The paper proposes a framework for sensitivity analyses of blackbox constrained optimization problems for which Lagrange multipliers are not available. Two strategies are developed to analyze the sensitivity of the optimal objective function value to general constraints. These are a simple method which may be performed immediately after a single optimization, and a detailed method performing biobjective optimization on the minimization of the objective versus the constraint of interest. The detailed method provides points on the Pareto front of the objective versus a chosen constraint. The proposed methods are tested on an academic test case and on an engineering problem using the mesh adaptive direct search algorithm.