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A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming

Author

Listed:
  • Arjan Berkelaar

    (Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands)

  • Cees Dert

    (ABN-AMRO Asset Management, and Faculty of Economic Sciences and Econometrics, Free University of Amsterdam, The Netherlands)

  • Bart Oldenkamp

    (ABN-AMRO Asset Management, and Econometric Institute, Erasmus University Rotterdam, The Netherlands)

  • Shuzhong Zhang

    (Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong)

Abstract

Decision making under uncertainty is a challenge faced by many decision makers. Stochastic programming is a major tool developed to deal with optimization with uncertainties which has found applications in, e.g., finance, such as asset--liability and bond--portfolio management. Computationally, however, many models in stochastic programming remain unsolvable because of overwhelming dimensionality. For a model to be well solvable, its special structure must be explored. Most of the solution methods are based on decomposing the data. In this paper we propose a new decomposition approach for two-stage stochastic programming, based on a direct application of the path-following method combined with the homogeneous self-dual technique. Numerical experiments show that our decomposition algorithm is very efficient for solving stochastic programs. In particular, we apply our decomposition method to a two-period portfolio selection problem using options on a stock index. In this model the investor can invest in a money-market account, a stock index, and European options on this index with different maturities. We experiment with our model with market prices of options on the S&P500.

Suggested Citation

  • Arjan Berkelaar & Cees Dert & Bart Oldenkamp & Shuzhong Zhang, 2002. "A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming," Operations Research, INFORMS, vol. 50(5), pages 904-915, October.
    • Handle: RePEc:inm:oropre:v:50:y:2002:i:5:p:904-915
    DOI: 10.1287/opre.50.5.904.360
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    References listed on IDEAS

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    1. John R. Birge & Liqun Qi, 1988. "Computing Block-Angular Karmarkar Projections with Applications to Stochastic Programming," Management Science, INFORMS, vol. 34(12), pages 1472-1479, December.
    2. Cees Dert & Bart Oldenkamp, 2000. "Optimal Guaranteed Return Portfolios and the Casino Effect," Operations Research, INFORMS, vol. 48(5), pages 768-775, October.
    3. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1996. "Duality and Self-Duality for Conic Convex Programming," Econometric Institute Research Papers EI 9620-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Irvin J. Lustig & John M. Mulvey & Tamra J. Carpenter, 1991. "Formulating Two-Stage Stochastic Programs for Interior Point Methods," Operations Research, INFORMS, vol. 39(5), pages 757-770, October.
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    Cited by:

    1. Miguel, Angel Víctor de & Nogales Martín, Francisco Javier, 2004. "On the relationship between bilevel decomposition algorithms and direct interior-point methods," DES - Working Papers. Statistics and Econometrics. WS ws042509, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Arjen Siegmann & André Lucas, 2005. "Discrete-Time Financial Planning Models Under Loss-Averse Preferences," Operations Research, INFORMS, vol. 53(3), pages 403-414, June.
    3. Ankur Kulkarni & Uday Shanbhag, 2012. "Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms," Computational Optimization and Applications, Springer, vol. 51(1), pages 77-123, January.
    4. Kuang-Yu Ding & Xin-Yee Lam & Kim-Chuan Toh, 2023. "On proximal augmented Lagrangian based decomposition methods for dual block-angular convex composite programming problems," Computational Optimization and Applications, Springer, vol. 86(1), pages 117-161, September.
    5. Jianfeng Liang & Shuzhong Zhang & Duan Li, 2008. "Optioned Portfolio Selection: Models And Analysis," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 569-593, October.
    6. X. W. Liu & M. Fukushima, 2006. "Parallelizable Preprocessing Method for Multistage Stochastic Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 327-346, December.
    7. Jie Sun & Xinwei Liu, 2006. "Scenario Formulation of Stochastic Linear Programs and the Homogeneous Self-Dual Interior-Point Method," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 444-454, November.

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