This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(β) ≥ 0 versus K: β ∈ RK. We extend these results to consider local hypotheses tests of the form H: h1(β) ≥ 0 and h2(β) = 0 versus K: β ∈ RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.