A Note on Duality in Inequality Systems

The purpose of this note is to present a proposition on a sort of duality relation concerning systems of nonlinear inequalities. A useful theorem due to A.W. Tucker(1956) on duality relations valid for systems of linear inequalities was partially generalized to systems of nonlinear inequalities in Fujimoto(1980), in which a given vector-valued mapping is assumed to be either pseudoconcave or homogeneous of a positive degree. Another partial extension of Tucker's result is made in Fujimoto-Ranade(1995), where the concavity of a given mapping is required. The assumption of concavity seems too strong in a model with externalities and variable returns, and so in this note a weaker assumption of pseudoconcavity is restored with an additional requirement