Pseudo-Concave Programming and Lagrange Regularity

For the mathematical programming problem max f(x) subject to G(x) ≧ 0, we show that if G(x) is pseudo-concave, a property weaker than concavity but stronger than quasi-concavity, and differentiable, then the constraint set is necessarily determined by the natural gradient (tangent) inequality system of G. We then apply the duality constructs of semi-infinite programming, in a manner which admits generalizations, to this special case to show that pseudo-concave constraint functions that have an interior point are convex Lagrange regular. Analogous to a theorem of Arrow-Hurwicz-Uzawa, we characterize functions that are both pseudo-concave and pseudo-convex, and for programming problems with objective functions of this form, we obtain equivalent problems having linear objective functions.