A.H. Dempster, Stochastic Programming, The Institute of Mathematics and its Applications Conference Series (Academic Press, London, 1980).

    Google Scholar 

  15. M.A.H. Dempster and R.T. Thompson, EVPI-based importance sampling solution procedures for multistage stochastic linear programmes on parallel mimd architectures, Technical report, Judge Institute of Management Studies, University of Cambridge, UK (1996).

  16. N.C.P. Edirisinghe, Bound-based approximations in stochastic programming, Ann. Oper. Res. 85 (1999) 103–127.

    Google Scholar 

  17. Yu. Ermoliev, A. Kryazhimskii and A. Ruszczynski, Constraint aggregation principle in convex optimization, Math. Program. 76 (1997) 353–372.

    Google Scholar 

  18. Yu. Ermoliev and R. Wets, eds., Numerical Techniques for Stochastic Optimization (Springer, Berlin, 1988).

    Google Scholar 

  19. K. Frauendorfer, Barycentric scenario trees in convex multistage stochastic programming, Math. Program. 75 (1996) 277–294.

    Google Scholar 

  20. H.I. Gassmann, MSLiP: A computer code for the multistage stochastic linear programming problem, Math. Program., Ser. A 47 (1990) 407–423.

    Google Scholar 

  21. J.L. Higle and S. Sen, Stochastic decomposition: An algorithm for two-stage linear programs with recourse, Math. Oper. Res. 16 (1991) 650–669.

    Google Scholar 

  22. J.L. Higle and S. Sen, Duality and statistical tests of optimality for two stage stochastic programs, Math. Program. 75 (1996) 257–276.

    Google Scholar 

  23. J.L. Higle and S. Sen, Stochastic Decomposition (Kluwer Academic, 1996).

  24. P. Kall, Solution methods in stochastic programming, in: System Modeling and Optimization, Proceedings of the 16th IFIP-TC7 Conference, Compiegne, France, July 5–9, 1993. Lect. Notes Control Inf. Sci., Vol. 197 (Springer, London, (994) pp. 3–22.

    Google Scholar 

  25. P. Kall, A. Ruszczynski and K. Frauendorfer, Approximation techniques in stochastic programming, in: Numerical Techniques for Stochastic Optimization, Springer Ser. Comput. Math., Vol. 10 (1988) pp. 33–64.

    Google Scholar 

  26. K. Kiwiel, Methods of Descent for Nondifferentiable Optimization (Springer, Berlin, 1985).

    Google Scholar 

  27. C. Lemaréchal, A.S. Nemirovskij and Y. Nesterov, New variants of bundle methods, Math. Prog. 69 (1995) 111–147.

    Google Scholar 

  28. C. Lemaréchal, J.J. Strodiot and A. Bihain, On a bundle algorithm for non-smooth optimization, in: Nonlinear Programming 4 (Academic Press, New York, 1981) pp. 245–282.

    Google Scholar 

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

    Google Scholar 

  30. H.M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (Yale University Press, 1959).

  31. R. Mifflin, A modification and an extension of Lemaréchal's algorithm for nonsmooth minimization, Math. Prog. Study 17 (1982) 77–90.

    Google Scholar 

  32. J.M. Mulvey and H. Vladimirou, Applying the progressive hedging algorithm to stochastic generalized networks, Ann. Oper. Res. 31 (1991) 399–424.

    Google Scholar 

  33. NAG C Library Manual, Mark 4, Numerical Algorithms Group Ltd., UK (1996).

    Google Scholar 

  34. A.S. Nemirovskij and D.B.Yudin, Problem Complexity andMethod Efficiency in Optimization (Wiley, New York, 1983).

    Google Scholar 

  35. M.-C. Noel and Y. Smeers, Nested decomposition of multistage nonlinear programs with recourse, Math. Program. 37 (1987) 131–152.

    Google Scholar 

  36. S.M. Robinson, Extended scenario analysis, Ann. Oper. Res. 31 (1991) 385–397.

    Google Scholar 

  37. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1973).

    Google Scholar 

  38. R.T. Rockafellar and R.J.-B. Wets, Scenarios and policy aggregation in optimization under uncertainty, Math. Oper. Res. 16 (1991) 119–147.

    Google Scholar 

  39. A. Ruszczynski, Parallel decomposition of multistage stochastic programming problems, Math. Program. Ser. A 58 (1993) 201–228.

    Google Scholar 

  40. H. Schramm and J. Zowe, A version of the bundle idea for minimizing a non-smooth function: conceptual idea, convergence analysis, numerical results, SIAM J. Opt. 2 (1992) 121–152.

    Google Scholar 

  41. A. Shapiro, Asymptotic behavior of optimal solutions in stochastic programming, Math. Oper. Res. 18 (1993) 829–845.

    Google Scholar 

  42. R.J.-B. Wets, Large scale linear programming techniques in stochastic programming, in: Numerical Techniques for Stochastic Optimization, Springer Ser. Comput. Math., Vol. 10 (1988) pp. 65–93.

    Google Scholar 

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  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Vorobievy Gory, 119899, Moscow, Russia

    M. Davidson

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Davidson, M. Primal—Dual Constraint Aggregation with Application to Stochastic Programming. Annals of Operations Research 99, 41–58 (2000). https://doi.org/10.1023/A:1019232731587

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