Pattern definition of the p-efficiency concept

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).

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