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A Note on Arrow-Enthoven's Sufficiency Theorem on Quasi-Concave Programming: Corrigendum to: Quasi-Concave Programming

The seminal contribution of Arrow and Enthoven (1961) on quasi-concave programming has enlarged the set of functions to which optimization theory can be applied. Arrow and Intriligator (1982), Intriligator (2000), and Samuelson (1983) put forward the importance of ordinal functions in economic theory. Koenker and Mizera (2010) used quasi-concavity density estimation in probability and statistics. In this note, we reconsider Theorem 1 of Arrow and Enthoven (1961) (henceforth AET), which provides a set of independent sufficient conditions so that the Kuhn and Tucker theorem holds in quasi-concave programming maximization problems. The AET was extended notably by Ferland (1972) to optimization problems with twice continuously differentiable objective functions. Some characterizations of quasi-concavity properties were provided in Diewert, Avriel, and Zang (1981). It is common knowledge that the Kuhn and Tucker first-order conditions are not sufficient to determine, unlike the first-order conditions in concave programming, global optima of quasi-convex optimization problems, that is, problems with a quasi-concave objective function and a convex constraint set. In AET, the objective function and the constraint functions are assumed to be differentiable and quasi-concave, so that the Kuhn and Tucker conditions are sufficient to determine global optima provided some additional regularity conditions are met at the optimum point. More specifically, we focus on condition (c) in AET. Condition (c) states that, in the neighborhood of the optimum point, if the objective function is twice differentiable, and its gradient is different from zero, then the stationary point that meets the Kuhn and Tucker conditions solves the inequality constrained maximization problem. Through an example, we show that this condition also requires the twice continuous differentiability of the quasi-concave objective function. The remainder of the paper is organized as follows. In Section 2, we restate and discuss the AET. In Section 3, we provide an example in which condition (c) fails to identify the maximum. In Section 4, we conclude.