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Pattern definition of the p-efficiency concept

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  • Miguel Lejeune

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 ). Copyright Springer Science+Business Media, LLC 2012

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  • Miguel Lejeune, 2012. "Pattern definition of the p-efficiency concept," Annals of Operations Research, Springer, vol. 200(1), pages 23-36, November.
    • Handle: RePEc:spr:annopr:v:200:y:2012:i:1:p:23-36:10.1007/s10479-010-0803-1
    DOI: 10.1007/s10479-010-0803-1
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    1. Darinka Dentcheva & Bogumila Lai & Andrzej Ruszczyński, 2004. "Dual methods for probabilistic optimization problems ," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(2), pages 331-346, October.
    2. Miguel A. Lejeune, 2012. "Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems," Operations Research, INFORMS, vol. 60(6), pages 1356-1372, December.
    3. M A Lejeune, 2008. "Preprocessing techniques and column generation algorithms for stochastically efficient demand," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(9), pages 1239-1252, September.
    4. Peter Hammer & Tibérius Bonates, 2006. "Logical analysis of data—An overview: From combinatorial optimization to medical applications," Annals of Operations Research, Springer, vol. 148(1), pages 203-225, November.
    5. Lejeune, Miguel & Noyan, Nilay, 2010. "Mathematical programming approaches for generating p-efficient points," European Journal of Operational Research, Elsevier, vol. 207(2), pages 590-600, December.
    6. A. Charnes & W. W. Cooper & G. H. Symonds, 1958. "Cost Horizons and Certainty Equivalents: An Approach to Stochastic Programming of Heating Oil," Management Science, INFORMS, vol. 4(3), pages 235-263, April.
    7. Miguel A. Lejeune & Andrzej Ruszczyński, 2007. "An Efficient Trajectory Method for Probabilistic Production-Inventory-Distribution Problems," Operations Research, INFORMS, vol. 55(2), pages 378-394, April.
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    1. Lejeune, Miguel & Lozin, Vadim & Lozina, Irina & Ragab, Ahmed & Yacout, Soumaya, 2019. "Recent advances in the theory and practice of Logical Analysis of Data," European Journal of Operational Research, Elsevier, vol. 275(1), pages 1-15.
    2. Miguel A. Lejeune, 2012. "Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems," Operations Research, INFORMS, vol. 60(6), pages 1356-1372, December.
    3. Lejeune, Miguel A. & Shen, Siqian, 2016. "Multi-objective probabilistically constrained programs with variable risk: Models for multi-portfolio financial optimization," European Journal of Operational Research, Elsevier, vol. 252(2), pages 522-539.
    4. Ran Ji & Miguel A. Lejeune, 2018. "Risk-budgeting multi-portfolio optimization with portfolio and marginal risk constraints," Annals of Operations Research, Springer, vol. 262(2), pages 547-578, March.
    5. Lejeune, Miguel A., 2013. "Probabilistic modeling of multiperiod service levels," European Journal of Operational Research, Elsevier, vol. 230(2), pages 299-312.
    6. Lukáš Adam & Martin Branda & Holger Heitsch & René Henrion, 2020. "Solving joint chance constrained problems using regularization and Benders’ decomposition," Annals of Operations Research, Springer, vol. 292(2), pages 683-709, September.

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