Skip to main content
SpringerLink
Log in
Book cover

Stochastic Programming pp 97–138Cite as

Stochastic Programming Approximations Using Limited Moment Information, with Application to Asset Allocation

  • Chapter
  • First Online:

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 150))

Abstract

In this chapter, stochastic programming solution methods that utilize bound-based approximations are discussed. The focus is on bounds that use limited moment information of the underlying random vectors to approximate the expected recourse function. Bounds that use first-order moments, as well as higher-order moments are described in the context of saddle functions and specialized for convex recourse functions. Relation of these bounds to generalized moment problems is also addressed. Bounds are applied within sequential approximation schemes to solve the underlying stochastic program to a user-specified degree of accuracy. Appropriate partitioning techniques for the domains of the random variables are described and applied in a financial asset allocation stochastic program. Useful solution characteristics as well as sensitivity analyses are discussed using the bound-based solution procedure.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    If the true distribution is indeed the approximating distribution, then the bound is achieved; hence, the bound is sharp.

References

  • Birge, J.R.: An l-shaped method computer code for multistage linear programs. In: Ermoliev, Y., Wets, R.J-B. (eds.) Numerical Techniques for Stochastic Optimization. Springer, New York, NY (1988)

    Google Scholar 

  • Birge, J.R., Dula, J.: Bounding separable recourse functions with limited moment information. Ann. Oper. Res. 30, 277–298 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Birge, J.R., Wallace, S.W.: Refining bounds for stochastic linear programs with linearly transformed independent random variables. Oper. Res. Lett. 5, 73–77 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Birge, J.R., Wallace, S.W.: A separable piecewise linear upper bound for stochastic linear programs. SIAM J. Control Optim. 26, 725–739 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Birge, J.R., Wets, R.J.-B.: Sublinear upper bounds for stochastic programs with recourse. Math. Program. 43, 131–149 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Casey, M.S., Sen, S.: The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res. 30, 615–631 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Dokov, S.P., Morton, D.P.: Higher-order upper bounds on the expectation of a convex function, University of Texas at Austin, Austin, TX (2002)

    Google Scholar 

  • Dokov, S.P., Morton, D.P.: Second-order lower bounds on the expectation of a convex function. Math. Oper. Res. 30, 662–677 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Dula, J.H.: An upper bound on the expectation of simplicial functions of multivariate random variables. Math. Program. 55, 69–80 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Dupačová, J.: On minimax solutions of stochastic linear programming problems. Časopis Pro Pětováni Matematiky. 91, 423–430 (1966) (as Žáčková)

    Google Scholar 

  • Dupačová, J.: Minimax stochastic programs with nonconvex nonseparable penalty functions. In: Prékopa, A. (ed.) Progress in Operations Research, pp. 303–316. North-Holland, Amsterdam (1976)

    Google Scholar 

  • Edirisinghe, C., Carino, D., Fan Y.-A., Murray, S., Stacey, C.: Implementation of stochastic programming approximations for asset allocation two-stage models, 1996. Invited talk at INFORMS national meeting, Atlanta, GA, (1996)

    Google Scholar 

  • Edirisinghe, N.C.P.: New second-order bounds on the expectation of saddle functions with applications to stochastic linear programming. Oper. Res. 44, 909–922 (1996)

    Article  MATH  Google Scholar 

  • Edirisinghe, N.C.P.: Bound-based approximations in multistage stochastic programming: Nonanticipativity aggregation. Ann. Oper. Res. 85, 103–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Edirisinghe, N.C.P.: A Linear-Time Algorithm for Enclosing Multivariate Data in a Compact Simplex. University of Tennessee, Knoxville, TN (2005)

    Google Scholar 

  • Edirisinghe, N.C.P., Patterson, E. I.: Multiperiod stochastic portfolio optimization: Block-separable decomposition. Ann. Oper. Res. 152, 367–394 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Edirisinghe, N.C.P., You, G-M.: Second-order scenario approximation and refinement in optimization under uncertainty. Ann. Oper. Res. 19, 314–340 (1996)

    MathSciNet  Google Scholar 

  • Edirisinghe, N.C.P., Ziemba, W.T.: Tight bounds for stochastic convex programs. Oper. Res. 40, 660–677 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Edirisinghe, N.C.P., Ziemba, W.T.: Bounds for two-stage stochastic programs with fixed recourse. Math. Oper. Res. 19, 292–313 (1994a)

    Article  MathSciNet  MATH  Google Scholar 

  • Edirisinghe, N.C.P., Ziemba, W.T.: Bounding the expectation of a saddle function with application to stochastic programming. Math. Oper. Res. 19, 314–340 (1994b)

    Article  MathSciNet  Google Scholar 

  • Edirisinghe, N.C.P., Ziemba, W.T.: Implementing bounds-based approximations in convex-concave two-stage stochastic programming. Math. Program. 19, 314–340 (1996)

    Google Scholar 

  • Ermoliev, Y.: Stochastic quasigradient methods and their application to systems optimization. Stochastics 9, 1–36 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Ermoliev, Y., Gaivoronsky, A.: Stochastic quasigradient methods and their implementation. In: Ermoliev, Y., Wets, R.J-B., (eds.) Numerical Techniques for Stochastic Optimization. Springer, New York, NY (1988)

    Chapter  Google Scholar 

  • Ermoliev, Y., Wets, R.J-B.: Numerical Techniques for Stochastic Optimization. Springer, New York, NY (1988)

    Book  Google Scholar 

  • Frauendorfer, K.: Slp problems: Objective and right-hand side being stochastically dependent—part ii, University of Zurich (1988a)

    Google Scholar 

  • Frauendorfer, K.: Slp recourse problems with arbitrary multivariate distributions—the dependent case. Math. Oper. Res. 13, 377–394 (1988b)

    Article  MathSciNet  MATH  Google Scholar 

  • Frauendorfer, K.: Stochastic Two-Stage Programming. Springer, Berlin (1992)

    Book  Google Scholar 

  • Frauendorfer, K.: Multistage stochastic programming: Error analysis for the convex case. Z. Oper. Res. 39, 93–122 (1994)

    MathSciNet  MATH  Google Scholar 

  • Frauendorfer, K.: Barycentric scenario trees in convex multistage stochastic programming. Math. Program. 75, 277–293 (1996)

    MathSciNet  MATH  Google Scholar 

  • Frauendorfer, K., Kall, P.: A solution method for slp problems with arbitrary multivariate distributions—the independent case. Prob. Cont. Inf. Theor. 17, 177–205 (1988)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gassmann, H.I., Ziemba, W.T.: A tight upper bound for the expectation of a convex function of a multivariate random variable. Math. Program. Study. 27, 39–53 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • Glashoff, K., Gustafson, S-A.: Linear Optimization and Approximation. Springer, New York, NY (1983)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Infanger, G.: Monte carlo (importance) sampling within a benders decomposition algorithm for stochastic linear programs. Ann. Oper. Res. 39, 69–95 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Infanger, G.: Planning Under Uncertainty, Solving Large-scale Stochastic Programs. The Scientific Press Series, Danvers, MA (1994)

    Google Scholar 

  • Jensen, J.L.: Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta. Math. 30, 173–177 (1906)

    Article  Google Scholar 

  • Kall, P.: Stochastic Linear Programming. Springer, Berlin (1976)

    Book  Google Scholar 

  • Kall, P.: Stochastic programs with recourse: An upper bound and the related moment problem. Z. Oper. Res. 31, A119–A141 (1987a)

    MathSciNet  Google Scholar 

  • Kall, P.: On approximations and stability in stochastic programming. In: Guddat, J., Jongen, H.Th., Kummer, B., Nozicka, F., (eds.) Parametric Optimization and Related Topics, pp. 387–407 Akademie, Berlin (1987b)

    Google Scholar 

  • Kall, P.: Stochastic programming with recourse : Upper bounds and moment problems—a review. In: Guddat, J., Bank, B., Hollatz, H., Kall, P., Klatte, D., Kummer, B., Lommatzsch, K., Vlach, M., Zimmermann, K. (eds.) Advances in Mathematical Optimization and Related Topics, pp. 86–103. Akademie, Berlin (1988)

    Google Scholar 

  • Kall, P.: An upper bound for slp using first and total second moments. Ann. Oper. Res. 30, 267–276 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Kall, P., Stoyan, D.: Solving stochastic programming problems with recourse including error bounds. Math. Oper. schung Stat. Ser. Optim. 13, 431–447 (1988)

    MathSciNet  Google Scholar 

  • Kall, P., Wallace, S.W.: Stochastic Programming. Wiley, New York, NY (1994)

    Google Scholar 

  • Karr, A.F.: Extreme points of certain sets of probability measures, with applications. Math. Oper. Res. 8, 74–85 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Kemperman, J.: The general moment problem. a geometric approach. Ann. Math. Stat. 39, 93–112 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  • Kuhn, D.: Generalized Bounds for Convex Multistage Stochastic Programs. Lecture Notes in Economics and Mathematical Systems, Vol. 548. Springer, Berlin (2004)

    Google Scholar 

  • Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea, New York, NY (1986)

    MATH  Google Scholar 

  • Madansky, A.: Bounds on the expectation of a convex function of a multivariate random variable. Ann. Math. Stat. 30, 743–746 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  • Morton, D.P., Wood, R.K.: Restricted-recourse bounds for stochastic linear programming. Oper. Res. 47, 943–956 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Van Slyke, R., Wets, R.J.-B.: L-shaped linear programs with application to optimal control and stochastic optimization. SIAM J. App. Math. 17, 638–663 (1969)

    Article  MATH  Google Scholar 

  • Wallace, S. W.: A piecewise linear upper bound on the network recourse function. Math. Program. 38, 133–146 (1987)

    Article  MATH  Google Scholar 

  • Wets, R.J-B.: Stochastic programs with fixed recourse: The equivalent deterministic problem. SIAM Rev. 16, 309–339 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Wets, R.J-B.: Convergence of convex functions, variational inequalities and convex optimization problems. In: Cottle, R.W., Giannesi, F., Lious, J-L., (eds.) Variational Inequalities and Complimentary Problems, pp. 375–403. Wiley, New York, NY (1980)

    Google Scholar 

  • Wets, R.J-B.: Stochastic programming: Solution techniques and approximation schemes. In: Bachem, A., Groetschel, M., Korte, B., (eds.) Mathematical Programming: State-of-the-Art, pp. 566–603 Springer, Berlin (1982)

    Google Scholar 

  • Ziemba, W.T.: Computational algorithms for convex stochastic programs with simple recourse. Oper. Res. 18, 414–431 (1970)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Operations, and Management Science, College of Business Administration, University of Tennessee, Knoxville, TN, 37996, USA

    N. Chanaka P. Edirisinghe

Authors
  1. N. Chanaka P. Edirisinghe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Chanaka P. Edirisinghe .

Editor information

Editors and Affiliations

  1. Dept. Management Science & Engineering, Stanford University, Stanford, 94305, California, USA

    Gerd Infanger

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Edirisinghe, N.C.P. (2010). Stochastic Programming Approximations Using Limited Moment Information, with Application to Asset Allocation. In: Infanger, G. (eds) Stochastic Programming. International Series in Operations Research & Management Science, vol 150. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1642-6_6

Download citation

Publish with us

Policies and ethics

Search

Navigation