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Epi-convergent discretizations of stochastic programs via integration quadratures

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Summary.

The simplest and the best-known method for numerical approximation of high-dimensional integrals is the Monte Carlo method (MC), i.e. random sampling. MC has also become the most popular method for constructing numerically solvable approximations of stochastic programs. However, certain modern integration quadratures are often superior to crude MC in high-dimensional integration, so it seems natural to try to use them also in discretization of stochastic programs. This paper derives conditions that guarantee the epi-convergence of the resulting objectives to the original one. Our epi-convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure. As examples, we prove epi-convergence of quadrature-based discretizations of three different models of portfolio management and we study their behavior numerically. Besides MC, our discretizations are the only existing ones with guaranteed epi-convergence for these problem classes. In our tests, modern quadratures seem to result in faster convergence of optimal values than MC.

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References

  1. Artstein, Z., Wets, R.J.-B.: Stability results for stochastic programs and sensors, allowing for discontinuous objective functions. SIAM J. Optim. 4(3), 537–550 (1994)

    Google Scholar 

  2. Artstein, Z., Wets, R.J.-B.: Consistency of minimizers and the SLLN for stochastic programs. J. Convex Anal. 2(1–2), 1–17 (1995)

    Google Scholar 

  3. Attouch, H.: Variational convergence for functions and operators. Pitman (Advanced Publishing Program), Boston, MA, 1984

  4. Attouch, H., Wets, R.J.-B.: Quantitative stability of variational systems. I. The epigraphical distance. Trans. Amer. Math. Soc. 328(2), 695–729 (1991)

    Google Scholar 

  5. Beer, G.: A geometric algorithm for approximating semicontinuous function. J. Approx. Theory 49(1), 31–40 (1987)

    Google Scholar 

  6. Bertsekas, D.P., Shreve, S.E.: Stochastic optimal control, volume 139 of Mathematics in Science and Engineering. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. The discrete time case

  7. Billingsley, P.: Convergence of probability measures. John Wiley & Sons Inc., New York, 1968

  8. Billingsley, P.: Convergence of probability measures. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication

  9. Birge, J.R., Wets, R.J.-B.: Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse. Math. Programming Stud., Stochastic programming 84. I. (27), 54–102 (1986)

  10. Bratley, P., Fox, B.L., Niederreiter, H.: Implementation and tests of low-discrepancy sequences. ACM Trans. Model. Comput. Simul. 2(3), 195–213 (1992)

    Google Scholar 

  11. Chen, X., Womersley, R.S.: A parallel inexact Newton method for stochastic programs with recourse. Ann. Oper. Res. 64, 113–141 (1996); Stochastic programming, algorithms and models (Lillehammer, 1994)

    Google Scholar 

  12. Deák, I.: Multidimensional integration and stochastic programming. In Numerical techniques for stochastic optimization, volume 10 of Springer Ser. Comput. Math. Springer, Berlin, 1988, pp. 187–200

  13. Dempster, M.A.H., Thompson, R.T.: EVPI-based importance sampling solution procedures for multistage stochastic linear programmes on parallel MIMD architectures. Ann. Oper. Res. 90, 161–184 (1999)

    Google Scholar 

  14. Dupačová, J., Wets, R.J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. 16(4), 1517–1549 (1988)

    Google Scholar 

  15. Faure, H.: Discrépance de suites associées à un système de numération (en dimension s). Acta Arith. 41(4), 337–351 (1982)

    Google Scholar 

  16. Fox, B.L.: Algorithm 647: Implementation and relative efficiency of quasirandom sequence generators. ACM Transactions on Mathematical Software 12(4), 362–376 (1986)

    Article  Google Scholar 

  17. Frauendorfer, K.: Stochastic two-stage programming, volume 392 of Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, 1992 Habilitationsschrift, University of Zürich, Zürich, 1992

  18. Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)

    Google Scholar 

  19. Hammersley, J.M.: Monte Carlo methods for solving multivariable problems. Ann. New York Acad. Sci. 86, 844–874 (1960) 1960

    Google Scholar 

  20. Høyland, K., Kaut, M., Wallace, S.W.: Heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24(2–3), 169–185 (2003)

    Google Scholar 

  21. Høyland, K., Wallace, S.W.: Generating scenario trees for multistage decision problems. Managament Science 47(2), 295–307 (2001)

    Google Scholar 

  22. Infanger, G.: Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs. Ann. Oper. Res. 39(1–4), 69–95 (1993, 1992)

    Google Scholar 

  23. Kall, P., Ruszczyński, A., Frauendorfer, K.: Approximation techniques in stochastic programming. In Numerical techniques for stochastic optimization, volume 10 of Springer Ser. Comput. Math. Springer, Berlin, 1988, pp. 33–64

  24. King, A.J.: Duality and martingales: a stochastic programming perspective on contingent claims. Math. Program. 91(3, Ser. B), 543–562 (2002); ISMP 2000, Part 1 (Atlanta, GA)

  25. Korf, L.A.: Stochastic programming duality: multipliers for unbounded constraints with an application to mathematical finance. Math. Program. 99(2, Ser. A), 241–259 (2004)

  26. Korobov, N.M.: Approximate evaluation of repeated integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959)

    Google Scholar 

  27. L’Ecuyer, P., Lemieux, C.: Variance reduction via lattice rules. Management Science 46(2), 1214–1235 (2000)

    Google Scholar 

  28. Lepp, R.: Approximations to stochastic programs with complete recourse. SIAM J. Control Optim. 28(2), 382–394 (1990)

    Google Scholar 

  29. Lucchetti, R., Salinetti, G., Wets, R.J.-B.: Uniform convergence of probability measures: topological criteria. J. Multivariate Anal. 51(2), 252–264 (1994)

    Article  Google Scholar 

  30. Lucchetti, R., Wets, R.J.-B.: Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. Statist. Decisions 11(1), 69–84 (1993)

    Google Scholar 

  31. Niederreiter, H.: Random number generation and quasi-Monte Carlo methods, volume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992

  32. Pennanen, T., Koivu, M.: Integration quadratures in discretization of stochastic programs. Stochastic Programming E-Print Series, 2002

  33. Pflug, G.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program., 89(2, Ser. B), 251–271 (2001); Mathematical programming and finance

  34. Pflug, G., Hochreiter, R.: Scenario generation for stochastic multi-stage decision processes as facility location problems. Technical report, Department of Statistics and Decision Support Systems, University of Vienna, 2003

  35. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in C. Cambridge University Press, Cambridge, second edition, 1992 The art of scientific computing

  36. Svetlozar T.: Rachev and Werner Römisch. Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27(4), 792–818 (2002)

    Google Scholar 

  37. Robinson, S.M., Wets, R.J.-B.: Stability in two-stage stochastic programming. SIAM J. Control Optim. 25(6), 1409–1416 (1987)

    Google Scholar 

  38. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1998

  39. Römisch, W., Schultz, R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18(3), 590–609 (1993)

    Google Scholar 

  40. Schultz, R.: Some aspects of stability in stochastic programming. Ann. Oper Res. 100, 55–84 (2001), 2000. Research in stochastic programming (Vancouver, BC, 1998)

    Google Scholar 

  41. Shapiro, A.: Stochastic programming by monte carlo simulation methods. Stochastic Programming E-Print Series, 2000

  42. Shapiro, A.: Quantitative stability in stochastic programming. Math. Programming 67(1, Ser. A), 99–108 (1994)

  43. Sloan, I.H., Joe, S.: Lattice methods for multiple integration. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1994

  44. Sobol’, I.M.: The distribution of points in a cube and the approximate evaluation of integrals. U.S.S.R. Comput. Math. And Math. Phys. (4), 86–112 (1967)

  45. Vogel, S., Lachout, P.: On continuous convergence and epi-convergence of random functions. II. Sufficient conditions and applications. Kybernetika (Prague) 39(1), 99–118 (2003)

    Google Scholar 

  46. Zervos, M.: On the epiconvergence of stochastic optimization problems. Math. Oper. Res. 24(2), 495–508 (1999)

    Google Scholar 

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Authors and Affiliations

  1. Department of Management Science, Helsinki School of Economics, PL 1210, 00101, Helsinki, Finland

    Teemu Pennanen & Matti Koivu

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  1. Teemu Pennanen
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  2. Matti Koivu
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Correspondence to Teemu Pennanen.

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Mathematics Subject Classification (2000): 90C15, 49M25

The work of this author was partially supported by The Finnish Foundation for Economic Education under grant no. 21599 and by Finnish Academy under contract no. 3385

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Pennanen, T., Koivu, M. Epi-convergent discretizations of stochastic programs via integration quadratures. Numer. Math. 100, 141–163 (2005). https://doi.org/10.1007/s00211-004-0571-4

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  • DOI: https://doi.org/10.1007/s00211-004-0571-4

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