Skip to main content
SpringerLink
Log in

Pattern definition of the p-efficiency concept

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

This study revisits the celebrated p-efficiency concept introduced by Prékopa (Z. Oper. Res. 34:441–461, 1990) and defines a p-efficient point (pLEP) as a combinatorial pattern. The new definition uses elements from the combinatorial pattern recognition field and is based on the combinatorial pattern framework for stochastic programming problems proposed in Lejeune (Stochastic programming e-print series (SPEPS) 2010-5, 2010). The approach is based on the binarization of the probability distribution, and the generation of a consistent partially defined Boolean function representing the combination (F,p) of the binarized probability distribution F and the enforced probability level p. A combinatorial pattern provides a compact representation of the defining characteristics of a pLEP and opens the door to new methods for the generation of pLEPs. We show that a combinatorial pattern representing a pLEP constitutes a strong and prime pattern and we derive it through the solution of an integer programming problem. Next, we demonstrate that the (finite) collection of pLEPs can be represented as a disjunctive normal form (DNF). We propose a mixed-integer programming formulation allowing for the construction of the DNF that is shown to be prime and irreducible. We illustrate the proposed method on a problem studied by Prékopa (Stochastic programming: handbook in operations research and management science, vol. 10, Elsevier, Amsterdam, 2003).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Alexe, G., & Hammer, P. L. (2006). Spanned patterns for the logical analysis of data. Discrete Applied Mathematics, 154(7), 1039–1049.

    Article  Google Scholar 

  • Alexe, S., & Hammer, P. L. (2007). Accelerated algorithm for pattern detection in logical analysis of data. Discrete Applied Mathematics, 154(7), 1050–1063.

    Article  Google Scholar 

  • Boros, E., Hammer, P. L., Ibaraki, T., & Kogan, A. (1997). Logical analysis of numerical data. Mathematical Programming, 79, 163–190.

    Google Scholar 

  • Boros, E., Hammer, P. L., Ibaraki, T., Kogan, A., Mayoraz, E., & Muchnik, I. (2000). An implementation of logical analysis of data. IEEE Transactions on Knowledge and Data Engineering, 12(2), 292–306.

    Article  Google Scholar 

  • Calafiore, G. C., & Campi, M. C. (2005). Uncertain convex programs: randomized solutions and confidence levels. Mathematical Programming, 102, 25–46.

    Article  Google Scholar 

  • Charnes, A., Cooper, W. W., & Symonds, G. H. (1958). Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Management Science, 4, 235–263.

    Article  Google Scholar 

  • Dentcheva, D., Lai, B., & Ruszczyński, A. (2004). Dual methods for probabilistic optimization problems. Mathematical Methods of Operations Research, 60(2), 331–346.

    Article  Google Scholar 

  • Dentcheva, D., Prékopa, A., & Ruszczyński, A. (2001). Concavity and efficient points of discrete distributions in probabilistic programming. Mathematical Programming, 47(3), 1997–2009.

    Google Scholar 

  • Hammer, P. L. (1986). Partially defined Boolean functions and cause-effect relationships. International conference on multi-attribute decision making via or-based expert systems. University of Passau, Passau, Germany.

  • Hammer, P. L., & Bonates, T. O. (2006). Logical analysis of data: from combinatorial optimization to medical applications. Annals of Operations Research, 148(1), 203–225.

    Article  Google Scholar 

  • Hammer, P. L., Kogan, A., Simeone, B., & Szedmak, S. (2004). Pareto-optimal patterns in logical analysis of data. Discrete Applied Mathematics, 144, 79–102.

    Article  Google Scholar 

  • Ibaraki, T. (2010). Partially defined Boolean functions. In Y. Crama & P. L. Hammer (Eds.), Boolean functions—theory, algorithms, and applications. Cambridge: Cambridge Press University (in press).

    Google Scholar 

  • Lejeune, M. A. (2008). Preprocessing techniques and column generation algorithms for p-efficiency. Journal of Operational Research Society, 59, 1239–1252.

    Article  Google Scholar 

  • Lejeune, M. A. (2009). Linear reformulation of probabilistically constrained optimization problems using combinatorial patterns. International colloquium on stochastic modeling and optimization. Dedicated to the 80th birthday of Professor András Prékopa. Piscataway, NJ 2009.

  • Lejeune, M. A. (2010). Pattern-based modeling and solution of probabilistically constrained optimization problems. Stochastic programming e-print series (SPEPS) 2010-5.

  • Lejeune, M. A., & Noyan, N. (2010). Mathematical programming approaches for generating p-efficient points. European Journal of Operational Research, 207(2), 590–600.

    Article  Google Scholar 

  • Lejeune, M. A., & Ruszczyński, A. (2007). An efficient trajectory method for probabilistic inventory-production-distribution problems. Operations Research, 55(2), 378–394.

    Article  Google Scholar 

  • Luedtke, J., Ahmed, S., & Nemhauser, G. (2010). An integer programming approach for linear programs with probabilistic constraints. Mathematical Programming, 122(2), 247–272.

    Article  Google Scholar 

  • Martinez-Trinidad, J. F., & Guzmán-Arenas, A. (2001). The logical combinatorial approach to pattern recognition, an overview through selected works. Pattern Recognition, 34(4), 741–751.

    Article  Google Scholar 

  • Prékopa, A. (1970). On probabilistic constrained programming. In Proceedings of the Princeton symposium on mathematical programming (pp. 113–138). Princeton: Princeton University Press.

    Google Scholar 

  • Prékopa, A. (1973). Contributions to the theory of stochastic programming. Mathematical Programming, 4, 202–221.

    Article  Google Scholar 

  • Prékopa, A. (1990). Dual method for a one-stage stochastic programming with random rhs obeying a discrete probability distribution. Zeitschrift of Operations Research, 34, 441–461.

    Article  Google Scholar 

  • Prékopa, A. (1995). Stochastic programming. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Prékopa, A. (2003). Probabilistic programming models. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming: handbook in operations research and management science (Vol. 10, pp. 267–351). Amsterdam: Elsevier. Chapter 5

    Google Scholar 

  • Prékopa, A., Vizvári, B., & Badics, T. (1998). Programming under probabilistic constraint with discrete random variable. In F. Giannessi, S. Komlósi, & T. Rapcsák (Eds.), New trends in mathematical programming. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Ruiz-Shulcloper, J., & Abidi, M. A. (2002). Logical combinatorial pattern recognition: a review. Transworld Research Networks, 3, 133–176.

    Google Scholar 

  • Ruszczyński, A. (2002). Probabilistic programming with discrete distribution and precedence constrained knapsack polyhedra. Mathematical Programming, 93, 195–215.

    Article  Google Scholar 

  • Torvik, V. I., & Triantaphyllou, E. (2009). Inference of monotone Boolean functions. Encyclopedia of Optimization, 1591–1598.

  • Urbano, R. H., & Mueller, R. K. (1956). A topological method for the determination of the minimal forms of a Boolean function. IRE Transactions on Electronic Computers, EC-5, 126–132.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. George Washington University, Washington, DC, USA

    Miguel A. Lejeune

Authors
  1. Miguel A. Lejeune
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel A. Lejeune.

Additional information

The author is supported by Grant # W911NF-09-1-0497 from the Army Research Office.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lejeune, M.A. Pattern definition of the p-efficiency concept. Ann Oper Res 200, 23–36 (2012). https://doi.org/10.1007/s10479-010-0803-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-010-0803-1

Keywords

Search

Navigation