In this paper we characterize the smallest production possibility set that contains a specified set of (input, output) combinations. In accordance with neoclassical production economics, this possibility set has convex projections into the input and output spaces (convex isoquants), and it satisfies the assumption of free disposability. We obtain it by means of a possibly infinite recursion which builds the possibility set as an ever larger union of convex sets. We remark on the nature of the approximations obtained by truncating the recursion, and we obtain a necessary and sufficient condition, checkable in one iteration for the recursion to stop in the next. For the case in which the recursion stops, we provide a succinct characterization of the dominance relations among the constituent sets produced by the procedure. Finally, we present examples of both finite and infinite cases. The example for the finite case illustrates the construction of the possibility set along with its associated production and consumption sets.data envelopment analysis, convexity
This paper is about a property of certain combinatorial structures, called sequential convexifiability, shown by Balas [1974, 1979] to hold for facial disjunctive programs.Sequential convexifiability means that the convex hull " of a nonconvex set defined by a collection of constraints can be generated by imposing the constraints one by one, sequentially, and generating each time the convex hull of the resulting set.Here we-extend the class of problems considered to disjunctive programs with infinitely many terms, also known as reverse convex programs, and give necessary and sufficient conditions for the ,-."solution sets of such problems to be sequentially convexifiable. We point out important classes of problems in addition to facial disjunctive programs (for instance, reverse convex programs with equations only) for which the conditions are always satisfied. Finally, we give examples of disjunctive programs for which the conditions are violated, and so the procedure breaks down..-. . T J %C-- QU-
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